Stokes wave - Wikipedia

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</div> </a> <ul id="toc-Third-order_Stokes_wave_on_deep_water-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Second-order_Stokes_wave_on_arbitrary_depth" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Second-order_Stokes_wave_on_arbitrary_depth"> <div class="vector-toc-text"> 1.2 Second-order Stokes wave on arbitrary depth </div> </a> <ul id="toc-Second-order_Stokes_wave_on_arbitrary_depth-sublist" class="vector-toc-list"> <li id="toc-Stokes_and_Ursell_parameters" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Stokes_and_Ursell_parameters"> <div class="vector-toc-text"> 1.2.1 Stokes and Ursell parameters </div> </a> <ul id="toc-Stokes_and_Ursell_parameters-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Third-order_dispersion_relation" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Third-order_dispersion_relation"> <div class="vector-toc-text"> 1.2.2 Third-order dispersion relation </div> </a> <ul id="toc-Third-order_dispersion_relation-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Overview" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Overview"> <div class="vector-toc-text"> 2 Overview </div> </a> <button aria-controls="toc-Overview-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Overview subsection </button> <ul id="toc-Overview-sublist" class="vector-toc-list"> <li id="toc-Stokes's_approach_to_the_nonlinear_wave_problem" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Stokes's_approach_to_the_nonlinear_wave_problem"> <div class="vector-toc-text"> 2.1 Stokes's approach to the nonlinear wave problem </div> </a> <ul id="toc-Stokes's_approach_to_the_nonlinear_wave_problem-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Applicability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Applicability"> <div class="vector-toc-text"> 2.2 Applicability </div> </a> <ul id="toc-Applicability-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Modern_extensions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Modern_extensions"> <div class="vector-toc-text"> 2.3 Modern extensions </div> </a> <ul id="toc-Modern_extensions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Convergence_and_instability" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Convergence_and_instability"> <div class="vector-toc-text"> 2.4 Convergence and instability </div> </a> <ul id="toc-Convergence_and_instability-sublist" class="vector-toc-list"> <li id="toc-Convergence" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Convergence"> <div class="vector-toc-text"> 2.4.1 Convergence </div> </a> <ul id="toc-Convergence-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Highest_wave" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Highest_wave"> <div class="vector-toc-text"> 2.4.2 Highest wave </div> </a> <ul id="toc-Highest_wave-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Instability" class="vector-toc-list-item vector-toc-level-3"> <a class="vector-toc-link" href="#Instability"> <div class="vector-toc-text"> 2.4.3 Instability </div> </a> <ul id="toc-Instability-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> </ul> </li> <li id="toc-Stokes_expansion" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stokes_expansion"> <div class="vector-toc-text"> 3 Stokes expansion </div> </a> <button aria-controls="toc-Stokes_expansion-sublist" class="cdx-button cdx-button--weight-quiet cdx-button--icon-only vector-toc-toggle"> Toggle Stokes expansion subsection </button> <ul id="toc-Stokes_expansion-sublist" class="vector-toc-list"> <li id="toc-Governing_equations_for_a_potential_flow" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Governing_equations_for_a_potential_flow"> <div class="vector-toc-text"> 3.1 Governing equations for a potential flow </div> </a> <ul id="toc-Governing_equations_for_a_potential_flow-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Taylor_series_in_the_free-surface_boundary_conditions" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Taylor_series_in_the_free-surface_boundary_conditions"> <div class="vector-toc-text"> 3.2 Taylor series in the free-surface boundary conditions </div> </a> <ul id="toc-Taylor_series_in_the_free-surface_boundary_conditions-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Perturbation-series_approach" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Perturbation-series_approach"> <div class="vector-toc-text"> 3.3 Perturbation-series approach </div> </a> <ul id="toc-Perturbation-series_approach-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Application_to_progressive_periodic_waves_of_permanent_form" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Application_to_progressive_periodic_waves_of_permanent_form"> <div class="vector-toc-text"> 3.4 Application to progressive periodic waves of permanent form </div> </a> <ul id="toc-Application_to_progressive_periodic_waves_of_permanent_form-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-Stokes's_two_definitions_of_wave_celerity" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#Stokes's_two_definitions_of_wave_celerity"> <div class="vector-toc-text"> 4 Stokes's two definitions of wave celerity </div> </a> <ul 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Stokes </div> </a> <ul id="toc-By_Sir_George_Gabriel_Stokes-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-Other_historical_references" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#Other_historical_references"> <div class="vector-toc-text"> 6.2 Other historical references </div> </a> <ul id="toc-Other_historical_references-sublist" class="vector-toc-list"> </ul> </li> <li id="toc-More_recent_(since_1960)" class="vector-toc-list-item vector-toc-level-2"> <a class="vector-toc-link" href="#More_recent_(since_1960)"> <div class="vector-toc-text"> 6.3 More recent (since 1960) </div> </a> <ul id="toc-More_recent_(since_1960)-sublist" class="vector-toc-list"> </ul> </li> </ul> </li> <li id="toc-External_links" class="vector-toc-list-item vector-toc-level-1 vector-toc-list-item-expanded"> <a class="vector-toc-link" href="#External_links"> <div class="vector-toc-text"> 7 External links </div> </a> <ul id="toc-External_links-sublist" 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src="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Stokes3_wave.svg/300px-Stokes3_wave.svg.png" decoding="async" width="300" height="39" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Stokes3_wave.svg/450px-Stokes3_wave.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/cb/Stokes3_wave.svg/600px-Stokes3_wave.svg.png 2x" data-file-width="512" data-file-height="66" /></a><figcaption>Surface elevation of a deep water wave according to <a href="/wiki/George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">Stokes</a>' third-order theory. The wave steepness is: ka = 0.3, with k the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a> and a the wave <a href="/wiki/Amplitude" title="Amplitude">amplitude</a>. Typical for these <a href="/wiki/Surface_gravity_wave" class="mw-redirect" title="Surface gravity wave">surface gravity waves</a> are the sharp <a href="/wiki/Crest_(physics)" class="mw-redirect" title="Crest (physics)">crests</a> and flat <a href="/wiki/Trough_(physics)" class="mw-redirect" title="Trough (physics)">troughs</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:JACOEL-Wave_Tow_Tank-Image16.png" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/JACOEL-Wave_Tow_Tank-Image16.png/300px-JACOEL-Wave_Tow_Tank-Image16.png" decoding="async" width="300" height="250" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/57/JACOEL-Wave_Tow_Tank-Image16.png/450px-JACOEL-Wave_Tow_Tank-Image16.png 1.5x, //upload.wikimedia.org/wikipedia/commons/5/57/JACOEL-Wave_Tow_Tank-Image16.png 2x" data-file-width="550" data-file-height="459" /></a><figcaption>Model testing with periodic waves in the wave–tow tank of the Jere A. Chase Ocean Engineering Laboratory, <a href="/wiki/University_of_New_Hampshire" title="University of New Hampshire">University of New Hampshire</a>.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Undular_bore_Araguari_River-Brazil-USGS-bws00026.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Undular_bore_Araguari_River-Brazil-USGS-bws00026.jpg/300px-Undular_bore_Araguari_River-Brazil-USGS-bws00026.jpg" decoding="async" width="300" height="200" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Undular_bore_Araguari_River-Brazil-USGS-bws00026.jpg/450px-Undular_bore_Araguari_River-Brazil-USGS-bws00026.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/b/b4/Undular_bore_Araguari_River-Brazil-USGS-bws00026.jpg/600px-Undular_bore_Araguari_River-Brazil-USGS-bws00026.jpg 2x" data-file-width="5964" data-file-height="3970" /></a><figcaption><a href="/wiki/Undular_bore" title="Undular bore">Undular bore</a> and <a href="/wiki/Whelp_(tidal_bore)" class="mw-redirect" title="Whelp (tidal bore)">whelps</a> near the mouth of <a href="/wiki/Araguari_River_(Amap%C3%A1)" title="Araguari River (Amapá)">Araguari River</a> in north-eastern Brazil. View is oblique toward mouth from airplane at approximately 100 ft (30 m) altitude.<a href="#cite_note-1">[1]</a> The undulations following behind the bore front appear as slowly <a href="/wiki/Modulation" title="Modulation">modulated</a> Stokes waves.</figcaption></figure> In <a href="/wiki/Fluid_dynamics" title="Fluid dynamics">fluid dynamics</a>, a Stokes wave is a <a href="/wiki/Nonlinear_system" title="Nonlinear system">nonlinear</a> and <a href="/wiki/Periodic_function" title="Periodic function">periodic</a> <a href="/wiki/Surface_wave" title="Surface wave">surface wave</a> on an <a href="/wiki/Inviscid_flow" title="Inviscid flow">inviscid fluid</a> layer of constant mean depth. This type of modelling has its origins in the mid 19th century when <a href="/wiki/Sir_George_Stokes" class="mw-redirect" title="Sir George Stokes">Sir George Stokes</a> – using a <a href="/wiki/Perturbation_series" class="mw-redirect" title="Perturbation series">perturbation series</a> approach, now known as the Stokes expansion – obtained approximate solutions for nonlinear wave motion. Stokes's wave theory is of direct practical use for waves on intermediate and deep water. It is used in the design of <a href="/wiki/Coastal_management" title="Coastal management">coastal</a> and <a href="/wiki/Offshore_construction" title="Offshore construction">offshore structures</a>, in order to determine the wave <a href="/wiki/Kinematics" title="Kinematics">kinematics</a> (<a href="/wiki/Free_surface" title="Free surface">free surface</a> elevation and <a href="/wiki/Flow_velocity" title="Flow velocity">flow velocities</a>). The wave kinematics are subsequently needed in the <a href="/wiki/Design_process" class="mw-redirect" title="Design process">design process</a> to determine the <a href="/wiki/Wave_load" class="mw-redirect" title="Wave load">wave loads</a> on a structure.<a href="#cite_note-2">[2]</a> For long waves (as compared to depth) – and using only a few terms in the Stokes expansion – its applicability is limited to waves of small <a href="/wiki/Amplitude" title="Amplitude">amplitude</a>. In such shallow water, a <a href="/wiki/Cnoidal_wave" title="Cnoidal wave">cnoidal wave</a> theory often provides better periodic-wave approximations. While, in the strict sense, Stokes wave refers to a progressive periodic wave of permanent form, the term is also used in connection with <a href="/wiki/Standing_wave" title="Standing wave">standing waves</a><a href="#cite_note-3">[3]</a> and even random waves.<a href="#cite_note-4">[4]</a><a href="#cite_note-5">[5]</a> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=1" title="Edit section: Examples">edit</a>]</div> The examples below describe Stokes waves under the action of gravity (without <a href="/wiki/Surface_tension" title="Surface tension">surface tension</a> effects) in case of pure wave motion, so without an ambient mean current. <div class="mw-heading mw-heading3"><h3 id="Third-order_Stokes_wave_on_deep_water"> Third-order Stokes wave on deep water</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=2" title="Edit section: Third-order Stokes wave on deep water">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stokes3_wave_definitions.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Stokes3_wave_definitions.svg/300px-Stokes3_wave_definitions.svg.png" decoding="async" width="300" height="98" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Stokes3_wave_definitions.svg/450px-Stokes3_wave_definitions.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/31/Stokes3_wave_definitions.svg/600px-Stokes3_wave_definitions.svg.png 2x" data-file-width="589" data-file-height="192" /></a><figcaption>Third-order Stokes wave in deep water under the action of gravity. The wave steepness is: ka = 0.3.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stokes3_harmonics.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Stokes3_harmonics.svg/300px-Stokes3_harmonics.svg.png" decoding="async" width="300" height="141" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/44/Stokes3_harmonics.svg/450px-Stokes3_harmonics.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/44/Stokes3_harmonics.svg/600px-Stokes3_harmonics.svg.png 2x" data-file-width="561" data-file-height="264" /></a><figcaption>The three <a href="/wiki/Harmonic" title="Harmonic">harmonics</a> contributing to the surface elevation of a deep water wave, according to Stokes's third-order theory. The wave steepness is: ka = 0.3. For visibility, the vertical scale is distorted by a factor of four, compared to the horizontal scale. Description: * the dark blue line is the surface elevation of the 3rd-order Stokes wave, * the black line is the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental</a> wave component, with wavenumber k (<a href="/wiki/Wavelength" title="Wavelength">wavelength</a> λ, k = 2π / λ), * the light blue line is the harmonic at 2 k (wavelength <style data-mw-deduplicate="TemplateStyles:r1154941027">.mw-parser-output .frac{white-space:nowrap}.mw-parser-output .frac .num,.mw-parser-output .frac .den{font-size:80%;line-height:0;vertical-align:super}.mw-parser-output .frac .den{vertical-align:sub}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>1⁄2 λ), and * the red line is the harmonic at 3 k (wavelength <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄3 λ).</figcaption></figure> According to Stokes's third-order theory, the <a href="/wiki/Free_surface" title="Free surface">free surface</a> elevation η, the <a href="/wiki/Velocity_potential" title="Velocity potential">velocity potential</a> Φ, the <a href="/wiki/Phase_speed" class="mw-redirect" title="Phase speed">phase speed</a> (or celerity) c and the wave <a href="/wiki/Phase_(waves)" title="Phase (waves)">phase</a> θ are, for a <a href="/wiki/Wave_propagation" class="mw-redirect" title="Wave propagation">progressive</a> <a href="/wiki/Surface_gravity_wave" class="mw-redirect" title="Surface gravity wave">surface gravity wave</a> on deep water – i.e. the fluid layer has infinite depth:<a href="#cite_note-Dingemans-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\eta (x,t)=&a\left\{\left[1-{\tfrac {1}{16}}(ka)^{2}\right]\cos \theta +{\tfrac {1}{2}}(ka)\,\cos 2\theta +{\tfrac {3}{8}}(ka)^{2}\,\cos 3\theta \right\}+{\mathcal {O}}\left((ka)^{4}\right),\\\Phi (x,z,t)=&a{\sqrt {\frac {g}{k}}}\,{\text{e}}^{kz}\,\sin \theta +{\mathcal {O}}\left((ka)^{4}\right),\\c=&{\frac {\omega }{k}}=\left(1+{\tfrac {1}{2}}(ka)^{2}\right)\,{\sqrt {\frac {g}{k}}}+{\mathcal {O}}\left((ka)^{4}\right),{\text{ and}}\\\theta (x,t)=&kx-\omega t,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> <mtd> <mi>a</mi> <mrow> <mo>{</mo> <mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>16</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>]</mo> </mrow> <mi>cos</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <mo stretchy="false">)</mo> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mn>2</mn> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>cos</mi> <mo>⁡</mo> <mn>3</mn> <mi>θ</mi> </mrow> <mo>}</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> <mtd> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>g</mi> <mi>k</mi> </mfrac> </msqrt> </mrow> <mspace width="thinmathspace" /> <msup> <mrow class="MJX-TeXAtom-ORD"> <mtext>e</mtext> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>z</mi> </mrow> </msup> <mspace width="thinmathspace" /> <mi>sin</mi> <mo>⁡</mo> <mi>θ</mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>g</mi> <mi>k</mi> </mfrac> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> <mrow class="MJX-TeXAtom-ORD"> <mtext> and</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> <mtd> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\eta (x,t)=&a\left\{\left[1-{\tfrac {1}{16}}(ka)^{2}\right]\cos \theta +{\tfrac {1}{2}}(ka)\,\cos 2\theta +{\tfrac {3}{8}}(ka)^{2}\,\cos 3\theta \right\}+{\mathcal {O}}\left((ka)^{4}\right),\\\Phi (x,z,t)=&a{\sqrt {\frac {g}{k}}}\,{\text{e}}^{kz}\,\sin \theta +{\mathcal {O}}\left((ka)^{4}\right),\\c=&{\frac {\omega }{k}}=\left(1+{\tfrac {1}{2}}(ka)^{2}\right)\,{\sqrt {\frac {g}{k}}}+{\mathcal {O}}\left((ka)^{4}\right),{\text{ and}}\\\theta (x,t)=&kx-\omega t,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/47a8c57e1966be8bbf1a9bf1dc5862cf3ad92513" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -9.671ex; width:80.841ex; height:20.509ex;" alt="{\displaystyle {\begin{aligned}\eta (x,t)=&a\left\{\left[1-{\tfrac {1}{16}}(ka)^{2}\right]\cos \theta +{\tfrac {1}{2}}(ka)\,\cos 2\theta +{\tfrac {3}{8}}(ka)^{2}\,\cos 3\theta \right\}+{\mathcal {O}}\left((ka)^{4}\right),\\\Phi (x,z,t)=&a{\sqrt {\frac {g}{k}}}\,{\text{e}}^{kz}\,\sin \theta +{\mathcal {O}}\left((ka)^{4}\right),\\c=&{\frac {\omega }{k}}=\left(1+{\tfrac {1}{2}}(ka)^{2}\right)\,{\sqrt {\frac {g}{k}}}+{\mathcal {O}}\left((ka)^{4}\right),{\text{ and}}\\\theta (x,t)=&kx-\omega t,\end{aligned}}}"> where <ul><li>x is the horizontal coordinate;</li> <li>z is the vertical coordinate, with the positive z-direction upward – opposing to the direction of the <a href="/wiki/Earth%27s_gravity" class="mw-redirect" title="Earth's gravity">Earth's gravity</a> – and z = 0 corresponding with the <a href="/wiki/Average" title="Average">mean</a> surface elevation;</li> <li>t is time;</li> <li>a is the first-order wave <a href="/wiki/Amplitude" title="Amplitude">amplitude</a>;</li> <li>k is the <a href="/wiki/Angular_wavenumber" class="mw-redirect" title="Angular wavenumber">angular wavenumber</a>, k = 2π / λ with λ being the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>;</li> <li>ω is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a>, ω = 2π / τ where τ is the <a href="/wiki/Period_(physics)" class="mw-redirect" title="Period (physics)">period</a>, and</li> <li>g is the <a href="/wiki/Field_strength" title="Field strength">strength</a> of the Earth's gravity, a <a href="/wiki/Constant_(mathematics)" title="Constant (mathematics)">constant</a> in this approximation.</li></ul> The expansion parameter ka is known as the wave steepness. The phase speed increases with increasing nonlinearity ka of the waves. The <a href="/wiki/Wave_height" title="Wave height">wave height</a> H, being the difference between the surface elevation η at a <a href="/wiki/Crest_(physics)" class="mw-redirect" title="Crest (physics)">crest</a> and a <a href="/wiki/Trough_(physics)" class="mw-redirect" title="Trough (physics)">trough</a>, is:<a href="#cite_note-7">[7]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H=2a\,\left(1+{\tfrac {3}{8}}\,k^{2}a^{2}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>H</mi> <mo>=</mo> <mn>2</mn> <mi>a</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>3</mn> <mn>8</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mi>a</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle H=2a\,\left(1+{\tfrac {3}{8}}\,k^{2}a^{2}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/61c107eef8e24b85a2fce2f2c91431105b937541" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:22.736ex; height:4.843ex;" alt="{\displaystyle H=2a\,\left(1+{\tfrac {3}{8}}\,k^{2}a^{2}\right).}"> Note that the second- and third-order terms in the velocity potential Φ are zero. Only at fourth order do contributions deviating from first-order theory – i.e. <a href="/wiki/Airy_wave_theory" title="Airy wave theory">Airy wave theory</a> – appear.<a href="#cite_note-Dingemans-6">[6]</a> Up to third order the orbital velocity <a href="/wiki/Vector_field" title="Vector field">field</a> u = ∇Φ consists of a circular motion of the velocity vector at each position (x,z). As a result, the surface elevation of deep-water waves is to a good approximation <a href="/wiki/Trochoidal_wave" title="Trochoidal wave">trochoidal</a>, as already noted by <a href="#CITEREFStokes1847">Stokes (1847)</a>.<a href="#cite_note-Toba-8">[8]</a> Stokes further observed, that although (in this <a href="/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Eulerian</a> description) the third-order orbital velocity field consists of a circular motion at each point, the <a href="/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Lagrangian</a> paths of <a href="/wiki/Fluid_parcel" title="Fluid parcel">fluid parcels</a> are not closed circles. This is due to the reduction of the velocity amplitude at increasing depth below the surface. This Lagrangian drift of the fluid parcels is known as the <a href="/wiki/Stokes_drift" title="Stokes drift">Stokes drift</a>.<a href="#cite_note-Toba-8">[8]</a> <div class="mw-heading mw-heading3"><h3 id="Second-order_Stokes_wave_on_arbitrary_depth">Second-order Stokes wave on arbitrary depth</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=3" title="Edit section: Second-order Stokes wave on arbitrary depth">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stokes3_amplitude_double_frequency.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Stokes3_amplitude_double_frequency.svg/300px-Stokes3_amplitude_double_frequency.svg.png" decoding="async" width="300" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/68/Stokes3_amplitude_double_frequency.svg/450px-Stokes3_amplitude_double_frequency.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/68/Stokes3_amplitude_double_frequency.svg/600px-Stokes3_amplitude_double_frequency.svg.png 2x" data-file-width="677" data-file-height="373" /></a><figcaption>The ratio S = a2 / a of the amplitude a2 of the <a href="/wiki/Harmonic" title="Harmonic">harmonic</a> with twice the wavenumber (2 k), to the amplitude a of the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental</a>, according to Stokes's second-order theory for surface gravity waves. On the horizontal axis is the relative water depth h / λ, with h the mean depth and λ the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>, while the vertical axis is the Stokes parameter S divided by the wave steepness ka (with k = 2π / λ). Description: * the blue line is valid for arbitrary water depth, while * the dashed red line is the shallow-water limit (water depth small compared to the wavelength), and * the dash-dot green line is the asymptotic limit for deep water waves.</figcaption></figure> The surface elevation η and the velocity potential Φ are, according to Stokes's second-order theory of surface gravity waves on a fluid layer of <a href="/wiki/Average" title="Average">mean</a> depth h:<a href="#cite_note-Dingemans-6">[6]</a><a href="#cite_note-Whitham_13.13-9">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\eta (x,t)=&a\left\{\cos \,\theta +ka\,{\frac {3-\sigma ^{2}}{4\,\sigma ^{3}}}\,\cos \,2\theta \right\}+{\mathcal {O}}\left((ka)^{3}\right),\\\Phi (x,z,t)=&a\,{\frac {\omega }{k}}\,{\frac {1}{\sinh \,kh}}\\&\times \left\{\cosh \,k(z+h)\sin \,\theta +ka\,{\frac {3\cosh \,2k(z+h)}{8\,\sinh ^{3}\,kh}}\,\sin \,2\theta \right\}\\&-(ka)^{2}\,{\frac {1}{2\,\sinh \,2kh}}\,{\frac {g\,t}{k}}+{\mathcal {O}}\left((ka)^{3}\right),\\c=&{\frac {\omega }{k}}={\sqrt {{\frac {g}{k}}\,\sigma }}+{\mathcal {O}}\left((ka)^{2}\right),\\\sigma =&\tanh \,kh\quad {\text{and}}\quad \theta (x,t)=kx-\omega t.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> <mtd> <mi>a</mi> <mrow> <mo>{</mo> <mrow> <mi>cos</mi> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mrow> <mn>4</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>cos</mi> <mspace width="thinmathspace" /> <mn>2</mn> <mi>θ</mi> </mrow> <mo>}</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> </mtd> <mtd> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>k</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi>sinh</mi> <mspace width="thinmathspace" /> <mi>k</mi> <mi>h</mi> </mrow> </mfrac> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>×</mo> <mrow> <mo>{</mo> <mrow> <mi>cosh</mi> <mspace width="thinmathspace" /> <mi>k</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> <mi>sin</mi> <mspace width="thinmathspace" /> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mi>cosh</mi> <mspace width="thinmathspace" /> <mn>2</mn> <mi>k</mi> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>8</mn> <mspace width="thinmathspace" /> <msup> <mi>sinh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>k</mi> <mi>h</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mspace width="thinmathspace" /> <mn>2</mn> <mi>θ</mi> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mspace width="thinmathspace" /> <mi>sinh</mi> <mspace width="thinmathspace" /> <mn>2</mn> <mi>k</mi> <mi>h</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>g</mi> <mspace width="thinmathspace" /> <mi>t</mi> </mrow> <mi>k</mi> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>c</mi> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>ω</mi> <mi>k</mi> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>g</mi> <mi>k</mi> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>σ</mi> </msqrt> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi>σ</mi> <mo>=</mo> </mtd> <mtd> <mi>tanh</mi> <mspace width="thinmathspace" /> <mi>k</mi> <mi>h</mi> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mi>θ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\eta (x,t)=&a\left\{\cos \,\theta +ka\,{\frac {3-\sigma ^{2}}{4\,\sigma ^{3}}}\,\cos \,2\theta \right\}+{\mathcal {O}}\left((ka)^{3}\right),\\\Phi (x,z,t)=&a\,{\frac {\omega }{k}}\,{\frac {1}{\sinh \,kh}}\\&\times \left\{\cosh \,k(z+h)\sin \,\theta +ka\,{\frac {3\cosh \,2k(z+h)}{8\,\sinh ^{3}\,kh}}\,\sin \,2\theta \right\}\\&-(ka)^{2}\,{\frac {1}{2\,\sinh \,2kh}}\,{\frac {g\,t}{k}}+{\mathcal {O}}\left((ka)^{3}\right),\\c=&{\frac {\omega }{k}}={\sqrt {{\frac {g}{k}}\,\sigma }}+{\mathcal {O}}\left((ka)^{2}\right),\\\sigma =&\tanh \,kh\quad {\text{and}}\quad \theta (x,t)=kx-\omega t.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d78fe320abfcc1d6d3129d99d295cea277c0efab" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -16.005ex; width:66.257ex; height:33.176ex;" alt="{\displaystyle {\begin{aligned}\eta (x,t)=&a\left\{\cos \,\theta +ka\,{\frac {3-\sigma ^{2}}{4\,\sigma ^{3}}}\,\cos \,2\theta \right\}+{\mathcal {O}}\left((ka)^{3}\right),\\\Phi (x,z,t)=&a\,{\frac {\omega }{k}}\,{\frac {1}{\sinh \,kh}}\\&\times \left\{\cosh \,k(z+h)\sin \,\theta +ka\,{\frac {3\cosh \,2k(z+h)}{8\,\sinh ^{3}\,kh}}\,\sin \,2\theta \right\}\\&-(ka)^{2}\,{\frac {1}{2\,\sinh \,2kh}}\,{\frac {g\,t}{k}}+{\mathcal {O}}\left((ka)^{3}\right),\\c=&{\frac {\omega }{k}}={\sqrt {{\frac {g}{k}}\,\sigma }}+{\mathcal {O}}\left((ka)^{2}\right),\\\sigma =&\tanh \,kh\quad {\text{and}}\quad \theta (x,t)=kx-\omega t.\end{aligned}}}"> Observe that for finite depth the velocity potential Φ contains a linear drift in time, independent of position (x and z). Both this temporal drift and the double-frequency term (containing sin 2θ) in Φ vanish for deep-water waves. <div class="mw-heading mw-heading4"><h4 id="Stokes_and_Ursell_parameters"> Stokes and Ursell parameters</h4>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=4" title="Edit section: Stokes and Ursell parameters">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1236090951">.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}</style><div role="note" class="hatnote navigation-not-searchable">Further information: <a href="/wiki/Ursell_parameter" class="mw-redirect" title="Ursell parameter">Ursell parameter</a></div> The ratio S of the free-surface amplitudes at second order and first order – according to Stokes's second-order theory – is:<a href="#cite_note-Dingemans-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {S}}=ka\,{\frac {3-\tanh ^{2}\,kh}{4\,\tanh ^{3}\,kh}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mi>k</mi> <mi>a</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>3</mn> <mo>−</mo> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>k</mi> <mi>h</mi> </mrow> <mrow> <mn>4</mn> <mspace width="thinmathspace" /> <msup> <mi>tanh</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>k</mi> <mi>h</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {S}}=ka\,{\frac {3-\tanh ^{2}\,kh}{4\,\tanh ^{3}\,kh}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bdb9f4b32a6a967cc3df15faffd3a3e36c1647bd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:21.935ex; height:6.176ex;" alt="{\displaystyle {\mathcal {S}}=ka\,{\frac {3-\tanh ^{2}\,kh}{4\,\tanh ^{3}\,kh}}.}"> In deep water, for large kh the ratio S has the <a href="/wiki/Asymptote" title="Asymptote">asymptote</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{kh\to \infty }{\mathcal {S}}={\frac {1}{2}}\,ka.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>h</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mi>k</mi> <mi>a</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{kh\to \infty }{\mathcal {S}}={\frac {1}{2}}\,ka.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/254227a627436060a1be39ad735cbf521d45c9a8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:15.541ex; height:5.343ex;" alt="{\displaystyle \lim _{kh\to \infty }{\mathcal {S}}={\frac {1}{2}}\,ka.}"> For long waves, i.e. small kh, the ratio S behaves as <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{kh\downarrow 0}{\mathcal {S}}={\frac {3}{4}}\,{\frac {ka}{(kh)^{3}}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>h</mi> <mo stretchy="false">↓</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mn>4</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>k</mi> <mi>a</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>h</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{kh\downarrow 0}{\mathcal {S}}={\frac {3}{4}}\,{\frac {ka}{(kh)^{3}}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0afd11fead22738e30d267941046ff0d01562716" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:17.707ex; height:6.176ex;" alt="{\displaystyle \lim _{kh\downarrow 0}{\mathcal {S}}={\frac {3}{4}}\,{\frac {ka}{(kh)^{3}}},}"> or, in terms of the wave height H = 2a and wavelength λ = 2π / k: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{kh\downarrow 0}{\mathcal {S}}={\frac {3}{32\,\pi ^{2}}}\,{\frac {H\,\lambda ^{2}}{h^{3}}}={\frac {3}{32\,\pi ^{2}}}\,{\mathcal {U}},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>h</mi> <mo stretchy="false">↓</mo> <mn>0</mn> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">S</mi> </mrow> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>32</mn> <mspace width="thinmathspace" /> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>H</mi> <mspace width="thinmathspace" /> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>3</mn> <mrow> <mn>32</mn> <mspace width="thinmathspace" /> <msup> <mi>π</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{kh\downarrow 0}{\mathcal {S}}={\frac {3}{32\,\pi ^{2}}}\,{\frac {H\,\lambda ^{2}}{h^{3}}}={\frac {3}{32\,\pi ^{2}}}\,{\mathcal {U}},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5a89000a9e956a7e7f1aaa34f883c52b60006dbd" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:32.111ex; height:6.176ex;" alt="{\displaystyle \lim _{kh\downarrow 0}{\mathcal {S}}={\frac {3}{32\,\pi ^{2}}}\,{\frac {H\,\lambda ^{2}}{h^{3}}}={\frac {3}{32\,\pi ^{2}}}\,{\mathcal {U}},}"> with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {U}}\equiv {\frac {H\,\lambda ^{2}}{h^{3}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">U</mi> </mrow> </mrow> <mo>≡</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>H</mi> <mspace width="thinmathspace" /> <msup> <mi>λ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <msup> <mi>h</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\mathcal {U}}\equiv {\frac {H\,\lambda ^{2}}{h^{3}}}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6781b2347e7d8bae42f0430f5d1b71c147d7918d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; margin-left: -0.038ex; width:11.077ex; height:6.009ex;" alt="{\displaystyle {\mathcal {U}}\equiv {\frac {H\,\lambda ^{2}}{h^{3}}}.}"> Here U is the <a href="/wiki/Ursell_parameter" class="mw-redirect" title="Ursell parameter">Ursell parameter</a> (or Stokes parameter). For long waves (λ ≫ h) of small height H, i.e. U ≪ 32π2/3 ≈ 100, second-order Stokes theory is applicable. Otherwise, for fairly long waves (λ > 7h) of appreciable height H a <a href="/wiki/Cnoidal_wave" title="Cnoidal wave">cnoidal wave</a> description is more appropriate.<a href="#cite_note-Dingemans-6">[6]</a> According to Hedges, fifth-order Stokes theory is applicable for U < 40, and otherwise fifth-order <a href="/wiki/Cnoidal_wave" title="Cnoidal wave">cnoidal wave</a> theory is preferable.<a href="#cite_note-Hedges-10">[10]</a><a href="#cite_note-Fenton-11">[11]</a> <div class="mw-heading mw-heading4"><h4 id="Third-order_dispersion_relation"> Third-order dispersion relation</h4>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=5" title="Edit section: Third-order dispersion relation">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stokes3_nonlin_celerity.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Stokes3_nonlin_celerity.svg/300px-Stokes3_nonlin_celerity.svg.png" decoding="async" width="300" height="158" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Stokes3_nonlin_celerity.svg/450px-Stokes3_nonlin_celerity.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7a/Stokes3_nonlin_celerity.svg/600px-Stokes3_nonlin_celerity.svg.png 2x" data-file-width="708" data-file-height="373" /></a><figcaption>Nonlinear enhancement of the <a href="/wiki/Phase_velocity" title="Phase velocity">phase speed</a> c = ω / k – according to Stokes's third-order theory for <a href="/wiki/Surface_gravity_wave" class="mw-redirect" title="Surface gravity wave">surface gravity waves</a>, and using Stokes's first definition of celerity – as compared to the linear-theory phase speed c0. On the horizontal axis is the relative water depth h / λ, with h the mean depth and λ the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>, while the vertical axis is the nonlinear phase-speed enhancement (c − c0) / c0 divided by the wave steepness ka squared. Description: * the solid blue line is valid for arbitrary water depth, * the dashed red line is the shallow-water limit (water depth small compared to the wavelength), and * the dash-dot green line is the asymptotic limit for deep water waves.</figcaption></figure> For Stokes waves under the action of gravity, the third-order <a href="/wiki/Dispersion_(water_waves)" title="Dispersion (water waves)">dispersion relation</a> is – according to <a href="#Stokes's_first_definition_of_wave_celerity">Stokes's first definition of celerity</a>:<a href="#cite_note-Whitham_13.13-9">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\omega ^{2}&=\left(gk\,\tanh \,kh\right)\;\left\{1+{\frac {9-10\,\sigma ^{2}+9\,\sigma ^{4}}{8\,\sigma ^{4}}}\,(ka)^{2}\right\}+{\mathcal {O}}\left((ka)^{4}\right),\\&\qquad {\text{with}}\\\sigma &=\tanh \,kh.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mtd> <mtd> <mi></mi> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>g</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mi>tanh</mi> <mspace width="thinmathspace" /> <mi>k</mi> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mspace width="thickmathspace" /> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mn>9</mn> <mo>−</mo> <mn>10</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mn>9</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mrow> <mn>8</mn> <mspace width="thinmathspace" /> <msup> <mi>σ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>}</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mtd> </mtr> <mtr> <mtd /> <mtd> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>with</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mi>σ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>tanh</mi> <mspace width="thinmathspace" /> <mi>k</mi> <mi>h</mi> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\omega ^{2}&=\left(gk\,\tanh \,kh\right)\;\left\{1+{\frac {9-10\,\sigma ^{2}+9\,\sigma ^{4}}{8\,\sigma ^{4}}}\,(ka)^{2}\right\}+{\mathcal {O}}\left((ka)^{4}\right),\\&\qquad {\text{with}}\\\sigma &=\tanh \,kh.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e09f3176b25ba6f43dd3f7cf6befae1ab11ab5ac" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.505ex; width:63.706ex; height:12.176ex;" alt="{\displaystyle {\begin{aligned}\omega ^{2}&=\left(gk\,\tanh \,kh\right)\;\left\{1+{\frac {9-10\,\sigma ^{2}+9\,\sigma ^{4}}{8\,\sigma ^{4}}}\,(ka)^{2}\right\}+{\mathcal {O}}\left((ka)^{4}\right),\\&\qquad {\text{with}}\\\sigma &=\tanh \,kh.\end{aligned}}}"> This third-order dispersion relation is a direct consequence of avoiding <a href="/wiki/Secular_term" class="mw-redirect" title="Secular term">secular terms</a>, when inserting the second-order Stokes solution into the third-order equations (of the perturbation series for the periodic wave problem). In deep water (short wavelength compared to the depth): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{kh\to \infty }\omega ^{2}=gk\,\left\{1+\left(ka\right)^{2}\right\}+{\mathcal {O}}\left((ka)^{4}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>h</mi> <mo stretchy="false">→</mo> <mi mathvariant="normal">∞</mi> </mrow> </munder> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>g</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>}</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{kh\to \infty }\omega ^{2}=gk\,\left\{1+\left(ka\right)^{2}\right\}+{\mathcal {O}}\left((ka)^{4}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7045a858180b5440991380cfd21620430dd06863" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.13ex; height:5.009ex;" alt="{\displaystyle \lim _{kh\to \infty }\omega ^{2}=gk\,\left\{1+\left(ka\right)^{2}\right\}+{\mathcal {O}}\left((ka)^{4}\right),}"> and in shallow water (long wavelengths compared to the depth): <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lim _{kh\downarrow 0}\omega ^{2}=k^{2}\,gh\,\left\{1+{\frac {9}{8}}\,{\frac {\left(ka\right)^{2}}{\left(kh\right)^{4}}}\right\}+{\mathcal {O}}\left((ka)^{4}\right).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <munder> <mo movablelimits="true" form="prefix">lim</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>h</mi> <mo stretchy="false">↓</mo> <mn>0</mn> </mrow> </munder> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mi>g</mi> <mi>h</mi> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <mn>1</mn> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>9</mn> <mn>8</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mi>a</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msup> <mrow> <mo>(</mo> <mrow> <mi>k</mi> <mi>h</mi> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <mrow> <mo stretchy="false">(</mo> <mi>k</mi> <mi>a</mi> <msup> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lim _{kh\downarrow 0}\omega ^{2}=k^{2}\,gh\,\left\{1+{\frac {9}{8}}\,{\frac {\left(ka\right)^{2}}{\left(kh\right)^{4}}}\right\}+{\mathcal {O}}\left((ka)^{4}\right).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c7db296007cfdbf931378761fcf3303e548de8ce" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:45.245ex; height:7.509ex;" alt="{\displaystyle \lim _{kh\downarrow 0}\omega ^{2}=k^{2}\,gh\,\left\{1+{\frac {9}{8}}\,{\frac {\left(ka\right)^{2}}{\left(kh\right)^{4}}}\right\}+{\mathcal {O}}\left((ka)^{4}\right).}"> As <a href="#Stokes_and_Ursell_parameters">shown above</a>, the long-wave Stokes expansion for the dispersion relation will only be valid for small enough values of the Ursell parameter: U ≪ 100. <div style="clear:both;" class=""></div> <div class="mw-heading mw-heading2"><h2 id="Overview">Overview</h2>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=6" title="Edit section: Overview">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Stokes's_approach_to_the_nonlinear_wave_problem">Stokes's approach to the nonlinear wave problem</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=7" title="Edit section: Stokes's approach to the nonlinear wave problem">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wavemaker.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Wavemaker.jpg/300px-Wavemaker.jpg" decoding="async" width="300" height="202" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/51/Wavemaker.jpg/450px-Wavemaker.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/51/Wavemaker.jpg/600px-Wavemaker.jpg 2x" data-file-width="2138" data-file-height="1437" /></a><figcaption>Waves in the <a href="/wiki/Kelvin_wake_pattern" title="Kelvin wake pattern">Kelvin wake pattern</a> generated by a ship on the <a href="/wiki/Maas%E2%80%93Waalkanaal" class="mw-redirect" title="Maas–Waalkanaal">Maas–Waalkanaal</a> in The Netherlands. The transverse waves in this Kelvin wake pattern are nearly plane Stokes waves.</figcaption></figure> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Wea00810.jpg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Wea00810.jpg/300px-Wea00810.jpg" decoding="async" width="300" height="383" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/63/Wea00810.jpg/450px-Wea00810.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/63/Wea00810.jpg/600px-Wea00810.jpg 2x" data-file-width="1204" data-file-height="1536" /></a><figcaption><a href="/wiki/NOAA" class="mw-redirect" title="NOAA">NOAA</a> ship Delaware II in bad weather on <a href="/wiki/Georges_Bank" title="Georges Bank">Georges Bank</a>. While these ocean waves are <a href="/wiki/Random" class="mw-redirect" title="Random">random</a>, and not Stokes waves (in the strict sense), they indicate the typical sharp <a href="/wiki/Crest_(physics)" class="mw-redirect" title="Crest (physics)">crests</a> and flat <a href="/wiki/Trough_(physics)" class="mw-redirect" title="Trough (physics)">troughs</a> as found in nonlinear surface gravity waves.</figcaption></figure> A fundamental problem in finding solutions for surface gravity waves is that <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a> have to be applied at the position of the <a href="/wiki/Free_surface" title="Free surface">free surface</a>, which is not known beforehand and is thus a part of the solution to be found. <a href="/wiki/Sir_George_Stokes" class="mw-redirect" title="Sir George Stokes">Sir George Stokes</a> solved this nonlinear wave problem in 1847 by expanding the relevant <a href="/wiki/Potential_flow" title="Potential flow">potential flow</a> quantities in a <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> around the mean (or still) surface elevation.<a href="#cite_note-12">[12]</a> As a result, the boundary conditions can be expressed in terms of quantities at the mean (or still) surface elevation (which is fixed and known). Next, a solution for the nonlinear wave problem (including the Taylor series expansion around the mean or still surface elevation) is sought by means of a perturbation series – known as the Stokes expansion – in terms of a small parameter, most often the wave steepness. The unknown terms in the expansion can be solved sequentially.<a href="#cite_note-Dingemans-6">[6]</a><a href="#cite_note-Toba-8">[8]</a> Often, only a small number of terms is needed to provide a solution of sufficient accuracy for engineering purposes.<a href="#cite_note-Fenton-11">[11]</a> Typical applications are in the design of <a href="/wiki/Coastal_management" title="Coastal management">coastal</a> and <a href="/wiki/Offshore_construction" title="Offshore construction">offshore structures</a>, and of <a href="/wiki/Naval_architecture" title="Naval architecture">ships</a>. Another property of nonlinear waves is that the <a href="/wiki/Phase_speed" class="mw-redirect" title="Phase speed">phase speed</a> of nonlinear waves depends on the <a href="/wiki/Wave_height" title="Wave height">wave height</a>. In a perturbation-series approach, this easily gives rise to a spurious <a href="/wiki/Secular_variation" title="Secular variation">secular variation</a> of the solution, in contradiction with the periodic behaviour of the waves. Stokes solved this problem by also expanding the <a href="/wiki/Dispersion_(water_waves)" title="Dispersion (water waves)">dispersion relationship</a> into a perturbation series, by a method now known as the <a href="/wiki/Lindstedt%E2%80%93Poincar%C3%A9_method" class="mw-redirect" title="Lindstedt–Poincaré method">Lindstedt–Poincaré method</a>.<a href="#cite_note-Dingemans-6">[6]</a> <div class="mw-heading mw-heading3"><h3 id="Applicability">Applicability</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=8" title="Edit section: Applicability">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Water_wave_theories.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Water_wave_theories.svg/300px-Water_wave_theories.svg.png" decoding="async" width="300" height="338" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Water_wave_theories.svg/450px-Water_wave_theories.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e0/Water_wave_theories.svg/600px-Water_wave_theories.svg.png 2x" data-file-width="640" data-file-height="720" /></a><figcaption>Validity of several theories for periodic water waves, according to Le Méhauté (1976).<a href="#cite_note-Le_Méhauté-13">[13]</a> The light-blue area gives the range of validity of <a href="/wiki/Cnoidal_wave" title="Cnoidal wave">cnoidal wave</a> theory; light-yellow for <a href="/wiki/Airy_wave_theory" title="Airy wave theory">Airy wave theory</a>; and the dashed blue lines demarcate between the required order in Stokes's wave theory. The light-gray shading gives the range extension by numerical approximations using fifth-order <a href="/wiki/Stream_function" title="Stream function">stream-function</a> theory, for high waves (H > <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1154941027">1⁄4 Hbreaking).</figcaption></figure> Stokes's wave theory, when using a low order of the perturbation expansion (e.g. up to second, third or fifth order), is valid for nonlinear waves on intermediate and deep water, that is for <a href="/wiki/Wavelength" title="Wavelength">wavelengths</a> (λ) not large as compared with the mean depth (h). In <a href="/wiki/Waves_and_shallow_water" title="Waves and shallow water">shallow water</a>, the low-order Stokes expansion breaks down (gives unrealistic results) for appreciable wave amplitude (as compared to the depth). Then, <a href="/wiki/Boussinesq_approximation_(water_waves)" title="Boussinesq approximation (water waves)">Boussinesq approximations</a> are more appropriate. Further approximations on Boussinesq-type (multi-directional) wave equations lead – for one-way wave propagation – to the <a href="/wiki/Korteweg%E2%80%93de_Vries_equation" class="mw-redirect" title="Korteweg–de Vries equation">Korteweg–de Vries equation</a> or the <a href="/wiki/Benjamin%E2%80%93Bona%E2%80%93Mahony_equation" title="Benjamin–Bona–Mahony equation">Benjamin–Bona–Mahony equation</a>. Like (near) exact Stokes-wave solutions,<a href="#cite_note-14">[14]</a> these two equations have <a href="/wiki/Solitary_wave_(water_waves)" class="mw-redirect" title="Solitary wave (water waves)">solitary wave</a> (<a href="/wiki/Soliton" title="Soliton">soliton</a>) solutions, besides periodic-wave solutions known as <a href="/wiki/Cnoidal_wave" title="Cnoidal wave">cnoidal waves</a>.<a href="#cite_note-Fenton-11">[11]</a> <div class="mw-heading mw-heading3"><h3 id="Modern_extensions">Modern extensions</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=9" title="Edit section: Modern extensions">edit</a>]</div> Already in 1914, Wilton extended the Stokes expansion for deep-water surface gravity waves to tenth order, although introducing errors at the eight order.<a href="#cite_note-15">[15]</a> A fifth-order theory for finite depth was derived by De in 1955.<a href="#cite_note-16">[16]</a> For engineering use, the fifth-order formulations of Fenton are convenient, applicable to both Stokes <a href="#Stokes_first_definition">first</a> and <a href="#Stokes_second_definition">second</a> definition of phase speed (celerity).<a href="#cite_note-17">[17]</a> The demarcation between when fifth-order Stokes theory is preferable over fifth-order <a href="/wiki/Cnoidal_wave" title="Cnoidal wave">cnoidal wave</a> theory is for <a href="/wiki/Ursell_parameter" class="mw-redirect" title="Ursell parameter">Ursell parameters</a> below about 40.<a href="#cite_note-Hedges-10">[10]</a><a href="#cite_note-Fenton-11">[11]</a> Different choices for the frame of reference and expansion parameters are possible in Stokes-like approaches to the nonlinear wave problem. In 1880, Stokes himself inverted the dependent and independent variables, by taking the <a href="/wiki/Velocity_potential" title="Velocity potential">velocity potential</a> and <a href="/wiki/Stream_function" title="Stream function">stream function</a> as the independent variables, and the coordinates (x,z) as the dependent variables, with x and z being the horizontal and vertical coordinates respectively.<a href="#cite_note-Stokes_1880-18">[18]</a> This has the advantage that the free surface, in a frame of reference in which the wave is steady (i.e. moving with the phase velocity), corresponds with a line on which the stream function is a constant. Then the free surface location is known beforehand, and not an unknown part of the solution. The disadvantage is that the <a href="/wiki/Radius_of_convergence" title="Radius of convergence">radius of convergence</a> of the rephrased series expansion reduces.<a href="#cite_note-Drennan-19">[19]</a> Another approach is by using the <a href="/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Lagrangian frame of reference</a>, following the <a href="/wiki/Fluid_parcel" title="Fluid parcel">fluid parcels</a>. The Lagrangian formulations show enhanced convergence, as compared to the formulations in both the <a href="/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Eulerian frame</a>, and in the frame with the potential and streamfunction as independent variables.<a href="#cite_note-Buldakov_et_al_2006-20">[20]</a><a href="#cite_note-Clamond_2007-21">[21]</a> An exact solution for nonlinear pure <a href="/wiki/Capillary_wave" title="Capillary wave">capillary waves</a> of permanent form, and for infinite fluid depth, was obtained by Crapper in 1957. Note that these capillary waves – being short waves forced by <a href="/wiki/Surface_tension" title="Surface tension">surface tension</a>, if gravity effects are negligible – have sharp troughs and flat crests. This contrasts with nonlinear surface gravity waves, which have sharp crests and flat troughs.<a href="#cite_note-22">[22]</a> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stokes_wave_energy_deep_water.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Stokes_wave_energy_deep_water.svg/300px-Stokes_wave_energy_deep_water.svg.png" decoding="async" width="300" height="165" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Stokes_wave_energy_deep_water.svg/450px-Stokes_wave_energy_deep_water.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Stokes_wave_energy_deep_water.svg/600px-Stokes_wave_energy_deep_water.svg.png 2x" data-file-width="963" data-file-height="529" /></a><figcaption>Several integral properties of Stokes waves on deep water as a function of wave steepness.<a href="#cite_note-23">[23]</a> The wave steepness is defined as the ratio of <a href="/wiki/Wave_height" title="Wave height">wave height</a> H to the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> λ. The wave properties are made <a href="/wiki/Dimensionless_quantity" title="Dimensionless quantity">dimensionless</a> using the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a> k = 2π / λ, <a href="/wiki/Earth%27s_gravity" class="mw-redirect" title="Earth's gravity">gravitational acceleration</a> g and the fluid <a href="/wiki/Density" title="Density">density</a> ρ. Shown are the <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic energy</a> density T, the <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a> density V, the total energy density E = T + V, the horizontal wave <a href="/wiki/Momentum" title="Momentum">momentum</a> density I, and the relative enhancement of the <a href="/wiki/Phase_speed" class="mw-redirect" title="Phase speed">phase speed</a> c. Wave energy densities T, V and E are integrated over depth and averaged over one wavelength, so they are energies per unit of horizontal area; the wave momentum density I is similar. The dashed black lines show 1/16 (kH)2 and 1/8 (kH)2, being the values of the integral properties as derived from (linear) <a href="/wiki/Airy_wave_theory" title="Airy wave theory">Airy wave theory</a>. The maximum wave height occurs for a wave steepness H / λ ≈ 0.1412, above which no periodic surface gravity waves exist.<a href="#cite_note-Schwartz_&_Fenton-24">[24]</a> Note that the shown wave properties have a maximum for a wave height less than the maximum wave height (see e.g. <a href="#CITEREFLonguet-Higgins1975">Longuet-Higgins 1975</a>; <a href="#CITEREFCokelet1977">Cokelet 1977</a>).</figcaption></figure> By use of computer models, the Stokes expansion for surface gravity waves has been continued, up to high (117th) order by <a href="#CITEREFSchwartz1974">Schwartz (1974)</a>. Schwartz has found that the amplitude a (or a1) of the first-order <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental</a> reaches a maximum before the maximum <a href="/wiki/Wave_height" title="Wave height">wave height</a> H is reached. Consequently, the wave steepness ka in terms of wave amplitude is not a monotone function up to the highest wave, and Schwartz utilizes instead kH as the expansion parameter. To estimate the highest wave in deep water, Schwartz has used <a href="/wiki/Pad%C3%A9_approximant" title="Padé approximant">Padé approximants</a> and <a href="/wiki/Domb%E2%80%93Sykes_plot" class="mw-redirect" title="Domb–Sykes plot">Domb–Sykes plots</a> in order to improve the convergence of the Stokes expansion. Extended tables of Stokes waves on various depths, computed by a different method (but in accordance with the results by others), are provided in Williams (<a href="#CITEREFWilliams1981">1981</a>, <a href="#CITEREFWilliams1985">1985</a>). Several exact relationships exist between integral properties – such as <a href="/wiki/Kinetic_energy" title="Kinetic energy">kinetic</a> and <a href="/wiki/Potential_energy" title="Potential energy">potential energy</a>, horizontal wave <a href="/wiki/Momentum" title="Momentum">momentum</a> and <a href="/wiki/Radiation_stress" title="Radiation stress">radiation stress</a> – as found by <a href="#CITEREFLonguet-Higgins1975">Longuet-Higgins (1975)</a>. He shows, for deep-water waves, that many of these integral properties have a maximum before the maximum wave height is reached (in support of Schwartz's findings). <a href="#CITEREFCokelet1978">Cokelet (1978)</a> harvtxt error: no target: CITEREFCokelet1978 (<a href="/wiki/Category:Harv_and_Sfn_template_errors" title="Category:Harv and Sfn template errors">help</a>), using a method similar to the one of Schwartz, computed and tabulated integral properties for a wide range of finite water depths (all reaching maxima below the highest wave height). Further, these integral properties play an important role in the <a href="/wiki/Conservation_law_(physics)" class="mw-redirect" title="Conservation law (physics)">conservation laws</a> for water waves, through <a href="/wiki/Noether%27s_theorem" title="Noether's theorem">Noether's theorem</a>.<a href="#cite_note-25">[25]</a> In 2005, Hammack, <a href="/wiki/Diane_Henderson" title="Diane Henderson">Henderson</a> and Segur have provided the first experimental evidence for the existence of three-dimensional progressive waves of permanent form in deep water – that is bi-periodic and two-dimensional progressive wave patterns of permanent form.<a href="#cite_note-26">[26]</a> The existence of these three-dimensional steady deep-water waves has been revealed in 2002, from a bifurcation study of two-dimensional Stokes waves by Craig and Nicholls, using numerical methods.<a href="#cite_note-27">[27]</a> <div class="mw-heading mw-heading3"><h3 id="Convergence_and_instability">Convergence and instability</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=10" title="Edit section: Convergence and instability">edit</a>]</div> <div class="mw-heading mw-heading4"><h4 id="Convergence">Convergence</h4>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=11" title="Edit section: Convergence">edit</a>]</div> Convergence of the Stokes expansion was first proved by <a href="#CITEREFLevi-Civita1925">Levi-Civita (1925)</a> for the case of small-amplitude waves – on the free surface of a fluid of infinite depth. This was extended shortly afterwards by <a href="#CITEREFStruik1926">Struik (1926)</a> for the case of finite depth and small-amplitude waves.<a href="#cite_note-28">[28]</a> Near the end of the 20th century, it was shown that for finite-amplitude waves the convergence of the Stokes expansion depends strongly on the formulation of the periodic wave problem. For instance, an inverse formulation of the periodic wave problem as used by Stokes – with the spatial coordinates as a function of <a href="/wiki/Velocity_potential" title="Velocity potential">velocity potential</a> and <a href="/wiki/Stream_function" title="Stream function">stream function</a> – does not converge for high-amplitude waves. While other formulations converge much more rapidly, e.g. in the <a href="/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Eulerian frame of reference</a> (with the velocity potential or stream function as a function of the spatial coordinates).<a href="#cite_note-Drennan-19">[19]</a> <div class="mw-heading mw-heading4"><h4 id="Highest_wave">Highest wave</h4>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=12" title="Edit section: Highest wave">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Stokes_wave_max_height.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Stokes_wave_max_height.svg/300px-Stokes_wave_max_height.svg.png" decoding="async" width="300" height="73" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Stokes_wave_max_height.svg/450px-Stokes_wave_max_height.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a5/Stokes_wave_max_height.svg/600px-Stokes_wave_max_height.svg.png 2x" data-file-width="512" data-file-height="124" /></a><figcaption>Stokes waves of maximum <a href="/wiki/Wave_height" title="Wave height">wave height</a> on deep water, under the action of gravity.</figcaption></figure> The maximum wave steepness, for periodic and propagating deep-water waves, is H / λ = 0.1410633 ± 4 · 10−7,<a href="#cite_note-29">[29]</a> so the wave height is about one-seventh (<style data-mw-deduplicate="TemplateStyles:r1214402035">.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}</style>⁠1/7⁠) of the wavelength λ.<a href="#cite_note-Schwartz_&_Fenton-24">[24]</a> And surface gravity waves of this maximum height have a sharp <a href="/wiki/Crest_(physics)" class="mw-redirect" title="Crest (physics)">wave crest</a> – with an angle of 120° (in the fluid domain) – also for finite depth, as shown by Stokes in 1880.<a href="#cite_note-Stokes_1880-18">[18]</a> An accurate estimate of the highest wave steepness in deep water (H / λ ≈ 0.142) was already made in 1893, by <a href="/wiki/John_Henry_Michell" title="John Henry Michell">John Henry Michell</a>, using a numerical method.<a href="#cite_note-30">[30]</a> A more detailed study of the behaviour of the highest wave near the sharp-cornered crest has been published by Malcolm A. Grant, in 1973.<a href="#cite_note-31">[31]</a> The existence of the highest wave on deep water with a sharp-angled crest of 120° was proved by <a href="/wiki/John_Toland_(mathematician)" title="John Toland (mathematician)">John Toland</a> in 1978.<a href="#cite_note-32">[32]</a> The convexity of η(x) between the successive maxima with a sharp-angled crest of 120° was independently proven by C.J. Amick et al. and Pavel I. Plotnikov in 1982 .<a href="#cite_note-33">[33]</a><a href="#cite_note-34">[34]</a> The highest Stokes wave – under the action of gravity – can be approximated with the following simple and accurate representation of the <a href="/wiki/Free_surface" title="Free surface">free surface</a> elevation η(x,t):<a href="#cite_note-Rainey_&_Longuet-Higgins-35">[35]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\eta }{\lambda }}=A\,\left[\cosh \,\left({\frac {x-ct}{\lambda }}\right)-1\right],}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>η</mi> <mi>λ</mi> </mfrac> </mrow> <mo>=</mo> <mi>A</mi> <mspace width="thinmathspace" /> <mrow> <mo>[</mo> <mrow> <mi>cosh</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mi>t</mi> </mrow> <mi>λ</mi> </mfrac> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mn>1</mn> </mrow> <mo>]</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\eta }{\lambda }}=A\,\left[\cosh \,\left({\frac {x-ct}{\lambda }}\right)-1\right],}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6c2bf098c0fc88ab0c53a0a5ff9c3d20f750013b" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:30.751ex; height:6.176ex;" alt="{\displaystyle {\frac {\eta }{\lambda }}=A\,\left[\cosh \,\left({\frac {x-ct}{\lambda }}\right)-1\right],}"> with <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle A={\frac {1}{{\sqrt {3}}\,\sinh \left({\frac {1}{2}}\right)}}\approx 1.108,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>A</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mn>3</mn> </msqrt> </mrow> <mspace width="thinmathspace" /> <mi>sinh</mi> <mo>⁡</mo> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> </mrow> <mo>≈</mo> <mn>1.108</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle A={\frac {1}{{\sqrt {3}}\,\sinh \left({\frac {1}{2}}\right)}}\approx 1.108,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b115e80be3c6af5bb0be7bdc2eea3cbda223b7f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:26.528ex; height:6.676ex;" alt="{\displaystyle A={\frac {1}{{\sqrt {3}}\,\sinh \left({\frac {1}{2}}\right)}}\approx 1.108,}"> for <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle -{\tfrac {1}{2}}\,\lambda \leq (x-ct)\leq {\tfrac {1}{2}}\,\lambda ,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>λ</mi> <mo>≤</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mo>≤</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mi>λ</mi> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\tfrac {1}{2}}\,\lambda \leq (x-ct)\leq {\tfrac {1}{2}}\,\lambda ,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9023e9da96bfebc890e8096b0141b05178b2a29f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:23.279ex; height:3.509ex;" alt="{\displaystyle -{\tfrac {1}{2}}\,\lambda \leq (x-ct)\leq {\tfrac {1}{2}}\,\lambda ,}"> and shifted horizontally over an <a href="/wiki/Integer" title="Integer">integer</a> number of wavelengths to represent the other waves in the regular wave train. This approximation is accurate to within 0.7% everywhere, as compared with the "exact" solution for the highest wave.<a href="#cite_note-Rainey_&_Longuet-Higgins-35">[35]</a> Another accurate approximation – however less accurate than the previous one – of the fluid motion on the surface of the steepest wave is by analogy with the swing of a <a href="/wiki/Pendulum" title="Pendulum">pendulum</a> in a <a href="/wiki/Grandfather_clock" title="Grandfather clock">grandfather clock</a>.<a href="#cite_note-36">[36]</a> Large library of Stokes waves computed with high precision for the case of infinite depth, represented with high accuracy (at least 27 digits after decimal point) as a <a href="/wiki/Pad%C3%A9_approximant" title="Padé approximant">Padé approximant</a> can be found at StokesWave.org<a href="#cite_note-37">[37]</a> <div class="mw-heading mw-heading4"><h4 id="Instability">Instability</h4>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=13" title="Edit section: Instability">edit</a>]</div> In deeper water, Stokes waves are unstable.<a href="#cite_note-Craik_WIFF-38">[38]</a> This was shown by <a href="/wiki/T._Brooke_Benjamin" class="mw-redirect" title="T. Brooke Benjamin">T. Brooke Benjamin</a> and Jim E. Feir in 1967.<a href="#cite_note-39">[39]</a><a href="#cite_note-40">[40]</a> The <a href="/wiki/Benjamin%E2%80%93Feir_instability" class="mw-redirect" title="Benjamin–Feir instability">Benjamin–Feir instability</a> is a side-band or modulational instability, with the side-band modulations propagating in the same direction as the <a href="/wiki/Carrier_wave" title="Carrier wave">carrier wave</a>; waves become unstable on deeper water for a relative depth kh > 1.363 (with k the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a> and h the mean water depth).<a href="#cite_note-41">[41]</a> The Benjamin–Feir instability can be described with the <a href="/wiki/Nonlinear_Schr%C3%B6dinger_equation" title="Nonlinear Schrödinger equation">nonlinear Schrödinger equation</a>, by inserting a Stokes wave with side bands.<a href="#cite_note-Craik_WIFF-38">[38]</a> Subsequently, with a more refined analysis, it has been shown – theoretically and experimentally – that the Stokes wave and its side bands exhibit <a href="/wiki/Fermi%E2%80%93Pasta%E2%80%93Ulam%E2%80%93Tsingou_recurrence" class="mw-redirect" title="Fermi–Pasta–Ulam–Tsingou recurrence">Fermi–Pasta–Ulam–Tsingou recurrence</a>: a cyclic alternation between modulation and demodulation.<a href="#cite_note-42">[42]</a> In 1978 <a href="/wiki/Michael_Longuet-Higgins" class="mw-redirect" title="Michael Longuet-Higgins">Longuet-Higgins</a>, by means of numerical modelling of fully non-linear waves and modulations (propagating in the carrier wave direction), presented a detailed analysis of the region of instability in deep water: both for superharmonics (for perturbations at the spatial scales smaller than the wavelength <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">) <a href="#cite_note-43">[43]</a> and subharmonics (for perturbations at the spatial scales larger than <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>λ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b43d0ea3c9c025af1be9128e62a18fa74bedda2a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }">).<a href="#cite_note-44">[44]</a> With increase of Stokes wave's amplitude, new modes of superharmonic instability appear. Appearance of a new branch of instability happens when the energy of the wave passes extremum. Detailed analysis of the mechanism of appearance of the new branches of instability has shown that their behavior follows closely a simple law, which allows to find with a good accuracy instability growth rates for all known and predicted branches.<a href="#cite_note-45">[45]</a> In Longuet-Higgins studies of two-dimensional wave motion, as well as the subsequent studies of three-dimensional modulations by McLean et al., new types of instabilities were found – these are associated with <a href="/wiki/Resonance" title="Resonance">resonant</a> wave interactions between five (or more) wave components.<a href="#cite_note-46">[46]</a><a href="#cite_note-47">[47]</a><a href="#cite_note-Dias_Kharif-48">[48]</a> <div class="mw-heading mw-heading2"><h2 id="Stokes_expansion"> Stokes expansion</h2>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=14" title="Edit section: Stokes expansion">edit</a>]</div> <div class="mw-heading mw-heading3"><h3 id="Governing_equations_for_a_potential_flow">Governing equations for a potential flow</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=15" title="Edit section: Governing equations for a potential flow">edit</a>]</div> In many instances, the oscillatory flow in the fluid interior of surface waves can be described accurately using <a href="/wiki/Potential_flow" title="Potential flow">potential flow</a> theory, apart from <a href="/wiki/Boundary_layer" title="Boundary layer">boundary layers</a> near the free surface and bottom (where <a href="/wiki/Vorticity" title="Vorticity">vorticity</a> is important, due to <a href="/wiki/Viscosity" title="Viscosity">viscous effects</a>, see <a href="/wiki/Stokes_boundary_layer" class="mw-redirect" title="Stokes boundary layer">Stokes boundary layer</a>).<a href="#cite_note-Phillips-49">[49]</a> Then, the <a href="/wiki/Flow_velocity" title="Flow velocity">flow velocity</a> u can be described as the <a href="/wiki/Gradient" title="Gradient">gradient</a> of a <a href="/wiki/Velocity_potential" title="Velocity potential">velocity potential</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }">: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} ={\boldsymbol {\nabla }}\Phi .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mi mathvariant="normal">Φ</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} ={\boldsymbol {\nabla }}\Phi .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/dc38425d1d26d9764468a09043d5e726ac66ba05" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.135ex; height:2.176ex;" alt="{\displaystyle \mathbf {u} ={\boldsymbol {\nabla }}\Phi .}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(A)</td></tr></tbody></table> Consequently, assuming <a href="/wiki/Incompressible_flow" title="Incompressible flow">incompressible flow</a>, the velocity field u is <a href="/wiki/Divergence-free" class="mw-redirect" title="Divergence-free">divergence-free</a> and the velocity potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> satisfies <a href="/wiki/Laplace%27s_equation" title="Laplace's equation">Laplace's equation</a><a href="#cite_note-Phillips-49">[49]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla ^{2}\Phi =0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla ^{2}\Phi =0}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ff3dc05852dd419ed76a30cfd3f97dd24878aef7" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.929ex; height:2.676ex;" alt="{\displaystyle \nabla ^{2}\Phi =0}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(B)</td></tr></tbody></table> in the fluid interior. The fluid region is described using three-dimensional <a href="/wiki/Cartesian_coordinate" class="mw-redirect" title="Cartesian coordinate">Cartesian coordinates</a> (x,y,z), with x and y the horizontal coordinates, and z the vertical coordinate – with the positive z-direction opposing the direction of the <a href="/wiki/Gravity_of_Earth" title="Gravity of Earth">gravitational acceleration</a>. Time is denoted with t. The free surface is located at z = η(x,y,t), and the bottom of the fluid region is at z = −h(x,y). The free-surface <a href="/wiki/Boundary_condition" class="mw-redirect" title="Boundary condition">boundary conditions</a> for <a href="/wiki/Surface_gravity_wave" class="mw-redirect" title="Surface gravity wave">surface gravity waves</a> – using a <a href="/wiki/Potential_flow" title="Potential flow">potential flow</a> description – consist of a kinematic and a dynamic boundary condition.<a href="#cite_note-Mei_4_6-50">[50]</a> The kinematic boundary condition ensures that the <a href="/wiki/Normal_component" class="mw-redirect" title="Normal component">normal component</a> of the fluid's <a href="/wiki/Flow_velocity" title="Flow velocity">flow velocity</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathbf {u} =[\partial \Phi /\partial x~~~\partial \Phi /\partial y~~~\partial \Phi /\partial z]^{\mathrm {T} }}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">[</mo> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> <mtext> </mtext> <mtext> </mtext> <mtext> </mtext> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> <msup> <mo stretchy="false">]</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">T</mi> </mrow> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mathbf {u} =[\partial \Phi /\partial x~~~\partial \Phi /\partial y~~~\partial \Phi /\partial z]^{\mathrm {T} }}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/edecd59b6933394e1af77f51ced0cd58d4ece37e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:30.783ex; height:3.176ex;" alt="{\displaystyle \mathbf {u} =[\partial \Phi /\partial x~~~\partial \Phi /\partial y~~~\partial \Phi /\partial z]^{\mathrm {T} }}"> in matrix notation, at the free surface equals the normal velocity component of the free-surface motion z = η(x,y,t): <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \eta }{\partial t}}+{\frac {\partial \Phi }{\partial x}}\,{\frac {\partial \eta }{\partial x}}+{\frac {\partial \Phi }{\partial y}}\,{\frac {\partial \eta }{\partial y}}={\frac {\partial \Phi }{\partial z}}\qquad {\text{ at }}z=\eta (x,y,t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>η</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> at </mtext> </mrow> <mi>z</mi> <mo>=</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \eta }{\partial t}}+{\frac {\partial \Phi }{\partial x}}\,{\frac {\partial \eta }{\partial x}}+{\frac {\partial \Phi }{\partial y}}\,{\frac {\partial \eta }{\partial y}}={\frac {\partial \Phi }{\partial z}}\qquad {\text{ at }}z=\eta (x,y,t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/20a33c1fd7719ba28a0b35ee908f32b65d96b953" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:52.26ex; height:6.176ex;" alt="{\displaystyle {\frac {\partial \eta }{\partial t}}+{\frac {\partial \Phi }{\partial x}}\,{\frac {\partial \eta }{\partial x}}+{\frac {\partial \Phi }{\partial y}}\,{\frac {\partial \eta }{\partial y}}={\frac {\partial \Phi }{\partial z}}\qquad {\text{ at }}z=\eta (x,y,t).}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(C)</td></tr></tbody></table> The dynamic boundary condition states that, without <a href="/wiki/Surface_tension" title="Surface tension">surface tension</a> effects, the atmospheric pressure just above the free surface equals the fluid <a href="/wiki/Pressure" title="Pressure">pressure</a> just below the surface. For an unsteady potential flow this means that the <a href="/wiki/Bernoulli%27s_principle" title="Bernoulli's principle">Bernoulli equation</a> is to be applied at the free surface. In case of a constant atmospheric pressure, the dynamic boundary condition becomes: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}\,\left|\mathbf {u} \right|^{2}+g\,\eta =0\qquad {\text{ at }}z=\eta (x,y,t),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mi>η</mi> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> at </mtext> </mrow> <mi>z</mi> <mo>=</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}\,\left|\mathbf {u} \right|^{2}+g\,\eta =0\qquad {\text{ at }}z=\eta (x,y,t),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e35fdd555b19cb7b7e99c6e00e58bb344d68cd23" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:43.403ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}\,\left|\mathbf {u} \right|^{2}+g\,\eta =0\qquad {\text{ at }}z=\eta (x,y,t),}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(D)</td></tr></tbody></table> where the constant atmospheric pressure has been taken equal to zero, <a href="/wiki/Without_loss_of_generality" title="Without loss of generality">without loss of generality</a>. Both boundary conditions contain the potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> as well as the surface elevation η. A (dynamic) boundary condition in terms of only the potential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.678ex; height:2.176ex;" alt="{\displaystyle \Phi }"> can be constructed by taking the <a href="/wiki/Material_derivative" title="Material derivative">material derivative</a> of the dynamic boundary condition, and using the kinematic boundary condition:<a href="#cite_note-Phillips-49">[49]</a><a href="#cite_note-Mei_4_6-50">[50]</a><a href="#cite_note-51">[51]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {Gray}{{\Bigl (}{\frac {\partial }{\partial t}}+\mathbf {u} \cdot {\boldsymbol {\nabla }}{\Bigr )}\,\left({\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}\,|\mathbf {u} |^{2}+g\,\eta \right)=0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.623em" minsize="1.623em">(</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.623em" minsize="1.623em">)</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mi>η</mi> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {Gray}{{\Bigl (}{\frac {\partial }{\partial t}}+\mathbf {u} \cdot {\boldsymbol {\nabla }}{\Bigr )}\,\left({\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}\,|\mathbf {u} |^{2}+g\,\eta \right)=0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/32ad1e7840a0fbe45ec825d11114bbdcfd8c1c89" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:40.521ex; height:6.176ex;" alt="{\displaystyle {\color {Gray}{{\Bigl (}{\frac {\partial }{\partial t}}+\mathbf {u} \cdot {\boldsymbol {\nabla }}{\Bigr )}\,\left({\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}\,|\mathbf {u} |^{2}+g\,\eta \right)=0}}}"> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {Gray}{\Rightarrow \quad {\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}+\mathbf {u} \cdot {\boldsymbol {\nabla }}{\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}\,{\frac {\partial }{\partial t}}\left(|\mathbf {u} |^{2}\right)+{\tfrac {1}{2}}\,\mathbf {u} \cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} |^{2}\right)=0}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">⇒</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> </mrow> </mstyle> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {Gray}{\Rightarrow \quad {\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}+\mathbf {u} \cdot {\boldsymbol {\nabla }}{\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}\,{\frac {\partial }{\partial t}}\left(|\mathbf {u} |^{2}\right)+{\tfrac {1}{2}}\,\mathbf {u} \cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} |^{2}\right)=0}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/3daf1fc5cb5baa334d2767e520428e5404503e40" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:66.85ex; height:6.009ex;" alt="{\displaystyle {\color {Gray}{\Rightarrow \quad {\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}+\mathbf {u} \cdot {\boldsymbol {\nabla }}{\frac {\partial \Phi }{\partial t}}+{\tfrac {1}{2}}\,{\frac {\partial }{\partial t}}\left(|\mathbf {u} |^{2}\right)+{\tfrac {1}{2}}\,\mathbf {u} \cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} |^{2}\right)=0}}}"> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\color {Gray}{\Rightarrow \quad }}{\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}+{\frac {\partial }{\partial t}}\left(|\mathbf {u} |^{2}\right)+{\tfrac {1}{2}}\,\mathbf {u} \cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} |^{2}\right)=0\qquad {\text{ at }}z=\eta (x,y,t).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mstyle mathcolor="#949698"> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">⇒</mo> <mspace width="1em" /> </mrow> </mstyle> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> at </mtext> </mrow> <mi>z</mi> <mo>=</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\color {Gray}{\Rightarrow \quad }}{\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}+{\frac {\partial }{\partial t}}\left(|\mathbf {u} |^{2}\right)+{\tfrac {1}{2}}\,\mathbf {u} \cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} |^{2}\right)=0\qquad {\text{ at }}z=\eta (x,y,t).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f664ea420138bcefe61d84a3fc9f2e795fbe0a46" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:73.175ex; height:6.009ex;" alt="{\displaystyle {\color {Gray}{\Rightarrow \quad }}{\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}+{\frac {\partial }{\partial t}}\left(|\mathbf {u} |^{2}\right)+{\tfrac {1}{2}}\,\mathbf {u} \cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} |^{2}\right)=0\qquad {\text{ at }}z=\eta (x,y,t).}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(E)</td></tr></tbody></table> At the bottom of the fluid layer, <a href="/wiki/Permeability_(earth_sciences)" class="mw-redirect" title="Permeability (earth sciences)">impermeability</a> requires the <a href="/wiki/Normal_component" class="mw-redirect" title="Normal component">normal component</a> of the flow velocity to vanish:<a href="#cite_note-Phillips-49">[49]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \Phi }{\partial n}}={\frac {1}{\sqrt {1+\left({\frac {\partial h}{\partial x}}\right)^{2}+\left({\frac {\partial h}{\partial y}}\right)^{2}}}}\,\left\{{\frac {\partial \Phi }{\partial z}}+{\frac {\partial h}{\partial x}}\,{\frac {\partial \Phi }{\partial x}}+{\frac {\partial h}{\partial y}}\,{\frac {\partial \Phi }{\partial y}}\right\}=0,\qquad {\text{ at }}z=-h(x,y),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msqrt> <mn>1</mn> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>h</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>+</mo> <msup> <mrow> <mo>(</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>h</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </msqrt> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>h</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>h</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>y</mi> </mrow> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> at </mtext> </mrow> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mi>h</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Phi }{\partial n}}={\frac {1}{\sqrt {1+\left({\frac {\partial h}{\partial x}}\right)^{2}+\left({\frac {\partial h}{\partial y}}\right)^{2}}}}\,\left\{{\frac {\partial \Phi }{\partial z}}+{\frac {\partial h}{\partial x}}\,{\frac {\partial \Phi }{\partial x}}+{\frac {\partial h}{\partial y}}\,{\frac {\partial \Phi }{\partial y}}\right\}=0,\qquad {\text{ at }}z=-h(x,y),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a5eb5c5e84eaf07a23bac1ef204dc674cff49c91" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.005ex; width:85.699ex; height:9.676ex;" alt="{\displaystyle {\frac {\partial \Phi }{\partial n}}={\frac {1}{\sqrt {1+\left({\frac {\partial h}{\partial x}}\right)^{2}+\left({\frac {\partial h}{\partial y}}\right)^{2}}}}\,\left\{{\frac {\partial \Phi }{\partial z}}+{\frac {\partial h}{\partial x}}\,{\frac {\partial \Phi }{\partial x}}+{\frac {\partial h}{\partial y}}\,{\frac {\partial \Phi }{\partial y}}\right\}=0,\qquad {\text{ at }}z=-h(x,y),}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(F)</td></tr></tbody></table> where h(x,y) is the depth of the bed below the <a href="/wiki/Datum_(geodesy)" class="mw-redirect" title="Datum (geodesy)">datum</a> z = 0 and n is the coordinate component in the direction <a href="/wiki/Surface_normal" class="mw-redirect" title="Surface normal">normal to the bed</a>. For permanent waves above a horizontal bed, the mean depth h is a constant and the boundary condition at the bed becomes: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \Phi }{\partial z}}=0\qquad {\text{ at }}z=-h.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> at </mtext> </mrow> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mi>h</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Phi }{\partial z}}=0\qquad {\text{ at }}z=-h.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/a4f4a564596e6c35a811c20144fadeb14c65eefa" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:23.947ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \Phi }{\partial z}}=0\qquad {\text{ at }}z=-h.}"> <div class="mw-heading mw-heading3"><h3 id="Taylor_series_in_the_free-surface_boundary_conditions">Taylor series in the free-surface boundary conditions</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=16" title="Edit section: Taylor series in the free-surface boundary conditions">edit</a>]</div> The free-surface boundary conditions <a href="#math_D">(D)</a> and <a href="#math_E">(E)</a> apply at the yet unknown free-surface elevation z = η(x,y,t). They can be transformed into boundary conditions at a fixed elevation z = constant by use of <a href="/wiki/Taylor_series" title="Taylor series">Taylor series</a> expansions of the flow field around that elevation.<a href="#cite_note-Phillips-49">[49]</a> Without loss of generality the mean surface elevation – around which the Taylor series are developed – can be taken at z = 0. This assures the expansion is around an elevation in the proximity of the actual free-surface elevation. Convergence of the Taylor series for small-amplitude steady-wave motion was proved by <a href="#CITEREFLevi-Civita1925">Levi-Civita (1925)</a>. The following notation is used: the Taylor series of some field f(x,y,z,t) around z = 0 – and evaluated at z = η(x,y,t) – is:<a href="#cite_note-Mei_607_608-52">[52]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x,y,\eta ,t)=\left[f\right]_{0}+\eta \,\left[{\frac {\partial f}{\partial z}}\right]_{0}+{\frac {1}{2}}\,\eta ^{2}\,\left[{\frac {\partial ^{2}f}{\partial z^{2}}}\right]_{0}+\cdots }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>η</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mrow> <mo>[</mo> <mi>f</mi> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>η</mi> <mspace width="thinmathspace" /> <msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mrow> <mspace width="thinmathspace" /> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f(x,y,\eta ,t)=\left[f\right]_{0}+\eta \,\left[{\frac {\partial f}{\partial z}}\right]_{0}+{\frac {1}{2}}\,\eta ^{2}\,\left[{\frac {\partial ^{2}f}{\partial z^{2}}}\right]_{0}+\cdots }</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/465aba21f0d6abae049ccb4b9ed2e9a479701d32" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:50.174ex; height:6.343ex;" alt="{\displaystyle f(x,y,\eta ,t)=\left[f\right]_{0}+\eta \,\left[{\frac {\partial f}{\partial z}}\right]_{0}+{\frac {1}{2}}\,\eta ^{2}\,\left[{\frac {\partial ^{2}f}{\partial z^{2}}}\right]_{0}+\cdots }"> with subscript zero meaning evaluation at z = 0, e.g.: [f]0 = f(x,y,0,t). Applying the Taylor expansion to free-surface boundary condition <a href="#math_E">Eq. (E)</a> in terms of the potential Φ gives:<a href="#cite_note-Phillips-49">[49]</a><a href="#cite_note-Mei_607_608-52">[52]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\left[{\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}\right]_{0}+\eta \left[{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}\right)\right]_{0}+\left[{\frac {\partial }{\partial t}}\left(|\mathbf {u} |^{2}\right)\right]_{0}\\&\quad +{\tfrac {1}{2}}\,\eta ^{2}\left[{\frac {\partial ^{2}}{\partial z^{2}}}\left({\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}\right)\right]_{0}+\eta \left[{\frac {\partial ^{2}}{\partial t\,\partial z}}\left(|\mathbf {u} |^{2}\right)\right]_{0}+\left[{\tfrac {1}{2}}\,\mathbf {u} \cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} |^{2}\right)\right]_{0}\\&\quad +\cdots =0,\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>η</mi> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>η</mi> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\left[{\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}\right]_{0}+\eta \left[{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}\right)\right]_{0}+\left[{\frac {\partial }{\partial t}}\left(|\mathbf {u} |^{2}\right)\right]_{0}\\&\quad +{\tfrac {1}{2}}\,\eta ^{2}\left[{\frac {\partial ^{2}}{\partial z^{2}}}\left({\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}\right)\right]_{0}+\eta \left[{\frac {\partial ^{2}}{\partial t\,\partial z}}\left(|\mathbf {u} |^{2}\right)\right]_{0}+\left[{\tfrac {1}{2}}\,\mathbf {u} \cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} |^{2}\right)\right]_{0}\\&\quad +\cdots =0,\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d01dcfd64cb3bec33751086f052cd96c8395392c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -7.338ex; width:75.714ex; height:15.843ex;" alt="{\displaystyle {\begin{aligned}&\left[{\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}\right]_{0}+\eta \left[{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}\right)\right]_{0}+\left[{\frac {\partial }{\partial t}}\left(|\mathbf {u} |^{2}\right)\right]_{0}\\&\quad +{\tfrac {1}{2}}\,\eta ^{2}\left[{\frac {\partial ^{2}}{\partial z^{2}}}\left({\frac {\partial ^{2}\Phi }{\partial t^{2}}}+g\,{\frac {\partial \Phi }{\partial z}}\right)\right]_{0}+\eta \left[{\frac {\partial ^{2}}{\partial t\,\partial z}}\left(|\mathbf {u} |^{2}\right)\right]_{0}+\left[{\tfrac {1}{2}}\,\mathbf {u} \cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} |^{2}\right)\right]_{0}\\&\quad +\cdots =0,\end{aligned}}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(G)</td></tr></tbody></table> showing terms up to triple products of η, Φ and u, as required for the construction of the Stokes expansion up to third-order O((ka)3). Here, ka is the wave steepness, with k a characteristic <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a> and a a characteristic wave <a href="/wiki/Amplitude" title="Amplitude">amplitude</a> for the problem under study. The fields η, Φ and u are assumed to be O(ka). The dynamic free-surface boundary condition <a href="#math_D">Eq. (D)</a> can be evaluated in terms of quantities at z = 0 as:<a href="#cite_note-Phillips-49">[49]</a><a href="#cite_note-Mei_607_608-52">[52]</a> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\left[{\frac {\partial \Phi }{\partial t}}+g\,\eta \right]_{0}+\eta \left[{\frac {\partial ^{2}\Phi }{\partial t\,\partial z}}\right]_{0}+{\biggl [}{\tfrac {1}{2}}\,\left|\mathbf {u} \right|^{2}{\biggr ]}_{0}\\&\quad +{\tfrac {1}{2}}\,\eta ^{2}\left[{\frac {\partial ^{3}\Phi }{\partial t\,\partial z^{2}}}\right]_{0}+\eta \left[{\frac {\partial }{\partial z}}\left({\tfrac {1}{2}}\,\left|\mathbf {u} \right|^{2}\right)\right]_{0}+\cdots =0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mi>η</mi> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>η</mi> <msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mspace width="1em" /> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mrow> <mo>[</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mi mathvariant="normal">Φ</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>η</mi> <msub> <mrow> <mo>[</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mrow> <mo>|</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>]</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\left[{\frac {\partial \Phi }{\partial t}}+g\,\eta \right]_{0}+\eta \left[{\frac {\partial ^{2}\Phi }{\partial t\,\partial z}}\right]_{0}+{\biggl [}{\tfrac {1}{2}}\,\left|\mathbf {u} \right|^{2}{\biggr ]}_{0}\\&\quad +{\tfrac {1}{2}}\,\eta ^{2}\left[{\frac {\partial ^{3}\Phi }{\partial t\,\partial z^{2}}}\right]_{0}+\eta \left[{\frac {\partial }{\partial z}}\left({\tfrac {1}{2}}\,\left|\mathbf {u} \right|^{2}\right)\right]_{0}+\cdots =0.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fdf6dbe6b8cc8e1e686a7de4f2cdf495717828e3" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.838ex; width:50.814ex; height:12.843ex;" alt="{\displaystyle {\begin{aligned}&\left[{\frac {\partial \Phi }{\partial t}}+g\,\eta \right]_{0}+\eta \left[{\frac {\partial ^{2}\Phi }{\partial t\,\partial z}}\right]_{0}+{\biggl [}{\tfrac {1}{2}}\,\left|\mathbf {u} \right|^{2}{\biggr ]}_{0}\\&\quad +{\tfrac {1}{2}}\,\eta ^{2}\left[{\frac {\partial ^{3}\Phi }{\partial t\,\partial z^{2}}}\right]_{0}+\eta \left[{\frac {\partial }{\partial z}}\left({\tfrac {1}{2}}\,\left|\mathbf {u} \right|^{2}\right)\right]_{0}+\cdots =0.\end{aligned}}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(H)</td></tr></tbody></table> The advantages of these Taylor-series expansions fully emerge in combination with a perturbation-series approach, for weakly non-linear waves (ka ≪ 1). <div class="mw-heading mw-heading3"><h3 id="Perturbation-series_approach">Perturbation-series approach</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=17" title="Edit section: Perturbation-series approach">edit</a>]</div> The <a href="/wiki/Perturbation_series" class="mw-redirect" title="Perturbation series">perturbation series</a> are in terms of a small ordering parameter ε ≪ 1 – which subsequently turns out to be proportional to (and of the order of) the wave slope ka, see the series solution in <a href="#Third-order_Stokes_wave_on_deep_water">this section</a>.<a href="#cite_note-53">[53]</a> So, take ε = ka: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}\eta &=\varepsilon \,\eta _{1}+\varepsilon ^{2}\,\eta _{2}+\varepsilon ^{3}\,\eta _{3}+\cdots ,\\\Phi &=\varepsilon \,\Phi _{1}+\varepsilon ^{2}\,\Phi _{2}+\varepsilon ^{3}\,\Phi _{3}+\cdots \quad {\text{and}}\\\mathbf {u} &=\varepsilon \,\mathbf {u} _{1}+\varepsilon ^{2}\,\mathbf {u} _{2}+\varepsilon ^{3}\,\mathbf {u} _{3}+\cdots .\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mi>η</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ε</mi> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mi mathvariant="normal">Φ</mi> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ε</mi> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> </mtd> </mtr> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> </mtd> <mtd> <mi></mi> <mo>=</mo> <mi>ε</mi> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}\eta &=\varepsilon \,\eta _{1}+\varepsilon ^{2}\,\eta _{2}+\varepsilon ^{3}\,\eta _{3}+\cdots ,\\\Phi &=\varepsilon \,\Phi _{1}+\varepsilon ^{2}\,\Phi _{2}+\varepsilon ^{3}\,\Phi _{3}+\cdots \quad {\text{and}}\\\mathbf {u} &=\varepsilon \,\mathbf {u} _{1}+\varepsilon ^{2}\,\mathbf {u} _{2}+\varepsilon ^{3}\,\mathbf {u} _{3}+\cdots .\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c586f808119dec09af742dd75a94b0d8c23368f1" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.171ex; width:37.947ex; height:9.509ex;" alt="{\displaystyle {\begin{aligned}\eta &=\varepsilon \,\eta _{1}+\varepsilon ^{2}\,\eta _{2}+\varepsilon ^{3}\,\eta _{3}+\cdots ,\\\Phi &=\varepsilon \,\Phi _{1}+\varepsilon ^{2}\,\Phi _{2}+\varepsilon ^{3}\,\Phi _{3}+\cdots \quad {\text{and}}\\\mathbf {u} &=\varepsilon \,\mathbf {u} _{1}+\varepsilon ^{2}\,\mathbf {u} _{2}+\varepsilon ^{3}\,\mathbf {u} _{3}+\cdots .\end{aligned}}}"> When applied in the flow equations, they should be valid independent of the particular value of ε. By equating in powers of ε, each term proportional to ε to a certain power has to equal to zero. As an example of how the perturbation-series approach works, consider the non-linear boundary condition <a href="#math_G">(G)</a>; it becomes:<a href="#cite_note-Dingemans-6">[6]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}&\varepsilon \,\left\{{\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right\}\\&+\varepsilon ^{2}\,\left\{{\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}+\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)+{\frac {\partial }{\partial t}}\left(|\mathbf {u} _{1}|^{2}\right)\right\}\\&+\varepsilon ^{3}\,\left\{{\frac {\partial ^{2}\Phi _{3}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{3}}{\partial z}}+\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}\right)\right.\\&\qquad \quad \left.+\eta _{2}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)+2\,{\frac {\partial }{\partial t}}\left(\mathbf {u} _{1}\cdot \mathbf {u} _{2}\right)\right.\\&\qquad \quad \left.+{\tfrac {1}{2}}\,\eta _{1}^{2}\,{\frac {\partial ^{2}}{\partial z^{2}}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)+\eta _{1}\,{\frac {\partial ^{2}}{\partial t\,\partial z}}\left(|\mathbf {u} _{1}|^{2}\right)+{\tfrac {1}{2}}\,\mathbf {u} _{1}\cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} _{1}|^{2}\right)\right\}\\&+{\mathcal {O}}\left(\varepsilon ^{4}\right)=0,\qquad {\text{at }}z=0.\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd /> <mtd> <mi>ε</mi> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <mspace width="thinmathspace" /> <mrow> <mo>{</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> 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class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> </mrow> <mo>}</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-ORD"> <mi class="MJX-tex-caligraphic" mathvariant="script">O</mi> </mrow> </mrow> <mrow> <mo>(</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>4</mn> </mrow> </msup> <mo>)</mo> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>at </mtext> </mrow> <mi>z</mi> <mo>=</mo> <mn>0.</mn> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&\varepsilon \,\left\{{\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right\}\\&+\varepsilon ^{2}\,\left\{{\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}+\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)+{\frac {\partial }{\partial t}}\left(|\mathbf {u} _{1}|^{2}\right)\right\}\\&+\varepsilon ^{3}\,\left\{{\frac {\partial ^{2}\Phi _{3}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{3}}{\partial z}}+\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}\right)\right.\\&\qquad \quad \left.+\eta _{2}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)+2\,{\frac {\partial }{\partial t}}\left(\mathbf {u} _{1}\cdot \mathbf {u} _{2}\right)\right.\\&\qquad \quad \left.+{\tfrac {1}{2}}\,\eta _{1}^{2}\,{\frac {\partial ^{2}}{\partial z^{2}}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)+\eta _{1}\,{\frac {\partial ^{2}}{\partial t\,\partial z}}\left(|\mathbf {u} _{1}|^{2}\right)+{\tfrac {1}{2}}\,\mathbf {u} _{1}\cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} _{1}|^{2}\right)\right\}\\&+{\mathcal {O}}\left(\varepsilon ^{4}\right)=0,\qquad {\text{at }}z=0.\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/cd44af06776c855a053095812782cc4e5f1fb5be" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -17.171ex; width:77.889ex; height:35.509ex;" alt="{\displaystyle {\begin{aligned}&\varepsilon \,\left\{{\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right\}\\&+\varepsilon ^{2}\,\left\{{\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}+\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)+{\frac {\partial }{\partial t}}\left(|\mathbf {u} _{1}|^{2}\right)\right\}\\&+\varepsilon ^{3}\,\left\{{\frac {\partial ^{2}\Phi _{3}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{3}}{\partial z}}+\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}\right)\right.\\&\qquad \quad \left.+\eta _{2}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)+2\,{\frac {\partial }{\partial t}}\left(\mathbf {u} _{1}\cdot \mathbf {u} _{2}\right)\right.\\&\qquad \quad \left.+{\tfrac {1}{2}}\,\eta _{1}^{2}\,{\frac {\partial ^{2}}{\partial z^{2}}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)+\eta _{1}\,{\frac {\partial ^{2}}{\partial t\,\partial z}}\left(|\mathbf {u} _{1}|^{2}\right)+{\tfrac {1}{2}}\,\mathbf {u} _{1}\cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} _{1}|^{2}\right)\right\}\\&+{\mathcal {O}}\left(\varepsilon ^{4}\right)=0,\qquad {\text{at }}z=0.\end{aligned}}}"> The resulting boundary conditions at z = 0 for the first three orders are: <dl><dt>First order:</dt></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/60479350cffa98593794fea0763038091f651f78" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:20.104ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}=0,}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(J1)</td></tr></tbody></table> <dl><dt>Second order:</dt></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}=-\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)-{\frac {\partial }{\partial t}}\left(|\mathbf {u} _{1}|^{2}\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}=-\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)-{\frac {\partial }{\partial t}}\left(|\mathbf {u} _{1}|^{2}\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7275e74b25b8286aed02072b9e38c33a8c2e17c0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:59.864ex; height:6.343ex;" alt="{\displaystyle {\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}=-\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)-{\frac {\partial }{\partial t}}\left(|\mathbf {u} _{1}|^{2}\right),}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(J2)</td></tr></tbody></table> <dl><dt>Third order:</dt></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\partial ^{2}\Phi _{3}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{3}}{\partial z}}=&-\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}\right)-\eta _{2}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)\\&-2\,{\frac {\partial }{\partial t}}\left(\mathbf {u} _{1}\cdot \mathbf {u} _{2}\right)-{\tfrac {1}{2}}\,\eta _{1}^{2}\,{\frac {\partial ^{2}}{\partial z^{2}}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)\\&-\eta _{1}\,{\frac {\partial ^{2}}{\partial t\,\partial z}}\left(|\mathbf {u} _{1}|^{2}\right)-{\tfrac {1}{2}}\,\mathbf {u} _{1}\cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} _{1}|^{2}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mn>2</mn> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msubsup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> </mrow> <mo>)</mo> </mrow> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <msup> <mrow class="MJX-TeXAtom-ORD"> <mo stretchy="false">|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial ^{2}\Phi _{3}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{3}}{\partial z}}=&-\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}\right)-\eta _{2}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)\\&-2\,{\frac {\partial }{\partial t}}\left(\mathbf {u} _{1}\cdot \mathbf {u} _{2}\right)-{\tfrac {1}{2}}\,\eta _{1}^{2}\,{\frac {\partial ^{2}}{\partial z^{2}}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)\\&-\eta _{1}\,{\frac {\partial ^{2}}{\partial t\,\partial z}}\left(|\mathbf {u} _{1}|^{2}\right)-{\tfrac {1}{2}}\,\mathbf {u} _{1}\cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} _{1}|^{2}\right).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1fc4f560b04632df20616efe879b6dc885fbc547" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.671ex; width:73.768ex; height:18.509ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial ^{2}\Phi _{3}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{3}}{\partial z}}=&-\eta _{1}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{2}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{2}}{\partial z}}\right)-\eta _{2}\,{\frac {\partial }{\partial z}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)\\&-2\,{\frac {\partial }{\partial t}}\left(\mathbf {u} _{1}\cdot \mathbf {u} _{2}\right)-{\tfrac {1}{2}}\,\eta _{1}^{2}\,{\frac {\partial ^{2}}{\partial z^{2}}}\left({\frac {\partial ^{2}\Phi _{1}}{\partial t^{2}}}+g\,{\frac {\partial \Phi _{1}}{\partial z}}\right)\\&-\eta _{1}\,{\frac {\partial ^{2}}{\partial t\,\partial z}}\left(|\mathbf {u} _{1}|^{2}\right)-{\tfrac {1}{2}}\,\mathbf {u} _{1}\cdot {\boldsymbol {\nabla }}\left(|\mathbf {u} _{1}|^{2}\right).\end{aligned}}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(J3)</td></tr></tbody></table> In a similar fashion – from the dynamic boundary condition <a href="#math_H">(H)</a> – the conditions at z = 0 at the orders 1, 2 and 3 become: <dl><dt>First order:</dt></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \Phi _{1}}{\partial t}}+g\,\eta _{1}=0,}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>=</mo> <mn>0</mn> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Phi _{1}}{\partial t}}+g\,\eta _{1}=0,}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ebd3cba4df2a5aece89e64293f456bba4b399ac6" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:16.347ex; height:5.509ex;" alt="{\displaystyle {\frac {\partial \Phi _{1}}{\partial t}}+g\,\eta _{1}=0,}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(K1)</td></tr></tbody></table> <dl><dt>Second order:</dt></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial \Phi _{2}}{\partial t}}+g\,\eta _{2}=-\eta _{1}\,{\frac {\partial ^{2}\Phi _{1}}{\partial t\,\partial z}}-{\tfrac {1}{2}}\,\left|\mathbf {u} _{1}\right|^{2},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>=</mo> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mrow> <mo>|</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial \Phi _{2}}{\partial t}}+g\,\eta _{2}=-\eta _{1}\,{\frac {\partial ^{2}\Phi _{1}}{\partial t\,\partial z}}-{\tfrac {1}{2}}\,\left|\mathbf {u} _{1}\right|^{2},}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c8da3714c1d901894b8a20676dbfb72827646224" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:35.329ex; height:5.843ex;" alt="{\displaystyle {\frac {\partial \Phi _{2}}{\partial t}}+g\,\eta _{2}=-\eta _{1}\,{\frac {\partial ^{2}\Phi _{1}}{\partial t\,\partial z}}-{\tfrac {1}{2}}\,\left|\mathbf {u} _{1}\right|^{2},}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(K2)</td></tr></tbody></table> <dl><dt>Third order:</dt></dl> <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\begin{aligned}{\frac {\partial \Phi _{3}}{\partial t}}+g\,\eta _{3}=&-\eta _{1}\,{\frac {\partial ^{2}\Phi _{2}}{\partial t\,\partial z}}-\eta _{2}\,{\frac {\partial ^{2}\Phi _{1}}{\partial t\,\partial z}}-\mathbf {u} _{1}\cdot \mathbf {u} _{2}\\&-{\tfrac {1}{2}}\,\eta _{1}^{2}\,{\frac {\partial ^{3}\Phi _{1}}{\partial t\,\partial z^{2}}}-\eta _{1}\,{\frac {\partial }{\partial z}}\left({\tfrac {1}{2}}\,\left|\mathbf {u} _{1}\right|^{2}\right).\end{aligned}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right left right left right left right left right left right left" rowspacing="3pt" columnspacing="0em 2em 0em 2em 0em 2em 0em 2em 0em 2em 0em" displaystyle="true"> <mtr> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>+</mo> <mi>g</mi> <mspace width="thinmathspace" /> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo>=</mo> </mtd> <mtd> <mi></mi> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mo>−</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>⋅</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> </mtd> </mtr> <mtr> <mtd /> <mtd> <mi></mi> <mo>−</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msubsup> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> <mspace width="thinmathspace" /> <mi mathvariant="normal">∂</mi> <msup> <mi>z</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>−</mo> <msub> <mi>η</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi mathvariant="normal">∂</mi> <mrow> <mi mathvariant="normal">∂</mi> <mi>z</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mspace width="thinmathspace" /> <msup> <mrow> <mo>|</mo> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>.</mo> </mtd> </mtr> </mtable> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}{\frac {\partial \Phi _{3}}{\partial t}}+g\,\eta _{3}=&-\eta _{1}\,{\frac {\partial ^{2}\Phi _{2}}{\partial t\,\partial z}}-\eta _{2}\,{\frac {\partial ^{2}\Phi _{1}}{\partial t\,\partial z}}-\mathbf {u} _{1}\cdot \mathbf {u} _{2}\\&-{\tfrac {1}{2}}\,\eta _{1}^{2}\,{\frac {\partial ^{3}\Phi _{1}}{\partial t\,\partial z^{2}}}-\eta _{1}\,{\frac {\partial }{\partial z}}\left({\tfrac {1}{2}}\,\left|\mathbf {u} _{1}\right|^{2}\right).\end{aligned}}}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/71cd0b768bd81054566d95840d80836d1574954a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -5.338ex; width:48.797ex; height:11.843ex;" alt="{\displaystyle {\begin{aligned}{\frac {\partial \Phi _{3}}{\partial t}}+g\,\eta _{3}=&-\eta _{1}\,{\frac {\partial ^{2}\Phi _{2}}{\partial t\,\partial z}}-\eta _{2}\,{\frac {\partial ^{2}\Phi _{1}}{\partial t\,\partial z}}-\mathbf {u} _{1}\cdot \mathbf {u} _{2}\\&-{\tfrac {1}{2}}\,\eta _{1}^{2}\,{\frac {\partial ^{3}\Phi _{1}}{\partial t\,\partial z^{2}}}-\eta _{1}\,{\frac {\partial }{\partial z}}\left({\tfrac {1}{2}}\,\left|\mathbf {u} _{1}\right|^{2}\right).\end{aligned}}}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(K3)</td></tr></tbody></table> For the linear equations <a href="#math_A">(A)</a>, <a href="#math_B">(B)</a> and <a href="#math_F">(F)</a> the perturbation technique results in a series of equations independent of the perturbation solutions at other orders: <table role="presentation" style="border-collapse:collapse; margin:0 0 0 0em; border:none;"><tbody><tr><td style="vertical-align:middle; border:none; padding:0;" class="nowrap"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left.{\begin{array}{rcl}\mathbf {u} _{j}&=&{\boldsymbol {\nabla }}\Phi _{j},\\[1ex]\nabla ^{2}\Phi _{j}&=&0,\\[1ex]\displaystyle {\frac {\partial \Phi _{j}}{\partial n}}&=&0\quad {\text{ at }}z=-h,\end{array}}\right\}\qquad {\text{for all orders }}j\in \mathbb {N} ^{+}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mtable columnalign="right center left" rowspacing="0.83em 0.83em 0.4em" columnspacing="1em"> <mtr> <mtd> <msub> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">∇</mi> </mrow> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>0</mn> <mo>,</mo> </mtd> </mtr> <mtr> <mtd> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <msub> <mi mathvariant="normal">Φ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>n</mi> </mrow> </mfrac> </mrow> </mstyle> </mtd> <mtd> <mo>=</mo> </mtd> <mtd> <mn>0</mn> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext> at </mtext> </mrow> <mi>z</mi> <mo>=</mo> <mo>−</mo> <mi>h</mi> <mo>,</mo> </mtd> </mtr> </mtable> </mrow> <mo>}</mo> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>for all orders </mtext> </mrow> <mi>j</mi> <mo>∈</mo> <msup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="double-struck">N</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mo>+</mo> </mrow> </msup> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left.{\begin{array}{rcl}\mathbf {u} _{j}&=&{\boldsymbol {\nabla }}\Phi _{j},\\[1ex]\nabla ^{2}\Phi _{j}&=&0,\\[1ex]\displaystyle {\frac {\partial \Phi _{j}}{\partial n}}&=&0\quad {\text{ at }}z=-h,\end{array}}\right\}\qquad {\text{for all orders }}j\in \mathbb {N} ^{+}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2922c0f346b895682f25d8c9d081b9437974da3d" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -6.838ex; width:55.42ex; height:14.843ex;" alt="{\displaystyle \left.{\begin{array}{rcl}\mathbf {u} _{j}&=&{\boldsymbol {\nabla }}\Phi _{j},\\[1ex]\nabla ^{2}\Phi _{j}&=&0,\\[1ex]\displaystyle {\frac {\partial \Phi _{j}}{\partial n}}&=&0\quad {\text{ at }}z=-h,\end{array}}\right\}\qquad {\text{for all orders }}j\in \mathbb {N} ^{+}.}"></td> <td style="vertical-align:middle; width:99%; border:none; padding:0;"></td> <td style="vertical-align:middle; border:none; padding:0;" class="nowrap">(L)</td></tr></tbody></table> The above perturbation equations can be solved sequentially, i.e. starting with first order, thereafter continuing with the second order, third order, etc. <div class="mw-heading mw-heading3"><h3 id="Application_to_progressive_periodic_waves_of_permanent_form">Application to progressive periodic waves of permanent form</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=18" title="Edit section: Application to progressive periodic waves of permanent form">edit</a>]</div> <figure class="mw-halign-right" typeof="mw:File/Thumb"><a href="/wiki/File:Deep_water_wave.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/4/4a/Deep_water_wave.gif/300px-Deep_water_wave.gif" decoding="async" width="300" height="197" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/4/4a/Deep_water_wave.gif 1.5x" data-file-width="390" data-file-height="256" /></a><figcaption>Animation of steep Stokes waves in deep water, with a <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> of about twice the water depth, for three successive wave <a href="/wiki/Period_(physics)" class="mw-redirect" title="Period (physics)">periods</a>. The <a href="/wiki/Wave_height" title="Wave height">wave height</a> is about 9.2% of the <a href="/wiki/Wavelength" title="Wavelength">wavelength</a>. Description of the animation: The white dots are fluid particles, followed in time. In the case shown here, the <a href="/wiki/Average" title="Average">mean</a> <a href="/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Eulerian</a> horizontal <a href="/wiki/Flow_velocity" title="Flow velocity">velocity</a> below the wave <a href="/wiki/Trough_(physics)" class="mw-redirect" title="Trough (physics)">trough</a> is zero.<a href="#cite_note-54">[54]</a></figcaption></figure> The waves of permanent form propagate with a constant <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a> (or <a href="https://en.wiktionary.org/wiki/celerity" class="extiw" title="wikt:celerity">celerity</a>), denoted as c. If the steady wave motion is in the horizontal x-direction, the flow quantities η and u are not separately dependent on x and time t, but are functions of x − ct:<a href="#cite_note-Wehausen_Laitone-55">[55]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta (x,t)=\eta (x-ct)\quad {\text{and}}\quad \mathbf {u} (x,z,t)=\mathbf {u} (x-ct,z).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>η</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mi>t</mi> <mo stretchy="false">)</mo> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="1em" /> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold">u</mi> </mrow> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta (x,t)=\eta (x-ct)\quad {\text{and}}\quad \mathbf {u} (x,z,t)=\mathbf {u} (x-ct,z).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/51d8eb167bffaed80b8b352614e941995802dd15" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:50.467ex; height:2.843ex;" alt="{\displaystyle \eta (x,t)=\eta (x-ct)\quad {\text{and}}\quad \mathbf {u} (x,z,t)=\mathbf {u} (x-ct,z).}"> Further the waves are periodic – and because they are also of permanent form – both in horizontal space x and in time t, with <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> λ and <a href="/wiki/Period_(physics)" class="mw-redirect" title="Period (physics)">period</a> τ respectively. Note that Φ(x,z,t) itself is not necessary periodic due to the possibility of a constant (linear) drift in x and/or t:<a href="#cite_note-Whitham_16.6-56">[56]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi (x,z,t)=\beta x-\gamma t+\varphi (x-ct,z),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>,</mo> <mi>z</mi> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>β</mi> <mi>x</mi> <mo>−</mo> <mi>γ</mi> <mi>t</mi> <mo>+</mo> <mi>φ</mi> <mo stretchy="false">(</mo> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mi>t</mi> <mo>,</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi (x,z,t)=\beta x-\gamma t+\varphi (x-ct,z),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f382d76320dbf4dcfddc3dc0fb0d997b3425abc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:34.47ex; height:2.843ex;" alt="{\displaystyle \Phi (x,z,t)=\beta x-\gamma t+\varphi (x-ct,z),}"> with φ(x,z,t) – as well as the derivatives ∂Φ/∂t and ∂Φ/∂x – being periodic. Here β is the mean flow velocity below <a href="/wiki/Trough_(physics)" class="mw-redirect" title="Trough (physics)">trough</a> level, and γ is related to the <a href="/wiki/Hydraulic_head" title="Hydraulic head">hydraulic head</a> as observed in a <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a> moving with the wave's phase velocity c (so the flow becomes <a href="/wiki/Steady_flow" class="mw-redirect" title="Steady flow">steady</a> in this reference frame). In order to apply the Stokes expansion to progressive periodic waves, it is advantageous to describe them through <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a> as a function of the <a href="/wiki/Wave_phase" class="mw-redirect" title="Wave phase">wave phase</a> θ(x,t):<a href="#cite_note-Dias_Kharif-48">[48]</a><a href="#cite_note-Whitham_16.6-56">[56]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \theta =kx-\omega t=k\left(x-ct\right),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>θ</mi> <mo>=</mo> <mi>k</mi> <mi>x</mi> <mo>−</mo> <mi>ω</mi> <mi>t</mi> <mo>=</mo> <mi>k</mi> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>−</mo> <mi>c</mi> <mi>t</mi> </mrow> <mo>)</mo> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \theta =kx-\omega t=k\left(x-ct\right),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/c74aabc95d94296767272bfb4ee8bbdd65c5b48f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:25.412ex; height:2.843ex;" alt="{\displaystyle \theta =kx-\omega t=k\left(x-ct\right),}"> assuming waves propagating in the x–direction. Here k = 2π / λ is the <a href="/wiki/Wavenumber" title="Wavenumber">wavenumber</a>, ω = 2π / τ is the <a href="/wiki/Angular_frequency" title="Angular frequency">angular frequency</a> and c = ω / k (= λ / τ) is the <a href="/wiki/Phase_velocity" title="Phase velocity">phase velocity</a>. Now, the free surface elevation η(x,t) of a periodic wave can be described as the <a href="/wiki/Fourier_series" title="Fourier series">Fourier series</a>:<a href="#cite_note-Fenton-11">[11]</a><a href="#cite_note-Whitham_16.6-56">[56]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta =\sum _{n=1}^{\infty }A_{n}\,\cos \,(n\theta ).}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> <mo>=</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>A</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mi>cos</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta =\sum _{n=1}^{\infty }A_{n}\,\cos \,(n\theta ).}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fbb5ee8868d31aa7e523007485d7d29154e5841" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:20.185ex; height:6.843ex;" alt="{\displaystyle \eta =\sum _{n=1}^{\infty }A_{n}\,\cos \,(n\theta ).}"> Similarly, the corresponding expression for the velocity potential Φ(x,z,t) is:<a href="#cite_note-Whitham_16.6-56">[56]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Phi =\beta x-\gamma t+\sum _{n=1}^{\infty }B_{n}\,{\biggl [}\cosh \,\left(nk\,(z+h)\right){\biggr ]}\,\sin \,(n\theta ),}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">Φ</mi> <mo>=</mo> <mi>β</mi> <mi>x</mi> <mo>−</mo> <mi>γ</mi> <mi>t</mi> <mo>+</mo> <munderover> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="normal">∞</mi> </mrow> </munderover> <msub> <mi>B</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="2.047em" minsize="2.047em">[</mo> </mrow> </mrow> <mi>cosh</mi> <mspace width="thinmathspace" /> <mrow> <mo>(</mo> <mrow> <mi>n</mi> <mi>k</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>z</mi> <mo>+</mo> <mi>h</mi> <mo stretchy="false">)</mo> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="2.047em" minsize="2.047em">]</mo> </mrow> </mrow> <mspace width="thinmathspace" /> <mi>sin</mi> <mspace width="thinmathspace" /> <mo stretchy="false">(</mo> <mi>n</mi> <mi>θ</mi> <mo stretchy="false">)</mo> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Phi =\beta x-\gamma t+\sum _{n=1}^{\infty }B_{n}\,{\biggl [}\cosh \,\left(nk\,(z+h)\right){\biggr ]}\,\sin \,(n\theta ),}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fc8c218cc52fdfd3bfaf58112bee0018c3013eb" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:51.577ex; height:6.843ex;" alt="{\displaystyle \Phi =\beta x-\gamma t+\sum _{n=1}^{\infty }B_{n}\,{\biggl [}\cosh \,\left(nk\,(z+h)\right){\biggr ]}\,\sin \,(n\theta ),}"> satisfying both the <a href="/wiki/Laplace_equation" class="mw-redirect" title="Laplace equation">Laplace equation</a> ∇2Φ = 0 in the fluid interior, as well as the boundary condition ∂Φ/∂z = 0 at the bed z = −h. For a given value of the wavenumber k, the parameters: An, Bn (with n = 1, 2, 3, ...), c, β and γ have yet to be determined. They all can be expanded as perturbation series in ε. <a href="#CITEREFFenton1990">Fenton (1990)</a> provides these values for fifth-order Stokes's wave theory. For progressive periodic waves, derivatives with respect to x and t of functions f(θ,z) of θ(x,t) can be expressed as derivatives with respect to θ: <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {\partial f}{\partial x}}=+k\,{\frac {\partial f}{\partial \theta }}\qquad {\text{and}}\qquad {\frac {\partial f}{\partial t}}=-\omega \,{\frac {\partial f}{\partial \theta }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>+</mo> <mi>k</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mtext>and</mtext> </mrow> <mspace width="2em" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>t</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>−</mo> <mi>ω</mi> <mspace width="thinmathspace" /> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">∂</mi> <mi>f</mi> </mrow> <mrow> <mi mathvariant="normal">∂</mi> <mi>θ</mi> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial f}{\partial x}}=+k\,{\frac {\partial f}{\partial \theta }}\qquad {\text{and}}\qquad {\frac {\partial f}{\partial t}}=-\omega \,{\frac {\partial f}{\partial \theta }}.}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6ce564c6f34fe16f9fccae131369e49777c65e30" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:40.711ex; height:5.676ex;" alt="{\displaystyle {\frac {\partial f}{\partial x}}=+k\,{\frac {\partial f}{\partial \theta }}\qquad {\text{and}}\qquad {\frac {\partial f}{\partial t}}=-\omega \,{\frac {\partial f}{\partial \theta }}.}"> The important point for non-linear waves – in contrast to linear <a href="/wiki/Airy_wave_theory" title="Airy wave theory">Airy wave theory</a> – is that the phase velocity c also depends on the <a href="/wiki/Wave_amplitude" class="mw-redirect" title="Wave amplitude">wave amplitude</a> a, besides its dependence on wavelength λ = 2π / k and mean depth h. Negligence of the dependence of c on wave amplitude results in the appearance of <a href="/wiki/Secular_term" class="mw-redirect" title="Secular term">secular terms</a>, in the higher-order contributions to the perturbation-series solution. <a href="#CITEREFStokes1847">Stokes (1847)</a> already applied the required non-linear correction to the phase speed c in order to prevent secular behaviour. A general approach to do so is now known as the <a href="/wiki/Lindstedt%E2%80%93Poincar%C3%A9_method" class="mw-redirect" title="Lindstedt–Poincaré method">Lindstedt–Poincaré method</a>. Since the wavenumber k is given and thus fixed, the non-linear behaviour of the phase velocity c = ω / k is brought into account by also expanding the angular frequency ω into a perturbation series:<a href="#cite_note-Whitham_13.13-9">[9]</a> <math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega =\omega _{0}+\varepsilon \,\omega _{1}+\varepsilon ^{2}\,\omega _{2}+\cdots .}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ω</mi> <mo>=</mo> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>ε</mi> <mspace width="thinmathspace" /> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>+</mo> <msup> <mi>ε</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mspace width="thinmathspace" /> <msub> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>+</mo> <mo>⋯</mo> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \omega =\omega _{0}+\varepsilon \,\omega _{1}+\varepsilon ^{2}\,\omega _{2}+\cdots .}</annotation> </semantics> </math><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eba520f362f9e396d0991f8a7fd1e620205fceb0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:28.318ex; height:3.009ex;" alt="{\displaystyle \omega =\omega _{0}+\varepsilon \,\omega _{1}+\varepsilon ^{2}\,\omega _{2}+\cdots .}"> Here ω0 will turn out to be related to the wavenumber k through the linear <a href="/wiki/Dispersion_(water_waves)" title="Dispersion (water waves)">dispersion relation</a>. However time derivatives, through ∂f/∂t = −ω ∂f/∂θ, now also give contributions – containing ω1, ω2, etc. – to the governing equations at higher orders in the perturbation series. By tuning ω1, ω2, etc., secular behaviour can be prevented. For surface gravity waves, it is found that ω1 = 0 and the first non-zero contribution to the dispersion relation comes from ω2 (see e.g. the sub-section "<a href="#Third-order_dispersion_relation">Third-order dispersion relation</a>" above).<a href="#cite_note-Whitham_13.13-9">[9]</a> <div class="mw-heading mw-heading2"><h2 id="Stokes's_two_definitions_of_wave_celerity">Stokes's two definitions of wave celerity </h2>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=19" title="Edit section: Stokes's two definitions of wave celerity">edit</a>]</div> For non-linear surface waves there is, in general, ambiguity in splitting the total motion into a wave part and a <a href="/wiki/Average" title="Average">mean</a> part. As a consequence, there is some freedom in choosing the phase speed (celerity) of the wave. <a href="#CITEREFStokes1847">Stokes (1847)</a> identified two logical definitions of phase speed, known as Stokes's first and second definition of wave celerity:<a href="#cite_note-Dingemans-6">[6]</a><a href="#cite_note-Fenton-11">[11]</a><a href="#cite_note-57">[57]</a> <ol><li>Stokes's first definition of wave celerity has, for a pure wave motion, the <a href="/wiki/Average" title="Average">mean value</a> of the horizontal <a href="/wiki/Lagrangian_and_Eulerian_specification_of_the_flow_field" title="Lagrangian and Eulerian specification of the flow field">Eulerian</a> flow-velocity ŪE at any location below <a href="/wiki/Trough_(physics)" class="mw-redirect" title="Trough (physics)">trough</a> level equal to zero. Due to the <a href="/wiki/Irrotational_flow" class="mw-redirect" title="Irrotational flow">irrotationality</a> of potential flow, together with the horizontal sea bed and periodicity the mean horizontal velocity, the mean horizontal velocity is a constant between bed and trough level. So in Stokes first definition the wave is considered from a <a href="/wiki/Frame_of_reference" title="Frame of reference">frame of reference</a> moving with the mean horizontal velocity ŪE. This is an advantageous approach when the mean Eulerian flow velocity ŪE is known, e.g. from measurements.</li> <li>Stokes's second definition of wave celerity is for a frame of reference where the mean horizontal <a href="/wiki/Mass_transfer" title="Mass transfer">mass transport</a> of the wave motion equal to zero. This is different from the first definition due to the mass transport in the <a href="/wiki/Splash_zone" title="Splash zone">splash zone</a>, i.e. between the trough and crest level, in the wave propagation direction. This wave-induced mass transport is caused by the positive <a href="/wiki/Correlation_(statistics)" class="mw-redirect" title="Correlation (statistics)">correlation</a> between surface elevation and horizontal velocity. In the reference frame for Stokes's second definition, the wave-induced mass transport is compensated by an opposing <a href="/wiki/Undertow_(wave_action)" class="mw-redirect" title="Undertow (wave action)">undertow</a> (so ŪE < 0 for waves propagating in the positive x-direction). This is the logical definition for waves generated in a <a href="/wiki/Wave_flume" class="mw-redirect" title="Wave flume">wave flume</a> in the laboratory, or waves moving perpendicular towards a beach.</li></ol> As pointed out by <a href="/wiki/Michael_E._McIntyre" title="Michael E. McIntyre">Michael E. McIntyre</a>, the mean horizontal mass transport will be (near) zero for a <a href="/wiki/Wave_group" class="mw-redirect" title="Wave group">wave group</a> approaching into still water, with also in deep water the mass transport caused by the waves balanced by an opposite mass transport in a return flow (undertow).<a href="#cite_note-58">[58]</a> This is due to the fact that otherwise a large mean force will be needed to accelerate the body of water into which the wave group is propagating. <div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=20" title="Edit section: Notes">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239543626">.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}</style><div class="reflist reflist-columns references-column-width reflist-columns-2"> <ol class="references"> <li id="cite_note-1"><a href="#cite_ref-1">^</a> Figure 5 in: <style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFSusan_Bartsch-WinklerDavid_K._Lynch1988" class="citation cs2">Susan Bartsch-Winkler; David K. Lynch (1988), <a rel="nofollow" class="external text" href="https://pubs.er.usgs.gov/#search:advance/page=1/page_size=100/advance=undefined/series_cd=CIR/report_number=1022:0">Catalog of worldwide tidal bore occurrences and characteristics</a> (Circular 1022), <a href="/wiki/United_States_Geological_Survey" title="United States Geological Survey">U. S. Geological Survey</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Catalog+of+worldwide+tidal+bore+occurrences+and+characteristics&rft.pub=U.+S.+Geological+Survey&rft.date=1988&rft.au=Susan+Bartsch-Winkler&rft.au=David+K.+Lynch&rft_id=https%3A%2F%2Fpubs.er.usgs.gov%2F%23search%3Aadvance%2Fpage%3D1%2Fpage_size%3D100%2Fadvance%3Dundefined%2Fseries_cd%3DCIR%2Freport_number%3D1022%3A0&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-2"><a href="#cite_ref-2">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFChakrabarti2005" class="citation cs2">Chakrabarti, S.K. (2005), Handbook of Offshore Engineering, Elsevier, p. 235, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780080445687" title="Special:BookSources/9780080445687"><bdi>9780080445687</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Handbook+of+Offshore+Engineering&rft.pages=235&rft.pub=Elsevier&rft.date=2005&rft.isbn=9780080445687&rft.aulast=Chakrabarti&rft.aufirst=S.K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-3"><a href="#cite_ref-3">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrant1973" class="citation cs2">Grant, M.A. (1973), "Standing Stokes waves of maximum height", Journal of Fluid Mechanics, 60 (3): 593–604, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973JFM....60..593G">1973JFM....60..593G</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112073000364">10.1017/S0022112073000364</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123179735">123179735</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Standing+Stokes+waves+of+maximum+height&rft.volume=60&rft.issue=3&rft.pages=593-604&rft.date=1973&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123179735%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112073000364&rft_id=info%3Abibcode%2F1973JFM....60..593G&rft.aulast=Grant&rft.aufirst=M.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-4"><a href="#cite_ref-4">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFOchi2003" class="citation cs2">Ochi, Michel K. (2003), Hurricane-generated seas, Elsevier, p. 119, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780080443126" title="Special:BookSources/9780080443126"><bdi>9780080443126</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Hurricane-generated+seas&rft.pages=119&rft.pub=Elsevier&rft.date=2003&rft.isbn=9780080443126&rft.aulast=Ochi&rft.aufirst=Michel+K.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-5"><a href="#cite_ref-5">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTayfun1980" class="citation cs2">Tayfun, M.A. (1980), "Narrow-band nonlinear sea waves", Journal of Geophysical Research, 85 (C3): 1548–1552, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1980JGR....85.1548T">1980JGR....85.1548T</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1029%2FJC085iC03p01548">10.1029/JC085iC03p01548</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Geophysical+Research&rft.atitle=Narrow-band+nonlinear+sea+waves&rft.volume=85&rft.issue=C3&rft.pages=1548-1552&rft.date=1980&rft_id=info%3Adoi%2F10.1029%2FJC085iC03p01548&rft_id=info%3Abibcode%2F1980JGR....85.1548T&rft.aulast=Tayfun&rft.aufirst=M.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Dingemans-6">^ <a href="#cite_ref-Dingemans_6-0">a</a> <a href="#cite_ref-Dingemans_6-1">b</a> <a href="#cite_ref-Dingemans_6-2">c</a> <a href="#cite_ref-Dingemans_6-3">d</a> <a href="#cite_ref-Dingemans_6-4">e</a> <a href="#cite_ref-Dingemans_6-5">f</a> <a href="#cite_ref-Dingemans_6-6">g</a> <a href="#cite_ref-Dingemans_6-7">h</a> <a href="#cite_ref-Dingemans_6-8">i</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDingemans1997" class="citation cs2">Dingemans, M.W. (1997), "Water wave propagation over uneven bottoms", NASA Sti/Recon Technical Report N, Advanced Series on Ocean Engineering, 13: 171–184, §2.8, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1985STIN...8525769K">1985STIN...8525769K</a>, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-981-02-0427-3" title="Special:BookSources/978-981-02-0427-3"><bdi>978-981-02-0427-3</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/36126836">36126836</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=NASA+Sti%2FRecon+Technical+Report+N&rft.atitle=Water+wave+propagation+over+uneven+bottoms&rft.volume=13&rft.pages=171-184%2C+%C2%A72.8&rft.date=1997&rft_id=info%3Aoclcnum%2F36126836&rft_id=info%3Abibcode%2F1985STIN...8525769K&rft.isbn=978-981-02-0427-3&rft.aulast=Dingemans&rft.aufirst=M.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-7"><a href="#cite_ref-7">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSvendsen2006" class="citation cs2">Svendsen, I.A. (2006), Introduction to nearshore hydrodynamics, World Scientific, p. 370, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789812561428" title="Special:BookSources/9789812561428"><bdi>9789812561428</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+nearshore+hydrodynamics&rft.pages=370&rft.pub=World+Scientific&rft.date=2006&rft.isbn=9789812561428&rft.aulast=Svendsen&rft.aufirst=I.A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Toba-8">^ <a href="#cite_ref-Toba_8-0">a</a> <a href="#cite_ref-Toba_8-1">b</a> <a href="#cite_ref-Toba_8-2">c</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFToba2003" class="citation cs2">Toba, Yoshiaki (2003), Ocean–atmosphere interactions, Springer, pp. 27–31, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781402011719" title="Special:BookSources/9781402011719"><bdi>9781402011719</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Ocean%E2%80%93atmosphere+interactions&rft.pages=27-31&rft.pub=Springer&rft.date=2003&rft.isbn=9781402011719&rft.aulast=Toba&rft.aufirst=Yoshiaki&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Whitham_13.13-9">^ <a href="#cite_ref-Whitham_13.13_9-0">a</a> <a href="#cite_ref-Whitham_13.13_9-1">b</a> <a href="#cite_ref-Whitham_13.13_9-2">c</a> <a href="#cite_ref-Whitham_13.13_9-3">d</a> <a href="#CITEREFWhitham1974">Whitham (1974</a>, pp. 471–476, §13.13) </li> <li id="cite_note-Hedges-10">^ <a href="#cite_ref-Hedges_10-0">a</a> <a href="#cite_ref-Hedges_10-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHedges1995" class="citation cs2">Hedges, T.S. (1995), "Regions of validity of analytical wave theories", Proceedings of the Institution of Civil Engineers - Water Maritime and Energy, 112 (2): 111–114, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1680%2Fiwtme.1995.27656">10.1680/iwtme.1995.27656</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Institution+of+Civil+Engineers+-+Water+Maritime+and+Energy&rft.atitle=Regions+of+validity+of+analytical+wave+theories&rft.volume=112&rft.issue=2&rft.pages=111-114&rft.date=1995&rft_id=info%3Adoi%2F10.1680%2Fiwtme.1995.27656&rft.aulast=Hedges&rft.aufirst=T.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Fenton-11">^ <a href="#cite_ref-Fenton_11-0">a</a> <a href="#cite_ref-Fenton_11-1">b</a> <a href="#cite_ref-Fenton_11-2">c</a> <a href="#cite_ref-Fenton_11-3">d</a> <a href="#cite_ref-Fenton_11-4">e</a> <a href="#cite_ref-Fenton_11-5">f</a> <a href="#CITEREFFenton1990">Fenton (1990)</a> </li> <li id="cite_note-12"><a href="#cite_ref-12">^</a> <a href="#CITEREFStokes1847">Stokes (1847)</a> </li> <li id="cite_note-Le_Méhauté-13"><a href="#cite_ref-Le_Méhauté_13-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLe_Méhauté1976" class="citation cs2">Le Méhauté, B. (1976), An introduction to hydrodynamics and water waves, Springer, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0387072326" title="Special:BookSources/978-0387072326"><bdi>978-0387072326</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=An+introduction+to+hydrodynamics+and+water+waves&rft.pub=Springer&rft.date=1976&rft.isbn=978-0387072326&rft.aulast=Le+M%C3%A9haut%C3%A9&rft.aufirst=B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-14"><a href="#cite_ref-14">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLonguet-HigginsFenton1974" class="citation cs2"><a href="/wiki/Michael_Longuet-Higgins" class="mw-redirect" title="Michael Longuet-Higgins">Longuet-Higgins, M.S.</a>; Fenton, J.D. (1974), "On the mass, momentum, energy and circulation of a solitary wave. II", <a href="/wiki/Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>, 340 (1623): 471–493, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974RSPSA.340..471L">1974RSPSA.340..471L</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1974.0166">10.1098/rspa.1974.0166</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:124253945">124253945</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A&rft.atitle=On+the+mass%2C+momentum%2C+energy+and+circulation+of+a+solitary+wave.+II&rft.volume=340&rft.issue=1623&rft.pages=471-493&rft.date=1974&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A124253945%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1974.0166&rft_id=info%3Abibcode%2F1974RSPSA.340..471L&rft.aulast=Longuet-Higgins&rft.aufirst=M.S.&rft.au=Fenton%2C+J.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-15"><a href="#cite_ref-15">^</a> <a href="#CITEREFWilton1914">Wilton (1914)</a> </li> <li id="cite_note-16"><a href="#cite_ref-16">^</a> <a href="#CITEREFDe1955">De (1955)</a> </li> <li id="cite_note-17"><a href="#cite_ref-17">^</a> <a href="#CITEREFFenton1985">Fenton (1985)</a>, also (including corrections) in <a href="#CITEREFFenton1990">Fenton (1990)</a> </li> <li id="cite_note-Stokes_1880-18">^ <a href="#cite_ref-Stokes_1880_18-0">a</a> <a href="#cite_ref-Stokes_1880_18-1">b</a> <a href="#CITEREFStokes1880b">Stokes (1880b)</a> </li> <li id="cite_note-Drennan-19">^ <a href="#cite_ref-Drennan_19-0">a</a> <a href="#cite_ref-Drennan_19-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDrennanHuiTenti1992" class="citation cs2">Drennan, W.M.; Hui, W.H.; Tenti, G. (1992), "Accurate calculations of Stokes water waves of large amplitude", Zeitschrift für Angewandte Mathematik und Physik, 43 (2): 367–384, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1992ZaMP...43..367D">1992ZaMP...43..367D</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF00946637">10.1007/BF00946637</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121134205">121134205</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Zeitschrift+f%C3%BCr+Angewandte+Mathematik+und+Physik&rft.atitle=Accurate+calculations+of+Stokes+water+waves+of+large+amplitude&rft.volume=43&rft.issue=2&rft.pages=367-384&rft.date=1992&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121134205%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1007%2FBF00946637&rft_id=info%3Abibcode%2F1992ZaMP...43..367D&rft.aulast=Drennan&rft.aufirst=W.M.&rft.au=Hui%2C+W.H.&rft.au=Tenti%2C+G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Buldakov_et_al_2006-20"><a href="#cite_ref-Buldakov_et_al_2006_20-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBuldakovTaylorEatock_Taylor2006" class="citation cs2">Buldakov, E.V.; Taylor, P.H.; Eatock Taylor, R. (2006), "New asymptotic description of nonlinear water waves in Lagrangian coordinates", Journal of Fluid Mechanics, 562: 431–444, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2006JFM...562..431B">2006JFM...562..431B</a>, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.492.5377">10.1.1.492.5377</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112006001443">10.1017/S0022112006001443</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:29506471">29506471</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=New+asymptotic+description+of+nonlinear+water+waves+in+Lagrangian+coordinates&rft.volume=562&rft.pages=431-444&rft.date=2006&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.492.5377%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A29506471%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112006001443&rft_id=info%3Abibcode%2F2006JFM...562..431B&rft.aulast=Buldakov&rft.aufirst=E.V.&rft.au=Taylor%2C+P.H.&rft.au=Eatock+Taylor%2C+R.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Clamond_2007-21"><a href="#cite_ref-Clamond_2007_21-0">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFClamond2007" class="citation cs2">Clamond, D. (2007), "On the Lagrangian description of steady surface gravity waves", Journal of Fluid Mechanics, 589: 433–454, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2007JFM...589..433C">2007JFM...589..433C</a>, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.526.5643">10.1.1.526.5643</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112007007811">10.1017/S0022112007007811</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123255841">123255841</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=On+the+Lagrangian+description+of+steady+surface+gravity+waves&rft.volume=589&rft.pages=433-454&rft.date=2007&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.526.5643%23id-name%3DCiteSeerX&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123255841%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112007007811&rft_id=info%3Abibcode%2F2007JFM...589..433C&rft.aulast=Clamond&rft.aufirst=D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-22"><a href="#cite_ref-22">^</a> <a href="#CITEREFCrapper1957">Crapper (1957)</a> </li> <li id="cite_note-23"><a href="#cite_ref-23">^</a> This figure is a remake and adaptation of Figure 1 in <a href="#CITEREFSchwartzFenton1982">Schwartz & Fenton (1982)</a> </li> <li id="cite_note-Schwartz_&_Fenton-24">^ <a href="#cite_ref-Schwartz_&_Fenton_24-0">a</a> <a href="#cite_ref-Schwartz_&_Fenton_24-1">b</a> <a href="#CITEREFSchwartzFenton1982">Schwartz & Fenton (1982)</a> </li> <li id="cite_note-25"><a href="#cite_ref-25">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenjaminOlver1982" class="citation cs2"><a href="/wiki/T._Brooke_Benjamin" class="mw-redirect" title="T. Brooke Benjamin">Benjamin, T.B.</a>; <a href="/wiki/Peter_J._Olver" title="Peter J. Olver">Olver, P.J.</a> (1982), "Hamiltonian structure, symmetries and conservation laws for water waves", Journal of Fluid Mechanics, 125: 137–185, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982JFM...125..137B">1982JFM...125..137B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112082003292">10.1017/S0022112082003292</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:11744174">11744174</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Hamiltonian+structure%2C+symmetries+and+conservation+laws+for+water+waves&rft.volume=125&rft.pages=137-185&rft.date=1982&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A11744174%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112082003292&rft_id=info%3Abibcode%2F1982JFM...125..137B&rft.aulast=Benjamin&rft.aufirst=T.B.&rft.au=Olver%2C+P.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-26"><a href="#cite_ref-26">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFHammackHendersonSegur2005" class="citation cs2">Hammack, J.L.; <a href="/wiki/Diane_Henderson" title="Diane Henderson">Henderson, D.M.</a>; Segur, H. (2005), "Progressive waves with persistent two-dimensional surface patterns in deep water", Journal of Fluid Mechanics, 532: 1–52, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005JFM...532....1H">2005JFM...532....1H</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112005003733">10.1017/S0022112005003733</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:53416586">53416586</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Progressive+waves+with+persistent+two-dimensional+surface+patterns+in+deep+water&rft.volume=532&rft.pages=1-52&rft.date=2005&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A53416586%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112005003733&rft_id=info%3Abibcode%2F2005JFM...532....1H&rft.aulast=Hammack&rft.aufirst=J.L.&rft.au=Henderson%2C+D.M.&rft.au=Segur%2C+H.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-27"><a href="#cite_ref-27">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCraigNicholls2002" class="citation cs2">Craig, W.; Nicholls, D.P. (2002), "Traveling gravity water waves in two and three dimensions", European Journal of Mechanics B, 21 (6): 615–641, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2002EJMF...21..615C">2002EJMF...21..615C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0997-7546%2802%2901207-4">10.1016/S0997-7546(02)01207-4</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=European+Journal+of+Mechanics+B&rft.atitle=Traveling+gravity+water+waves+in+two+and+three+dimensions&rft.volume=21&rft.issue=6&rft.pages=615-641&rft.date=2002&rft_id=info%3Adoi%2F10.1016%2FS0997-7546%2802%2901207-4&rft_id=info%3Abibcode%2F2002EJMF...21..615C&rft.aulast=Craig&rft.aufirst=W.&rft.au=Nicholls%2C+D.P.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-28"><a href="#cite_ref-28">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDebnath2005" class="citation cs2">Debnath, L. (2005), Nonlinear partial differential equations for scientists and engineers, Birkhäuser, pp. 181 & 418–419, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780817643232" title="Special:BookSources/9780817643232"><bdi>9780817643232</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Nonlinear+partial+differential+equations+for+scientists+and+engineers&rft.pages=181+%26+418-419&rft.pub=Birkh%C3%A4user&rft.date=2005&rft.isbn=9780817643232&rft.aulast=Debnath&rft.aufirst=L.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-29"><a href="#cite_ref-29">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDyachenkoLushnikovKorotkevich2016" class="citation cs2">Dyachenko, S.A.; Lushnikov, P.M.; Korotkevich, A.O. (2016), <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/abs/10.1111/sapm.12128">"Branch Cuts of Stokes Wave on Deep Water. Part I: Numerical Solution and Padé Approximation"</a>, Studies in Applied Mathematics, 137 (4): 419–472, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/1507.02784">1507.02784</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fsapm.12128">10.1111/sapm.12128</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:52104285">52104285</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studies+in+Applied+Mathematics&rft.atitle=Branch+Cuts+of+Stokes+Wave+on+Deep+Water.+Part+I%3A+Numerical+Solution+and+Pad%C3%A9+Approximation&rft.volume=137&rft.issue=4&rft.pages=419-472&rft.date=2016&rft_id=info%3Aarxiv%2F1507.02784&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A52104285%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1111%2Fsapm.12128&rft.aulast=Dyachenko&rft.aufirst=S.A.&rft.au=Lushnikov%2C+P.M.&rft.au=Korotkevich%2C+A.O.&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2Fabs%2F10.1111%2Fsapm.12128&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-30"><a href="#cite_ref-30">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMichell1893" class="citation cs2"><a href="/wiki/John_Henry_Michell" title="John Henry Michell">Michell, J.H.</a> (1893), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1431203">"The highest waves in water"</a>, Philosophical Magazine, Series 5, 36 (222): 430–437, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786449308620499">10.1080/14786449308620499</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=The+highest+waves+in+water&rft.volume=36&rft.issue=222&rft.pages=430-437&rft.date=1893&rft_id=info%3Adoi%2F10.1080%2F14786449308620499&rft.aulast=Michell&rft.aufirst=J.H.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1431203&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-31"><a href="#cite_ref-31">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGrant1973" class="citation cs2">Grant, Malcolm A. (1973), "The singularity at the crest of a finite amplitude progressive Stokes wave", Journal of Fluid Mechanics, 59 (2): 257–262, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1973JFM....59..257G">1973JFM....59..257G</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112073001552">10.1017/S0022112073001552</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119356016">119356016</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=The+singularity+at+the+crest+of+a+finite+amplitude+progressive+Stokes+wave&rft.volume=59&rft.issue=2&rft.pages=257-262&rft.date=1973&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119356016%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112073001552&rft_id=info%3Abibcode%2F1973JFM....59..257G&rft.aulast=Grant&rft.aufirst=Malcolm+A.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-32"><a href="#cite_ref-32">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFToland1978" class="citation cs2"><a href="/wiki/John_Toland_(mathematician)" title="John Toland (mathematician)">Toland, J.F.</a> (1978), "On the existence of a wave of greatest height and Stokes's conjecture", <a href="/wiki/Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>, 363 (1715): 469–485, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978RSPSA.363..469T">1978RSPSA.363..469T</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1978.0178">10.1098/rspa.1978.0178</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120444295">120444295</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A&rft.atitle=On+the+existence+of+a+wave+of+greatest+height+and+Stokes%27s+conjecture&rft.volume=363&rft.issue=1715&rft.pages=469-485&rft.date=1978&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120444295%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1978.0178&rft_id=info%3Abibcode%2F1978RSPSA.363..469T&rft.aulast=Toland&rft.aufirst=J.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-33"><a href="#cite_ref-33">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlotnikov1982" class="citation cs2"><a href="/w/index.php?title=Pavel_I._Plotnikov_(mathematician)&action=edit&redlink=1" class="new" title="Pavel I. Plotnikov (mathematician) (page does not exist)">Plotnikov, P.I.</a> (1982), "A proof of the Stokes conjecture in the theory of surface waves.", Dinamika Splosh. Sredy [in Russian], 57: 41–76</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Dinamika+Splosh.+Sredy+%5Bin+Russian%5D&rft.atitle=A+proof+of+the+Stokes+conjecture+in+the+theory+of+surface+waves.&rft.volume=57&rft.pages=41-76&rft.date=1982&rft.aulast=Plotnikov&rft.aufirst=P.I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> <dl><dd>Reprinted in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPlotnikov2002" class="citation cs2">Plotnikov, P.I. (2002), "A proof of the Stokes conjecture in the theory of surface waves.", <a href="/wiki/Studies_in_Applied_Mathematics" title="Studies in Applied Mathematics">Studies in Applied Mathematics</a>, 3 (2): 217–244, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2F1467-9590.01408">10.1111/1467-9590.01408</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studies+in+Applied+Mathematics&rft.atitle=A+proof+of+the+Stokes+conjecture+in+the+theory+of+surface+waves.&rft.volume=3&rft.issue=2&rft.pages=217-244&rft.date=2002&rft_id=info%3Adoi%2F10.1111%2F1467-9590.01408&rft.aulast=Plotnikov&rft.aufirst=P.I.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></dd></dl> </li> <li id="cite_note-34"><a href="#cite_ref-34">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAmickFraenkelToland1982" class="citation cs2">Amick, C.J.; Fraenkel, L.E.; Toland, J.F. (1982), "On the Stokes conjecture for the wave of extreme form", <a href="/wiki/Acta_Mathematica" title="Acta Mathematica">Acta Mathematica</a>, 148: 193–214, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF02392728">10.1007/BF02392728</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Acta+Mathematica&rft.atitle=On+the+Stokes+conjecture+for+the+wave+of+extreme+form&rft.volume=148&rft.pages=193-214&rft.date=1982&rft_id=info%3Adoi%2F10.1007%2FBF02392728&rft.aulast=Amick&rft.aufirst=C.J.&rft.au=Fraenkel%2C+L.E.&rft.au=Toland%2C+J.F.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Rainey_&_Longuet-Higgins-35">^ <a href="#cite_ref-Rainey_&_Longuet-Higgins_35-0">a</a> <a href="#cite_ref-Rainey_&_Longuet-Higgins_35-1">b</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRaineyLonguet-Higgins2006" class="citation cs2">Rainey, R.C.T.; <a href="/wiki/Michael_S._Longuet-Higgins" title="Michael S. Longuet-Higgins">Longuet-Higgins, M.S.</a> (2006), "A close one-term approximation to the highest Stokes wave on deep water", Ocean Engineering, 33 (14–15): 2012–2024, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.oceaneng.2005.09.014">10.1016/j.oceaneng.2005.09.014</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ocean+Engineering&rft.atitle=A+close+one-term+approximation+to+the+highest+Stokes+wave+on+deep+water&rft.volume=33&rft.issue=14%E2%80%9315&rft.pages=2012-2024&rft.date=2006&rft_id=info%3Adoi%2F10.1016%2Fj.oceaneng.2005.09.014&rft.aulast=Rainey&rft.aufirst=R.C.T.&rft.au=Longuet-Higgins%2C+M.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-36"><a href="#cite_ref-36">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLonguet‐Higgins1979" class="citation cs2"><a href="/wiki/Michael_S._Longuet-Higgins" title="Michael S. Longuet-Higgins">Longuet‐Higgins, M.S.</a> (1979), "Why is a water wave like a grandfather clock?", Physics of Fluids, 22 (9): 1828–1829, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1979PhFl...22.1828L">1979PhFl...22.1828L</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1063%2F1.862789">10.1063/1.862789</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physics+of+Fluids&rft.atitle=Why+is+a+water+wave+like+a+grandfather+clock%3F&rft.volume=22&rft.issue=9&rft.pages=1828-1829&rft.date=1979&rft_id=info%3Adoi%2F10.1063%2F1.862789&rft_id=info%3Abibcode%2F1979PhFl...22.1828L&rft.aulast=Longuet%E2%80%90Higgins&rft.aufirst=M.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-37"><a href="#cite_ref-37">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDyachenkoKorotkevichLushnikovSemenova2013–2022" class="citation cs2">Dyachenko, S.A.; Korotkevich, A.O.; Lushnikov, P.M.; Semenova, A.A.; Silantyev, D.A. (2013–2022), <a rel="nofollow" class="external text" href="http://stokeswave.org/">StokesWave.org</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=StokesWave.org&rft.date=2013%2F2022&rft.aulast=Dyachenko&rft.aufirst=S.A.&rft.au=Korotkevich%2C+A.O.&rft.au=Lushnikov%2C+P.M.&rft.au=Semenova%2C+A.A.&rft.au=Silantyev%2C+D.A.&rft_id=http%3A%2F%2Fstokeswave.org%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Craik_WIFF-38">^ <a href="#cite_ref-Craik_WIFF_38-0">a</a> <a href="#cite_ref-Craik_WIFF_38-1">b</a> For a review of the instability of Stokes waves see e.g.: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCraik1988" class="citation cs2">Craik, A.D.D. (1988), Wave interactions and fluid flows, Cambridge University Press, pp. 199–219, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-36829-2" title="Special:BookSources/978-0-521-36829-2"><bdi>978-0-521-36829-2</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Wave+interactions+and+fluid+flows&rft.pages=199-219&rft.pub=Cambridge+University+Press&rft.date=1988&rft.isbn=978-0-521-36829-2&rft.aulast=Craik&rft.aufirst=A.D.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-39"><a href="#cite_ref-39">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenjaminFeir1967" class="citation cs2"><a href="/wiki/T._Brooke_Benjamin" class="mw-redirect" title="T. Brooke Benjamin">Benjamin, T. Brooke</a>; Feir, J.E. (1967), "The disintegration of wave trains on deep water. Part 1. Theory", Journal of Fluid Mechanics, 27 (3): 417–430, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967JFM....27..417B">1967JFM....27..417B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS002211206700045X">10.1017/S002211206700045X</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121996479">121996479</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=The+disintegration+of+wave+trains+on+deep+water.+Part+1.+Theory&rft.volume=27&rft.issue=3&rft.pages=417-430&rft.date=1967&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121996479%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS002211206700045X&rft_id=info%3Abibcode%2F1967JFM....27..417B&rft.aulast=Benjamin&rft.aufirst=T.+Brooke&rft.au=Feir%2C+J.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-40"><a href="#cite_ref-40">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFZakharovOstrovsky2009" class="citation journal cs1"><a href="/wiki/Vladimir_E._Zakharov" title="Vladimir E. Zakharov">Zakharov, V.E.</a>; Ostrovsky, L.A. (2009). <a rel="nofollow" class="external text" href="http://people.math.umass.edu/~kevrekid/math697wa/sdarticle_ZO.pdf">"Modulation instability: The beginning"</a> (PDF). Physica D. 238 (5): 540–548. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2009PhyD..238..540Z">2009PhyD..238..540Z</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2Fj.physd.2008.12.002">10.1016/j.physd.2008.12.002</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physica+D&rft.atitle=Modulation+instability%3A+The+beginning&rft.volume=238&rft.issue=5&rft.pages=540-548&rft.date=2009&rft_id=info%3Adoi%2F10.1016%2Fj.physd.2008.12.002&rft_id=info%3Abibcode%2F2009PhyD..238..540Z&rft.aulast=Zakharov&rft.aufirst=V.E.&rft.au=Ostrovsky%2C+L.A.&rft_id=http%3A%2F%2Fpeople.math.umass.edu%2F~kevrekid%2Fmath697wa%2Fsdarticle_ZO.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-41"><a href="#cite_ref-41">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBenjamin1967" class="citation cs2"><a href="/wiki/T._Brooke_Benjamin" class="mw-redirect" title="T. Brooke Benjamin">Benjamin, T.B.</a> (1967), "Instability of periodic wavetrains in nonlinear dispersive systems", <a href="/wiki/Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>, 299 (1456): 59–76, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1967RSPSA.299...59B">1967RSPSA.299...59B</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1967.0123">10.1098/rspa.1967.0123</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121661209">121661209</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A&rft.atitle=Instability+of+periodic+wavetrains+in+nonlinear+dispersive+systems&rft.volume=299&rft.issue=1456&rft.pages=59-76&rft.date=1967&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121661209%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1967.0123&rft_id=info%3Abibcode%2F1967RSPSA.299...59B&rft.aulast=Benjamin&rft.aufirst=T.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> Concluded with a discussion by <a href="/wiki/Klaus_Hasselmann" title="Klaus Hasselmann">Klaus Hasselmann</a>. </li> <li id="cite_note-42"><a href="#cite_ref-42">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLakeYuenRungaldierFerguson1977" class="citation cs2">Lake, B.M.; Yuen, H.C.; Rungaldier, H.; Ferguson, W.E. (1977), "Nonlinear deep-water waves: theory and experiment. Part 2. Evolution of a continuous wave train", Journal of Fluid Mechanics, 83 (1): 49–74, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1977JFM....83...49L">1977JFM....83...49L</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112077001037">10.1017/S0022112077001037</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123014293">123014293</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Nonlinear+deep-water+waves%3A+theory+and+experiment.+Part+2.+Evolution+of+a+continuous+wave+train&rft.volume=83&rft.issue=1&rft.pages=49-74&rft.date=1977&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123014293%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112077001037&rft_id=info%3Abibcode%2F1977JFM....83...49L&rft.aulast=Lake&rft.aufirst=B.M.&rft.au=Yuen%2C+H.C.&rft.au=Rungaldier%2C+H.&rft.au=Ferguson%2C+W.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-43"><a href="#cite_ref-43">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLonguet-Higgins1978" class="citation cs2"><a href="/wiki/Michael_Longuet-Higgins" class="mw-redirect" title="Michael Longuet-Higgins">Longuet-Higgins, M.S.</a> (1978), "The instabilities of gravity waves of finite amplitude in deep water. I. Superharmonics", <a href="/wiki/Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>, 360 (1703): 471–488, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978RSPSA.360..471L">1978RSPSA.360..471L</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1978.0080">10.1098/rspa.1978.0080</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:202575377">202575377</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A&rft.atitle=The+instabilities+of+gravity+waves+of+finite+amplitude+in+deep+water.+I.+Superharmonics&rft.volume=360&rft.issue=1703&rft.pages=471-488&rft.date=1978&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A202575377%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1978.0080&rft_id=info%3Abibcode%2F1978RSPSA.360..471L&rft.aulast=Longuet-Higgins&rft.aufirst=M.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-44"><a href="#cite_ref-44">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLonguet-Higgins1978" class="citation cs2"><a href="/wiki/Michael_Longuet-Higgins" class="mw-redirect" title="Michael Longuet-Higgins">Longuet-Higgins, M.S.</a> (1978), "The instabilities of gravity waves of finite amplitude in deep water. II. Subharmonics", <a href="/wiki/Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>, 360 (1703): 489–505, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1978RSPSA.360..471L">1978RSPSA.360..471L</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1978.0080">10.1098/rspa.1978.0080</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:202575377">202575377</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A&rft.atitle=The+instabilities+of+gravity+waves+of+finite+amplitude+in+deep+water.+II.+Subharmonics&rft.volume=360&rft.issue=1703&rft.pages=489-505&rft.date=1978&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A202575377%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1978.0080&rft_id=info%3Abibcode%2F1978RSPSA.360..471L&rft.aulast=Longuet-Higgins&rft.aufirst=M.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-45"><a href="#cite_ref-45">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKorotkevichLushnikovSemenovaDyachenko2022" class="citation cs2">Korotkevich, A.O.; Lushnikov, P.M.; Semenova, A.; Dyachenko, S.A. (2022), <a rel="nofollow" class="external text" href="https://onlinelibrary.wiley.com/doi/abs/10.1111/sapm.12535">"Superharmonic instability of stokes waves"</a>, Studies in Applied Mathematics, 150: 119–134, <a href="/wiki/ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/2206.00725">2206.00725</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1111%2Fsapm.12535">10.1111/sapm.12535</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:249282423">249282423</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Studies+in+Applied+Mathematics&rft.atitle=Superharmonic+instability+of+stokes+waves&rft.volume=150&rft.pages=119-134&rft.date=2022&rft_id=info%3Aarxiv%2F2206.00725&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A249282423%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1111%2Fsapm.12535&rft.aulast=Korotkevich&rft.aufirst=A.O.&rft.au=Lushnikov%2C+P.M.&rft.au=Semenova%2C+A.&rft.au=Dyachenko%2C+S.A.&rft_id=https%3A%2F%2Fonlinelibrary.wiley.com%2Fdoi%2Fabs%2F10.1111%2Fsapm.12535&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-46"><a href="#cite_ref-46">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcLeanMaMartinSaffman1981" class="citation cs2">McLean, J.W.; Ma, Y.C.; Martin, D.U.; <a href="/wiki/Philip_Saffman" title="Philip Saffman">Saffman, P.G.</a>; Yuen, H.C. (1981), <a rel="nofollow" class="external text" href="http://authors.library.caltech.edu/10155/1/MCLprl81.pdf">"Three-dimensional instability of finite-amplitude water waves"</a> (PDF), Physical Review Letters, 46 (13): 817–820, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981PhRvL..46..817M">1981PhRvL..46..817M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1103%2FPhysRevLett.46.817">10.1103/PhysRevLett.46.817</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Physical+Review+Letters&rft.atitle=Three-dimensional+instability+of+finite-amplitude+water+waves&rft.volume=46&rft.issue=13&rft.pages=817-820&rft.date=1981&rft_id=info%3Adoi%2F10.1103%2FPhysRevLett.46.817&rft_id=info%3Abibcode%2F1981PhRvL..46..817M&rft.aulast=McLean&rft.aufirst=J.W.&rft.au=Ma%2C+Y.C.&rft.au=Martin%2C+D.U.&rft.au=Saffman%2C+P.G.&rft.au=Yuen%2C+H.C.&rft_id=http%3A%2F%2Fauthors.library.caltech.edu%2F10155%2F1%2FMCLprl81.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-47"><a href="#cite_ref-47">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcLean1982" class="citation cs2">McLean, J.W. (1982), "Instabilities of finite-amplitude water waves", Journal of Fluid Mechanics, 114: 315–330, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982JFM...114..315M">1982JFM...114..315M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112082000172">10.1017/S0022112082000172</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122511104">122511104</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Instabilities+of+finite-amplitude+water+waves&rft.volume=114&rft.pages=315-330&rft.date=1982&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122511104%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112082000172&rft_id=info%3Abibcode%2F1982JFM...114..315M&rft.aulast=McLean&rft.aufirst=J.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Dias_Kharif-48">^ <a href="#cite_ref-Dias_Kharif_48-0">a</a> <a href="#cite_ref-Dias_Kharif_48-1">b</a> <a href="#CITEREFDiasKharif1999">Dias & Kharif (1999)</a> </li> <li id="cite_note-Phillips-49">^ <a href="#cite_ref-Phillips_49-0">a</a> <a href="#cite_ref-Phillips_49-1">b</a> <a href="#cite_ref-Phillips_49-2">c</a> <a href="#cite_ref-Phillips_49-3">d</a> <a href="#cite_ref-Phillips_49-4">e</a> <a href="#cite_ref-Phillips_49-5">f</a> <a href="#cite_ref-Phillips_49-6">g</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPhillips1980" class="citation cs2"><a href="/wiki/Owen_Martin_Phillips" title="Owen Martin Phillips">Phillips, O.M.</a> (1980), Dynamics of the upper ocean (2nd ed.), Cambridge University Press, pp. 33–37, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-521-29801-8" title="Special:BookSources/978-0-521-29801-8"><bdi>978-0-521-29801-8</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Dynamics+of+the+upper+ocean&rft.pages=33-37&rft.edition=2nd&rft.pub=Cambridge+University+Press&rft.date=1980&rft.isbn=978-0-521-29801-8&rft.aulast=Phillips&rft.aufirst=O.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Mei_4_6-50">^ <a href="#cite_ref-Mei_4_6_50-0">a</a> <a href="#cite_ref-Mei_4_6_50-1">b</a> <a href="#CITEREFMei1989">Mei (1989</a>, pp. 4–6) </li> <li id="cite_note-51"><a href="#cite_ref-51">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLonguet-Higgins1962" class="citation cs2"><a href="/wiki/Michael_Longuet-Higgins" class="mw-redirect" title="Michael Longuet-Higgins">Longuet-Higgins, M.S.</a> (1962), "Resonant interactions between two trains of gravity waves", Journal of Fluid Mechanics, 12 (3): 321–332, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1962JFM....12..321L">1962JFM....12..321L</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112062000233">10.1017/S0022112062000233</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122810532">122810532</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Resonant+interactions+between+two+trains+of+gravity+waves&rft.volume=12&rft.issue=3&rft.pages=321-332&rft.date=1962&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122810532%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112062000233&rft_id=info%3Abibcode%2F1962JFM....12..321L&rft.aulast=Longuet-Higgins&rft.aufirst=M.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-Mei_607_608-52">^ <a href="#cite_ref-Mei_607_608_52-0">a</a> <a href="#cite_ref-Mei_607_608_52-1">b</a> <a href="#cite_ref-Mei_607_608_52-2">c</a> <a href="#CITEREFMei1989">Mei (1989</a>, pp. 607–608) </li> <li id="cite_note-53"><a href="#cite_ref-53">^</a> By <a href="/wiki/Non-dimensionalization" class="mw-redirect" title="Non-dimensionalization">non-dimensionalization</a> of the flow equations and boundary conditions, different regimes may be identified, depending on the scaling of the coordinates and flow quantities. In deep(er) water, the characteristic <a href="/wiki/Wavelength" title="Wavelength">wavelength</a> is the only length scale available. So, the horizontal and vertical coordinates are all non-dimensionalized with the wavelength. This leads to Stokes wave theory. However, in shallow water, the water depth is the appropriate characteristic scale to make the vertical coordinate non-dimensional, while the horizontal coordinates are scaled with the wavelength – resulting in the <a href="/wiki/Boussinesq_approximation_(water_waves)" title="Boussinesq approximation (water waves)">Boussinesq approximation</a>. For a discussion, see: <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeji1995" class="citation cs2">Beji, S. (1995), "Note on a nonlinearity parameter of surface waves", Coastal Engineering, 25 (1–2): 81–85, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0378-3839%2894%2900031-R">10.1016/0378-3839(94)00031-R</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Coastal+Engineering&rft.atitle=Note+on+a+nonlinearity+parameter+of+surface+waves&rft.volume=25&rft.issue=1%E2%80%932&rft.pages=81-85&rft.date=1995&rft_id=info%3Adoi%2F10.1016%2F0378-3839%2894%2900031-R&rft.aulast=Beji&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988">;</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFKirby1998" class="citation cs2">Kirby, J.T. (1998), "Discussion of 'Note on a nonlinearity parameter of surface waves' by S. Beji", Coastal Engineering, 34 (1–2): 163–168, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0378-3839%2898%2900024-6">10.1016/S0378-3839(98)00024-6</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Coastal+Engineering&rft.atitle=Discussion+of+%27Note+on+a+nonlinearity+parameter+of+surface+waves%27+by+S.+Beji&rft.volume=34&rft.issue=1%E2%80%932&rft.pages=163-168&rft.date=1998&rft_id=info%3Adoi%2F10.1016%2FS0378-3839%2898%2900024-6&rft.aulast=Kirby&rft.aufirst=J.T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> and</li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFBeji1998" class="citation cs2">Beji, S. (1998), "Author's closure to J.T. Kirby's discussion 'Note on a nonlinearity parameter of surface waves'", Coastal Engineering, 34 (1–2): 169–171, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2FS0378-3839%2898%2900018-0">10.1016/S0378-3839(98)00018-0</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Coastal+Engineering&rft.atitle=Author%27s+closure+to+J.T.+Kirby%27s+discussion+%27Note+on+a+nonlinearity+parameter+of+surface+waves%27&rft.volume=34&rft.issue=1%E2%80%932&rft.pages=169-171&rft.date=1998&rft_id=info%3Adoi%2F10.1016%2FS0378-3839%2898%2900018-0&rft.aulast=Beji&rft.aufirst=S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li></ul> </li> <li id="cite_note-54"><a href="#cite_ref-54">^</a> The wave physics are computed with the Rienecker & Fenton (R&F) <a href="/wiki/Streamfunction" class="mw-redirect" title="Streamfunction">streamfunction</a> theory. For a computer code to compute these see: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFenton1988" class="citation cs2">Fenton, J.D. (1988), "The numerical solution of steady water wave problems", Computers & Geosciences, 14 (3): 357–368, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1988CG.....14..357F">1988CG.....14..357F</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1016%2F0098-3004%2888%2990066-0">10.1016/0098-3004(88)90066-0</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Computers+%26+Geosciences&rft.atitle=The+numerical+solution+of+steady+water+wave+problems&rft.volume=14&rft.issue=3&rft.pages=357-368&rft.date=1988&rft_id=info%3Adoi%2F10.1016%2F0098-3004%2888%2990066-0&rft_id=info%3Abibcode%2F1988CG.....14..357F&rft.aulast=Fenton&rft.aufirst=J.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> The animations are made from the R&F results with a series of <a href="/wiki/MATLAB" title="MATLAB">Matlab</a> scripts and <a href="/wiki/Shell_script" title="Shell script">shell scripts</a>. </li> <li id="cite_note-Wehausen_Laitone-55"><a href="#cite_ref-Wehausen_Laitone_55-0">^</a> <a href="#CITEREFWehausenLaitone1960">Wehausen & Laitone (1960</a>, pp. 653–667, §27) </li> <li id="cite_note-Whitham_16.6-56">^ <a href="#cite_ref-Whitham_16.6_56-0">a</a> <a href="#cite_ref-Whitham_16.6_56-1">b</a> <a href="#cite_ref-Whitham_16.6_56-2">c</a> <a href="#cite_ref-Whitham_16.6_56-3">d</a> <a href="#CITEREFWhitham1974">Whitham (1974</a>, pp. 553–556, §16.6) </li> <li id="cite_note-57"><a href="#cite_ref-57">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSarpkayaIsaacson1981" class="citation cs2">Sarpkaya, Turgut; Isaacson, Michael (1981), Mechanics of wave forces on offshore structures, Van Nostrand Reinhold, p. 183, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780442254025" title="Special:BookSources/9780442254025"><bdi>9780442254025</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Mechanics+of+wave+forces+on+offshore+structures&rft.pages=183&rft.pub=Van+Nostrand+Reinhold&rft.date=1981&rft.isbn=9780442254025&rft.aulast=Sarpkaya&rft.aufirst=Turgut&rft.au=Isaacson%2C+Michael&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> <li id="cite_note-58"><a href="#cite_ref-58">^</a> <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMcIntyre1981" class="citation cs2">McIntyre, M.E. (1981), "On the 'wave momentum' myth", Journal of Fluid Mechanics, 106: 331–347, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981JFM...106..331M">1981JFM...106..331M</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112081001626">10.1017/S0022112081001626</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:18232994">18232994</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=On+the+%27wave+momentum%27+myth&rft.volume=106&rft.pages=331-347&rft.date=1981&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A18232994%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112081001626&rft_id=info%3Abibcode%2F1981JFM...106..331M&rft.aulast=McIntyre&rft.aufirst=M.E.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="References">References</h2>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=21" title="Edit section: References">edit</a>]</div> <style data-mw-deduplicate="TemplateStyles:r1239549316">.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}</style><div class="refbegin" style=""> <div class="mw-heading mw-heading3"><h3 id="By_Sir_George_Gabriel_Stokes">By Sir George Gabriel Stokes</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=22" title="Edit section: By Sir George Gabriel Stokes">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStokes1847" class="citation cs2"><a href="/wiki/George_Gabriel_Stokes" class="mw-redirect" title="George Gabriel Stokes">Stokes, G.G.</a> (1847), "On the theory of oscillatory waves", Transactions of the Cambridge Philosophical Society, 8: 441–455.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Transactions+of+the+Cambridge+Philosophical+Society&rft.atitle=On+the+theory+of+oscillatory+waves&rft.volume=8&rft.pages=441-455&rft.date=1847&rft.aulast=Stokes&rft.aufirst=G.G.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li></ul> <dl><dd>Reprinted in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStokes1880a" class="citation cs2">Stokes, G.G. (1880a), "On the theory of oscillatory waves", <a rel="nofollow" class="external text" href="https://archive.org/details/mathphyspapers01stokrich">Mathematical and Physical Papers, Volume I</a>, Cambridge University Press, pp. 197–229, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781001435534" title="Special:BookSources/9781001435534"><bdi>9781001435534</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/314316422">314316422</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=On+the+theory+of+oscillatory+waves&rft.btitle=Mathematical+and+Physical+Papers%2C+Volume+I&rft.pages=197-229&rft.pub=Cambridge+University+Press&rft.date=1880&rft_id=info%3Aoclcnum%2F314316422&rft.isbn=9781001435534&rft.aulast=Stokes&rft.aufirst=G.G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathphyspapers01stokrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStokes1880b" class="citation cs2">Stokes, G.G. (1880b), "Supplement to a paper on the theory of oscillatory waves", <a rel="nofollow" class="external text" href="https://archive.org/details/mathphyspapers01stokrich">Mathematical and Physical Papers, Volume I</a>, Cambridge University Press, pp. 314–326, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9781001435534" title="Special:BookSources/9781001435534"><bdi>9781001435534</bdi></a>, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/314316422">314316422</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Supplement+to+a+paper+on+the+theory+of+oscillatory+waves&rft.btitle=Mathematical+and+Physical+Papers%2C+Volume+I&rft.pages=314-326&rft.pub=Cambridge+University+Press&rft.date=1880&rft_id=info%3Aoclcnum%2F314316422&rft.isbn=9781001435534&rft.aulast=Stokes&rft.aufirst=G.G.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fmathphyspapers01stokrich&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="Other_historical_references">Other historical references</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=23" title="Edit section: Other historical references">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCrapper1957" class="citation cs2">Crapper, G.D. (1957), "An exact solution for progressive capillary waves of arbitrary amplitude", Journal of Fluid Mechanics, 2 (6): 532–540, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1957JFM.....2..532C">1957JFM.....2..532C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112057000348">10.1017/S0022112057000348</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120377950">120377950</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=An+exact+solution+for+progressive+capillary+waves+of+arbitrary+amplitude&rft.volume=2&rft.issue=6&rft.pages=532-540&rft.date=1957&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120377950%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112057000348&rft_id=info%3Abibcode%2F1957JFM.....2..532C&rft.aulast=Crapper&rft.aufirst=G.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDe1955" class="citation cs2">De, S.C. (1955), "Contributions to the theory of Stokes waves", Mathematical Proceedings of the Cambridge Philosophical Society, 51 (4): 713–736, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1955PCPS...51..713D">1955PCPS...51..713D</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0305004100030796">10.1017/S0305004100030796</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122721263">122721263</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematical+Proceedings+of+the+Cambridge+Philosophical+Society&rft.atitle=Contributions+to+the+theory+of+Stokes+waves&rft.volume=51&rft.issue=4&rft.pages=713-736&rft.date=1955&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122721263%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0305004100030796&rft_id=info%3Abibcode%2F1955PCPS...51..713D&rft.aulast=De&rft.aufirst=S.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLevi-Civita1925" class="citation cs2"><a href="/wiki/Tullio_Levi-Civita" title="Tullio Levi-Civita">Levi-Civita, T.</a> (1925), "Détermination rigoureuse des ondes permanentes d'ampleur finie", Mathematische Annalen, 93: 264–314, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01449965">10.1007/BF01449965</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:121341503">121341503</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=D%C3%A9termination+rigoureuse+des+ondes+permanentes+d%27ampleur+finie&rft.volume=93&rft.pages=264-314&rft.date=1925&rft_id=info%3Adoi%2F10.1007%2FBF01449965&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A121341503%23id-name%3DS2CID&rft.aulast=Levi-Civita&rft.aufirst=T.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStruik1926" class="citation cs2"><a href="/wiki/Dirk_Jan_Struik" title="Dirk Jan Struik">Struik, D.J.</a> (1926), "Détermination rigoureuse des ondes irrotationelles périodiques dans un canal à profondeur finie", Mathematische Annalen, 95: 595–634, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1007%2FBF01206629">10.1007/BF01206629</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122656179">122656179</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Mathematische+Annalen&rft.atitle=D%C3%A9termination+rigoureuse+des+ondes+irrotationelles+p%C3%A9riodiques+dans+un+canal+%C3%A0+profondeur+finie&rft.volume=95&rft.pages=595-634&rft.date=1926&rft_id=info%3Adoi%2F10.1007%2FBF01206629&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122656179%23id-name%3DS2CID&rft.aulast=Struik&rft.aufirst=D.J.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLord_Rayleigh1917" class="citation cs2"><a href="/wiki/John_Strutt,_3rd_Baron_Rayleigh" class="mw-redirect" title="John Strutt, 3rd Baron Rayleigh">Lord Rayleigh</a> (1917), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1430750">"On periodic irrotational waves at the surface of deep water"</a>, Philosophical Magazine, Series 6, 33 (197): 381–389, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786440508635653">10.1080/14786440508635653</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+periodic+irrotational+waves+at+the+surface+of+deep+water&rft.volume=33&rft.issue=197&rft.pages=381-389&rft.date=1917&rft_id=info%3Adoi%2F10.1080%2F14786440508635653&rft.au=Lord+Rayleigh&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1430750&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li></ul> <dl><dd>Reprinted in: <link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFStrutt1920" class="citation cs2">Strutt, John William (Lord Rayleigh) (1920), <a rel="nofollow" class="external text" href="https://archive.org/details/scientificpapers06rayliala">Scientific Papers</a>, vol. 6, Cambridge University Press, pp. 478–485, §419, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/2316730">2316730</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Scientific+Papers&rft.pages=478-485%2C+%C2%A7419&rft.pub=Cambridge+University+Press&rft.date=1920&rft_id=info%3Aoclcnum%2F2316730&rft.aulast=Strutt&rft.aufirst=John+William+%28Lord+Rayleigh%29&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fscientificpapers06rayliala&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilton1914" class="citation cs2">Wilton, J.R. (1914), <a rel="nofollow" class="external text" href="https://zenodo.org/record/1430642">"On deep water waves"</a>, Philosophical Magazine, Series 6, 27 (158): 385–394, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1080%2F14786440208635100">10.1080/14786440208635100</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Magazine&rft.atitle=On+deep+water+waves&rft.volume=27&rft.issue=158&rft.pages=385-394&rft.date=1914&rft_id=info%3Adoi%2F10.1080%2F14786440208635100&rft.aulast=Wilton&rft.aufirst=J.R.&rft_id=https%3A%2F%2Fzenodo.org%2Frecord%2F1430642&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li></ul> <div class="mw-heading mw-heading3"><h3 id="More_recent_(since_1960)">More recent (since 1960)</h3>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=24" title="Edit section: More recent (since 1960)">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCokelet1977" class="citation cs2">Cokelet, E.D. (1977), "Steep gravity waves in water of arbitrary uniform depth", Philosophical Transactions of the Royal Society, 286 (1335): 183–230, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1977RSPTA.286..183C">1977RSPTA.286..183C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1977.0113">10.1098/rsta.1977.0113</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:119957640">119957640</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society&rft.atitle=Steep+gravity+waves+in+water+of+arbitrary+uniform+depth&rft.volume=286&rft.issue=1335&rft.pages=183-230&rft.date=1977&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A119957640%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frsta.1977.0113&rft_id=info%3Abibcode%2F1977RSPTA.286..183C&rft.aulast=Cokelet&rft.aufirst=E.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFCraik2005" class="citation cs2">Craik, A.D.D. (2005), "George Gabriel Stokes on water wave theory", Annual Review of Fluid Mechanics, 37 (1): 23–42, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2005AnRFM..37...23C">2005AnRFM..37...23C</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.fluid.37.061903.175836">10.1146/annurev.fluid.37.061903.175836</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Fluid+Mechanics&rft.atitle=George+Gabriel+Stokes+on+water+wave+theory&rft.volume=37&rft.issue=1&rft.pages=23-42&rft.date=2005&rft_id=info%3Adoi%2F10.1146%2Fannurev.fluid.37.061903.175836&rft_id=info%3Abibcode%2F2005AnRFM..37...23C&rft.aulast=Craik&rft.aufirst=A.D.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFDiasKharif1999" class="citation cs2">Dias, F.; Kharif, C. (1999), "Nonlinear gravity and capillary–gravity waves", Annual Review of Fluid Mechanics, 31: 301–346, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1999AnRFM..31..301D">1999AnRFM..31..301D</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.fluid.31.1.301">10.1146/annurev.fluid.31.1.301</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Fluid+Mechanics&rft.atitle=Nonlinear+gravity+and+capillary%E2%80%93gravity+waves&rft.volume=31&rft.pages=301-346&rft.date=1999&rft_id=info%3Adoi%2F10.1146%2Fannurev.fluid.31.1.301&rft_id=info%3Abibcode%2F1999AnRFM..31..301D&rft.aulast=Dias&rft.aufirst=F.&rft.au=Kharif%2C+C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFenton1985" class="citation cs2">Fenton, J.D. (1985), "A fifth-order Stokes theory for steady waves", Journal of Waterway, Port, Coastal, and Ocean Engineering, 111 (2): 216–234, <a href="/wiki/CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.461.6157">10.1.1.461.6157</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1061%2F%28ASCE%290733-950X%281985%29111%3A2%28216%29">10.1061/(ASCE)0733-950X(1985)111:2(216)</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Waterway%2C+Port%2C+Coastal%2C+and+Ocean+Engineering&rft.atitle=A+fifth-order+Stokes+theory+for+steady+waves&rft.volume=111&rft.issue=2&rft.pages=216-234&rft.date=1985&rft_id=https%3A%2F%2Fciteseerx.ist.psu.edu%2Fviewdoc%2Fsummary%3Fdoi%3D10.1.1.461.6157%23id-name%3DCiteSeerX&rft_id=info%3Adoi%2F10.1061%2F%28ASCE%290733-950X%281985%29111%3A2%28216%29&rft.aulast=Fenton&rft.aufirst=J.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> </li></ul> <dl><dd>And in (including corrections):</dd> <dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFenton1990" class="citation cs2">Fenton, J.D. (1990), "Nonlinear wave theories", in LeMéhauté, B.; Hanes, D.M. (eds.), <a rel="nofollow" class="external text" href="http://www.johndfenton.com/Papers/Fenton90b-Nonlinear-wave-theories.pdf">Ocean Engineering Science</a> (PDF), The Sea, vol. 9A, Wiley Interscience, pp. 3–25, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9780674017399" title="Special:BookSources/9780674017399"><bdi>9780674017399</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=bookitem&rft.atitle=Nonlinear+wave+theories&rft.btitle=Ocean+Engineering+Science&rft.series=The+Sea&rft.pages=3-25&rft.pub=Wiley+Interscience&rft.date=1990&rft.isbn=9780674017399&rft.aulast=Fenton&rft.aufirst=J.D.&rft_id=http%3A%2F%2Fwww.johndfenton.com%2FPapers%2FFenton90b-Nonlinear-wave-theories.pdf&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></dd></dl> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFLonguet-Higgins1975" class="citation cs2"><a href="/wiki/Michael_Longuet-Higgins" class="mw-redirect" title="Michael Longuet-Higgins">Longuet-Higgins, M.S.</a> (1975), "Integral properties of periodic gravity waves of finite amplitude", <a href="/wiki/Proceedings_of_the_Royal_Society_A" class="mw-redirect" title="Proceedings of the Royal Society A">Proceedings of the Royal Society A</a>, 342 (1629): 157–174, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1975RSPSA.342..157L">1975RSPSA.342..157L</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frspa.1975.0018">10.1098/rspa.1975.0018</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:123723040">123723040</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Proceedings+of+the+Royal+Society+A&rft.atitle=Integral+properties+of+periodic+gravity+waves+of+finite+amplitude&rft.volume=342&rft.issue=1629&rft.pages=157-174&rft.date=1975&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A123723040%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frspa.1975.0018&rft_id=info%3Abibcode%2F1975RSPSA.342..157L&rft.aulast=Longuet-Higgins&rft.aufirst=M.S.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFMei1989" class="citation cs2"><a href="/wiki/Chiang_C._Mei" title="Chiang C. Mei">Mei, C.C.</a> (1989), The Applied Dynamics of Ocean Surface Waves, World Scientific, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/9789971507893" title="Special:BookSources/9789971507893"><bdi>9789971507893</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+Applied+Dynamics+of+Ocean+Surface+Waves&rft.pub=World+Scientific&rft.date=1989&rft.isbn=9789971507893&rft.aulast=Mei&rft.aufirst=C.C.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartz1974" class="citation cs2">Schwartz, L.W. (1974), "Computer extension and analytic continuation of Stokes's expansion for gravity waves", Journal of Fluid Mechanics, 62 (3): 553–578, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1974JFM....62..553S">1974JFM....62..553S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1017%2FS0022112074000802">10.1017/S0022112074000802</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:120140832">120140832</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Journal+of+Fluid+Mechanics&rft.atitle=Computer+extension+and+analytic+continuation+of+Stokes%27s+expansion+for+gravity+waves&rft.volume=62&rft.issue=3&rft.pages=553-578&rft.date=1974&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A120140832%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1017%2FS0022112074000802&rft_id=info%3Abibcode%2F1974JFM....62..553S&rft.aulast=Schwartz&rft.aufirst=L.W.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFSchwartzFenton1982" class="citation cs2">Schwartz, L.W.; Fenton, J.D. (1982), "Strongly nonlinear waves", Annual Review of Fluid Mechanics, 14: 39–60, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1982AnRFM..14...39S">1982AnRFM..14...39S</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1146%2Fannurev.fl.14.010182.000351">10.1146/annurev.fl.14.010182.000351</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Annual+Review+of+Fluid+Mechanics&rft.atitle=Strongly+nonlinear+waves&rft.volume=14&rft.pages=39-60&rft.date=1982&rft_id=info%3Adoi%2F10.1146%2Fannurev.fl.14.010182.000351&rft_id=info%3Abibcode%2F1982AnRFM..14...39S&rft.aulast=Schwartz&rft.aufirst=L.W.&rft.au=Fenton%2C+J.D.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWehausenLaitone1960" class="citation cs2"><a href="/wiki/John_V._Wehausen" title="John V. Wehausen">Wehausen, J. V.</a> & Laitone, E. V. (1960), <a href="/wiki/Siegfried_Fl%C3%BCgge" title="Siegfried Flügge">Flügge, S.</a> & <a href="/wiki/Clifford_Truesdell" title="Clifford Truesdell">Truesdell, C.</a> (eds.), <a rel="nofollow" class="external text" href="http://coe.berkeley.edu/SurfaceWaves/">"Surface Waves"</a>, Encyclopaedia of Physics, 9: 653–667, §27, <a href="/wiki/OCLC_(identifier)" class="mw-redirect" title="OCLC (identifier)">OCLC</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/oclc/612422741">612422741</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Encyclopaedia+of+Physics&rft.atitle=Surface+Waves&rft.volume=9&rft.pages=653-667%2C+%C2%A727&rft.date=1960&rft_id=info%3Aoclcnum%2F612422741&rft.aulast=Wehausen&rft.aufirst=J.+V.&rft.au=Laitone%2C+E.+V.&rft_id=http%3A%2F%2Fcoe.berkeley.edu%2FSurfaceWaves%2F&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWhitham1974" class="citation cs2"><a href="/wiki/Gerald_B._Whitham" title="Gerald B. Whitham">Whitham, G.B.</a> (1974), Linear and nonlinear waves, Wiley-Interscience, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-471-94090-6" title="Special:BookSources/978-0-471-94090-6"><bdi>978-0-471-94090-6</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Linear+and+nonlinear+waves&rft.pub=Wiley-Interscience&rft.date=1974&rft.isbn=978-0-471-94090-6&rft.aulast=Whitham&rft.aufirst=G.B.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams1981" class="citation cs2">Williams, J.M. (1981), "Limiting gravity waves in water of finite depth", Philosophical Transactions of the Royal Society, Series A, 302 (1466): 139–188, <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1981RSPTA.302..139W">1981RSPTA.302..139W</a>, <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1098%2Frsta.1981.0159">10.1098/rsta.1981.0159</a>, <a href="/wiki/S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:122673867">122673867</a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Philosophical+Transactions+of+the+Royal+Society&rft.atitle=Limiting+gravity+waves+in+water+of+finite+depth&rft.volume=302&rft.issue=1466&rft.pages=139-188&rft.date=1981&rft_id=https%3A%2F%2Fapi.semanticscholar.org%2FCorpusID%3A122673867%23id-name%3DS2CID&rft_id=info%3Adoi%2F10.1098%2Frsta.1981.0159&rft_id=info%3Abibcode%2F1981RSPTA.302..139W&rft.aulast=Williams&rft.aufirst=J.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"> and </li></ul> <dl><dd><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFWilliams1985" class="citation cs2">Williams, J.M. (1985), Tables of progressive gravity waves, Pitman, <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0273087335" title="Special:BookSources/978-0273087335"><bdi>978-0273087335</bdi></a></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Tables+of+progressive+gravity+waves&rft.pub=Pitman&rft.date=1985&rft.isbn=978-0273087335&rft.aulast=Williams&rft.aufirst=J.M.&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></dd></dl> </div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2>[<a href="/w/index.php?title=Stokes_wave&action=edit&section=25" title="Edit section: External links">edit</a>]</div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFJun_Zhang" class="citation cs2">Jun Zhang, <a rel="nofollow" class="external text" href="http://ceprofs.civil.tamu.edu/jzhang/oe671class/nonLinear.html">Stokes waves applet</a>, Texas A&M University, retrieved 2012-08-09</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Stokes+waves+applet&rft.pub=Texas+A%26M+University&rft.au=Jun+Zhang&rft_id=http%3A%2F%2Fceprofs.civil.tamu.edu%2Fjzhang%2Foe671class%2FnonLinear.html&rfr_id=info%3Asid%2Fen.wikipedia.org%3AStokes+wave" class="Z3988"></li></ul> <div class="navbox-styles"><style data-mw-deduplicate="TemplateStyles:r1129693374">.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl 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title="Boussinesq approximation (water waves)">Boussinesq approximation</a></li> <li><a href="/wiki/Breaking_wave" title="Breaking wave">Breaking wave</a></li> <li><a href="/wiki/Clapotis" title="Clapotis">Clapotis</a></li> <li><a href="/wiki/Cnoidal_wave" title="Cnoidal wave">Cnoidal wave</a></li> <li><a href="/wiki/Cross_sea" title="Cross sea">Cross sea</a></li> <li><a href="/wiki/Dispersion_(water_waves)" title="Dispersion (water waves)">Dispersion</a></li> <li><a href="/wiki/Edge_wave" title="Edge wave">Edge wave</a></li> <li><a href="/wiki/Equatorial_wave" title="Equatorial wave">Equatorial waves</a></li> <li><a href="/wiki/Gravity_wave" title="Gravity wave">Gravity wave</a></li> <li><a href="/wiki/Green%27s_law" title="Green's law">Green's law</a></li> <li><a href="/wiki/Infragravity_wave" title="Infragravity wave">Infragravity wave</a></li> <li><a href="/wiki/Internal_wave" title="Internal wave">Internal wave</a></li> <li><a href="/wiki/Iribarren_number" title="Iribarren number">Iribarren number</a></li> <li><a href="/wiki/Kelvin_wave" title="Kelvin wave">Kelvin wave</a></li> <li><a href="/wiki/Kinematic_wave" title="Kinematic wave">Kinematic wave</a></li> <li><a href="/wiki/Longshore_drift" title="Longshore drift">Longshore drift</a></li> <li><a href="/wiki/Luke%27s_variational_principle" title="Luke's variational principle">Luke's variational principle</a></li> <li><a href="/wiki/Mild-slope_equation" title="Mild-slope equation">Mild-slope equation</a></li> <li><a href="/wiki/Radiation_stress" title="Radiation stress">Radiation stress</a></li> <li><a href="/wiki/Rogue_wave" title="Rogue wave">Rogue wave</a></li> <li><a href="/wiki/Rossby_wave" title="Rossby wave">Rossby wave</a></li> <li><a href="/wiki/Rossby-gravity_waves" title="Rossby-gravity waves">Rossby-gravity waves</a></li> <li><a href="/wiki/Sea_state" title="Sea state">Sea state</a></li> <li><a href="/wiki/Seiche" title="Seiche">Seiche</a></li> <li><a href="/wiki/Significant_wave_height" 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water equations">shallow water equations</a></li></ul></li> <li><a href="/wiki/Wind_fetch" title="Wind fetch">Wind fetch</a></li> <li><a href="/wiki/Wind_setup" title="Wind setup">Wind setup</a></li> <li><a href="/wiki/Wind_wave" title="Wind wave">Wind wave</a> <ul><li><a href="/wiki/Wind_wave_model" title="Wind wave model">model</a></li></ul></li></ul> </div></td><td class="noviewer navbox-image" rowspan="10" style="width:1px;padding:0 0 0 2px"><div><a href="/wiki/File:Upwelling.svg" class="mw-file-description" title="Upwelling"><img alt="Upwelling" src="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/120px-Upwelling.svg.png" decoding="async" width="120" height="80" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/180px-Upwelling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/ee/Upwelling.svg/240px-Upwelling.svg.png 2x" data-file-width="365" data-file-height="242" /></a> <a 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circulation</a></li> <li><a href="/wiki/Baroclinity" title="Baroclinity">Baroclinity</a></li> <li><a href="/wiki/Boundary_current" title="Boundary current">Boundary current</a></li> <li><a href="/wiki/Coriolis_force" title="Coriolis force">Coriolis force</a></li> <li><a href="/wiki/Coriolis%E2%80%93Stokes_force" title="Coriolis–Stokes force">Coriolis–Stokes force</a></li> <li><a href="/wiki/Craik%E2%80%93Leibovich_vortex_force" title="Craik–Leibovich vortex force">Craik–Leibovich vortex force</a></li> <li><a href="/wiki/Downwelling" title="Downwelling">Downwelling</a></li> <li><a href="/wiki/Eddy_(fluid_dynamics)" title="Eddy (fluid dynamics)">Eddy</a></li> <li><a href="/wiki/Ekman_layer" title="Ekman layer">Ekman layer</a></li> <li><a href="/wiki/Ekman_spiral" title="Ekman spiral">Ekman spiral</a></li> <li><a href="/wiki/Ekman_transport" title="Ekman transport">Ekman transport</a></li> <li><a href="/wiki/El_Ni%C3%B1o%E2%80%93Southern_Oscillation" title="El Niño–Southern Oscillation">El Niño–Southern Oscillation</a></li> <li><a href="/wiki/General_circulation_model" title="General circulation model">General circulation model</a></li> <li><a href="/wiki/Geochemical_Ocean_Sections_Study" title="Geochemical Ocean Sections Study">Geochemical Ocean Sections Study</a></li> <li><a href="/wiki/Geostrophic_current" title="Geostrophic current">Geostrophic current</a></li> <li><a href="/wiki/Global_Ocean_Data_Analysis_Project" title="Global Ocean Data Analysis Project">Global Ocean Data Analysis Project</a></li> <li><a href="/wiki/Gulf_Stream" title="Gulf Stream">Gulf Stream</a></li> <li><a href="/wiki/Humboldt_Current" title="Humboldt Current">Humboldt Current</a></li> <li><a href="/wiki/Hydrothermal_circulation" title="Hydrothermal circulation">Hydrothermal circulation</a></li> <li><a href="/wiki/Langmuir_circulation" title="Langmuir circulation">Langmuir circulation</a></li> <li><a href="/wiki/Longshore_drift" title="Longshore drift">Longshore drift</a></li> <li><a href="/wiki/Loop_Current" title="Loop Current">Loop Current</a></li> <li><a href="/wiki/Modular_Ocean_Model" title="Modular Ocean Model">Modular Ocean Model</a></li> <li><a href="/wiki/Ocean_current" title="Ocean current">Ocean current</a></li> <li><a href="/wiki/Ocean_dynamical_thermostat" title="Ocean dynamical thermostat">Ocean dynamical thermostat</a></li> <li><a href="/wiki/Ocean_dynamics" title="Ocean dynamics">Ocean dynamics</a></li> <li><a href="/wiki/Ocean_gyre" title="Ocean gyre">Ocean gyre</a></li> <li><a href="/wiki/Overflow_(oceanography)" title="Overflow (oceanography)">Overflow</a></li> <li><a href="/wiki/Princeton_Ocean_Model" title="Princeton Ocean Model">Princeton Ocean Model</a></li> <li><a href="/wiki/Rip_current" title="Rip current">Rip current</a></li> <li><a href="/wiki/Subsurface_ocean_current" title="Subsurface ocean current">Subsurface ocean current</a></li> <li><a href="/wiki/Sverdrup_balance" title="Sverdrup balance">Sverdrup balance</a></li> <li><a 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href="/wiki/Earth_tide" title="Earth tide">Earth tide</a></li> <li><a href="/wiki/Head_of_tide" title="Head of tide">Head of tide</a></li> <li><a href="/wiki/Internal_tide" title="Internal tide">Internal tide</a></li> <li><a href="/wiki/Lunitidal_interval" title="Lunitidal interval">Lunitidal interval</a></li> <li><a href="/wiki/Perigean_spring_tide" title="Perigean spring tide">Perigean spring tide</a></li> <li><a href="/wiki/Rip_tide" title="Rip tide">Rip tide</a></li> <li><a href="/wiki/Rule_of_twelfths" title="Rule of twelfths">Rule of twelfths</a></li> <li><a href="/wiki/Slack_tide" title="Slack tide">Slack tide</a></li> <li><a href="/wiki/Theory_of_tides" title="Theory of tides">Theory of tides</a></li> <li><a href="/wiki/Tidal_bore" title="Tidal bore">Tidal bore</a></li> <li><a href="/wiki/Tidal_force" title="Tidal force">Tidal force</a></li> <li><a href="/wiki/Tidal_power" title="Tidal power">Tidal power</a></li> <li><a href="/wiki/Tidal_race" title="Tidal race">Tidal race</a></li> <li><a href="/wiki/Tidal_range" title="Tidal range">Tidal range</a></li> <li><a href="/wiki/Tidal_resonance" title="Tidal resonance">Tidal resonance</a></li> <li><a href="/wiki/Tide_gauge" title="Tide gauge">Tide gauge</a></li> <li><a href="/wiki/Tideline" title="Tideline">Tideline</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Landform" title="Landform">Landforms</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Abyssal_fan" title="Abyssal fan">Abyssal fan</a></li> <li><a href="/wiki/Abyssal_plain" title="Abyssal plain">Abyssal plain</a></li> <li><a href="/wiki/Atoll" title="Atoll">Atoll</a></li> <li><a href="/wiki/Bathymetric_chart" title="Bathymetric chart">Bathymetric chart</a></li> <li><a href="/wiki/Carbonate_platform" title="Carbonate platform">Carbonate platform</a></li> <li><a href="/wiki/Coastal_geography" title="Coastal geography">Coastal geography</a></li> <li><a href="/wiki/Cold_seep" title="Cold seep">Cold seep</a></li> <li><a href="/wiki/Continental_margin" title="Continental margin">Continental margin</a></li> <li><a href="/wiki/Continental_rise" title="Continental rise">Continental rise</a></li> <li><a href="/wiki/Continental_shelf" title="Continental shelf">Continental shelf</a></li> <li><a href="/wiki/Contourite" title="Contourite">Contourite</a></li> <li><a href="/wiki/Guyot" title="Guyot">Guyot</a></li> <li><a href="/wiki/Hydrography" title="Hydrography">Hydrography</a></li> <li><a href="/wiki/Knoll_(oceanography)" title="Knoll (oceanography)">Knoll</a></li> <li><a href="/wiki/Ocean_bank" title="Ocean bank">Ocean bank</a></li> <li><a href="/wiki/Oceanic_basin" title="Oceanic basin">Oceanic basin</a></li> <li><a href="/wiki/Oceanic_plateau" title="Oceanic plateau">Oceanic plateau</a></li> <li><a href="/wiki/Oceanic_trench" title="Oceanic trench">Oceanic trench</a></li> <li><a href="/wiki/Passive_margin" title="Passive margin">Passive margin</a></li> <li><a href="/wiki/Seabed" title="Seabed">Seabed</a></li> <li><a href="/wiki/Seamount" title="Seamount">Seamount</a></li> <li><a href="/wiki/Submarine_canyon" title="Submarine canyon">Submarine canyon</a></li> <li><a href="/wiki/Submarine_volcano" title="Submarine volcano">Submarine volcano</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Plate_tectonics" title="Plate tectonics">Plate tectonics</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Convergent_boundary" title="Convergent boundary">Convergent boundary</a></li> <li><a href="/wiki/Divergent_boundary" title="Divergent boundary">Divergent boundary</a></li> <li><a href="/wiki/Fracture_zone" title="Fracture zone">Fracture zone</a></li> <li><a href="/wiki/Hydrothermal_vent" title="Hydrothermal vent">Hydrothermal vent</a></li> <li><a href="/wiki/Marine_geology" title="Marine geology">Marine geology</a></li> <li><a href="/wiki/Mid-ocean_ridge" title="Mid-ocean ridge">Mid-ocean ridge</a></li> <li><a href="/wiki/Mohorovi%C4%8Di%C4%87_discontinuity" title="Mohorovičić discontinuity">Mohorovičić discontinuity</a></li> <li><a href="/wiki/Oceanic_crust" title="Oceanic crust">Oceanic crust</a></li> <li><a href="/wiki/Outer_trench_swell" title="Outer trench swell">Outer trench swell</a></li> <li><a href="/wiki/Ridge_push" title="Ridge push">Ridge push</a></li> <li><a href="/wiki/Seafloor_spreading" title="Seafloor spreading">Seafloor spreading</a></li> <li><a href="/wiki/Slab_pull" title="Slab pull">Slab pull</a></li> <li><a href="/wiki/Slab_suction" title="Slab suction">Slab suction</a></li> <li><a href="/wiki/Slab_window" title="Slab window">Slab window</a></li> <li><a href="/wiki/Subduction" title="Subduction">Subduction</a></li> <li><a href="/wiki/Transform_fault" title="Transform fault">Transform fault</a></li> <li><a href="/wiki/Vine%E2%80%93Matthews%E2%80%93Morley_hypothesis" title="Vine–Matthews–Morley hypothesis">Vine–Matthews–Morley hypothesis</a></li> <li><a href="/wiki/Volcanic_arc" title="Volcanic arc">Volcanic arc</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ocean zones</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Benthic_zone" title="Benthic zone">Benthic</a></li> <li><a href="/wiki/Deep_ocean_water" title="Deep ocean water">Deep ocean water</a></li> <li><a href="/wiki/Deep_sea" title="Deep sea">Deep sea</a></li> <li><a href="/wiki/Littoral_zone" title="Littoral zone">Littoral</a></li> <li><a href="/wiki/Mesopelagic_zone" title="Mesopelagic zone">Mesopelagic</a></li> <li><a href="/wiki/Oceanic_zone" title="Oceanic zone">Oceanic</a></li> <li><a href="/wiki/Pelagic_zone" title="Pelagic zone">Pelagic</a></li> <li><a href="/wiki/Photic_zone" title="Photic zone">Photic</a></li> <li><a href="/wiki/Surf_zone" title="Surf zone">Surf</a></li> <li><a href="/wiki/Swash" title="Swash">Swash</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Sea_level" title="Sea level">Sea level</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Deep-ocean_Assessment_and_Reporting_of_Tsunamis" title="Deep-ocean Assessment and Reporting of Tsunamis">Deep-ocean Assessment and Reporting of Tsunamis</a></li> <li><a href="/wiki/Global_Sea_Level_Observing_System" title="Global Sea Level Observing System">Global Sea Level Observing System</a></li> <li><a href="/wiki/North_West_Shelf_Operational_Oceanographic_System" title="North West Shelf Operational Oceanographic System">North West Shelf Operational Oceanographic System</a></li> <li><a href="/wiki/Sea-level_curve" title="Sea-level curve">Sea-level curve</a></li> <li><a href="/wiki/Sea_level_drop" title="Sea level drop">Sea level drop</a></li> <li><a href="/wiki/Sea_level_rise" title="Sea level rise">Sea level rise</a></li> <li><a href="/wiki/World_Geodetic_System" title="World Geodetic System">World Geodetic System</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Acoustical_oceanography" class="mw-redirect" title="Acoustical oceanography">Acoustics</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Deep_scattering_layer" title="Deep scattering layer">Deep scattering layer</a></li> <li><a href="/wiki/Ocean_acoustic_tomography" title="Ocean acoustic tomography">Ocean acoustic tomography</a></li> <li><a href="/wiki/Sofar_bomb" title="Sofar bomb">Sofar bomb</a></li> <li><a href="/wiki/SOFAR_channel" title="SOFAR channel">SOFAR channel</a></li> <li><a href="/wiki/Underwater_acoustics" title="Underwater acoustics">Underwater acoustics</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Satellites</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Jason-1" title="Jason-1">Jason-1</a></li> <li><a href="/wiki/OSTM/Jason-2" title="OSTM/Jason-2">OSTM/Jason-2</a></li> <li><a href="/wiki/Jason-3" title="Jason-3">Jason-3</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Ocean_acidification" title="Ocean acidification">Acidification</a></li> <li><a href="/wiki/Argo_(oceanography)" title="Argo (oceanography)">Argo</a></li> <li><a href="/wiki/Benthic_lander" title="Benthic lander">Benthic lander</a></li> <li><a href="/wiki/Color_of_water" title="Color of water">Color of water</a></li> <li><a href="/wiki/DSV_Alvin" title="DSV Alvin">DSV Alvin</a></li> <li><a href="/wiki/Marginal_sea" class="mw-redirect" title="Marginal sea">Marginal sea</a></li> <li><a href="/wiki/Marine_energy" title="Marine energy">Marine energy</a></li> <li><a href="/wiki/Marine_pollution" title="Marine pollution">Marine pollution</a></li> <li><a href="/wiki/Mooring_(oceanography)" title="Mooring (oceanography)">Mooring</a></li> <li><a href="/wiki/National_Oceanographic_Data_Center" title="National Oceanographic Data Center">National Oceanographic Data Center</a></li> <li><a href="/wiki/Ocean" title="Ocean">Ocean</a></li> <li><a href="/wiki/Ocean_exploration" title="Ocean exploration">Explorations</a></li> <li><a href="/wiki/Ocean_observations" title="Ocean observations">Observations</a></li> <li><a href="/wiki/Ocean_reanalysis" title="Ocean reanalysis">Reanalysis</a></li> <li><a href="/wiki/Ocean_surface_topography" title="Ocean surface topography">Ocean surface topography</a></li> <li><a href="/wiki/Ocean_temperature" title="Ocean temperature">Ocean temperature</a></li> <li><a href="/wiki/Ocean_thermal_energy_conversion" title="Ocean thermal energy conversion">Ocean thermal energy conversion</a></li> <li><a href="/wiki/Oceanography" title="Oceanography">Oceanography</a> <ul><li><a href="/wiki/Outline_of_oceanography" title="Outline of oceanography">Outline of oceanography</a></li></ul></li> <li><a href="/wiki/Pelagic_sediment" title="Pelagic sediment">Pelagic sediment</a></li> <li><a href="/wiki/Sea_surface_microlayer" title="Sea surface microlayer">Sea surface microlayer</a></li> <li><a href="/wiki/Sea_surface_temperature" title="Sea surface temperature">Sea surface temperature</a></li> <li><a href="/wiki/Seawater" title="Seawater">Seawater</a></li> <li><a href="/wiki/Science_On_a_Sphere" title="Science On a Sphere">Science On a Sphere</a></li> <li><a href="/wiki/Ocean_stratification" title="Ocean stratification">Stratification</a></li> <li><a href="/wiki/Thermocline" title="Thermocline">Thermocline</a></li> <li><a href="/wiki/Underwater_glider" title="Underwater glider">Underwater glider</a></li> <li><a href="/wiki/Water_column" title="Water column">Water column</a></li> <li><a href="/wiki/World_Ocean_Atlas" title="World Ocean Atlas">World Ocean Atlas</a></li></ul> </div></td></tr><tr><td class="navbox-abovebelow" colspan="3"><div> <ul><li><img alt="" src="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/16px-Symbol_category_class.svg.png" decoding="async" width="16" height="16" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/23px-Symbol_category_class.svg.png 1.5x, //upload.wikimedia.org/wikipedia/en/thumb/9/96/Symbol_category_class.svg/31px-Symbol_category_class.svg.png 2x" data-file-width="180" data-file-height="185" /> <a href="/wiki/Category:Physical_oceanography" 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srcset="//upload.wikimedia.org/wikipedia/commons/thumb/4/45/Waves_in_pacifica_1.jpg/24px-Waves_in_pacifica_1.jpg 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/4/45/Waves_in_pacifica_1.jpg/32px-Waves_in_pacifica_1.jpg 2x" data-file-width="2000" data-file-height="1358" /></a> <a href="/wiki/Portal:Oceans" title="Portal:Oceans">Oceans portal</a></li></ul> </div></td></tr></tbody></table></div>    </div><noscript><img src="https://login.wikimedia.org/wiki/Special:CentralAutoLogin/start?type=1x1" alt="" width="1" height="1" style="border: none; position: absolute;"></noscript> <div class="printfooter" data-nosnippet="">Retrieved from "<a dir="ltr" href="https://en.wikipedia.org/w/index.php?title=Stokes_wave&oldid=1230946249">https://en.wikipedia.org/w/index.php?title=Stokes_wave&oldid=1230946249</a>"</div></div> <div id="catlinks" class="catlinks" data-mw="interface"><div id="mw-normal-catlinks" class="mw-normal-catlinks"><a href="/wiki/Help:Category" 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Stokes wave - Wikipedia