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href="https://donate.wikimedia.org/wiki/Special:FundraiserRedirector?utm_source=donate&utm_medium=sidebar&utm_campaign=C13_en.wikipedia.org&uselang=en" class=""><span>Donate</span></a> </li> <li id="pt-createaccount-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:CreateAccount&returnto=S+wave&returntoquery=oldid%3D1256618779" title="You are encouraged to create an account and log in; however, it is not mandatory" class=""><span>Create account</span></a> </li> <li id="pt-login-2" class="user-links-collapsible-item mw-list-item user-links-collapsible-item"><a data-mw="interface" href="/w/index.php?title=Special:UserLogin&returnto=S+wave&returntoquery=oldid%3D1256618779" title="You're encouraged to log in; however, it's not mandatory. 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Available in 19 languages" > <label id="p-lang-btn-label" for="p-lang-btn-checkbox" class="vector-dropdown-label cdx-button cdx-button--fake-button cdx-button--fake-button--enabled cdx-button--weight-quiet cdx-button--action-progressive mw-portlet-lang-heading-19" aria-hidden="true" ><span class="vector-icon mw-ui-icon-language-progressive mw-ui-icon-wikimedia-language-progressive"></span> <span class="vector-dropdown-label-text">19 languages</span> </label> <div class="vector-dropdown-content"> <div class="vector-menu-content"> <ul class="vector-menu-content-list"> <li class="interlanguage-link interwiki-ar mw-list-item"><a href="https://ar.wikipedia.org/wiki/%D9%85%D9%88%D8%AC%D8%A9_S" title="موجة S – Arabic" lang="ar" hreflang="ar" data-title="موجة S" data-language-autonym="العربية" data-language-local-name="Arabic" class="interlanguage-link-target"><span>العربية</span></a></li><li class="interlanguage-link interwiki-ca mw-list-item"><a href="https://ca.wikipedia.org/wiki/Ones_S" title="Ones S – Catalan" lang="ca" hreflang="ca" data-title="Ones S" data-language-autonym="Català" data-language-local-name="Catalan" class="interlanguage-link-target"><span>Català</span></a></li><li class="interlanguage-link interwiki-de badge-Q70894304 mw-list-item" title=""><a href="https://de.wikipedia.org/wiki/S-Welle" title="S-Welle – German" lang="de" hreflang="de" data-title="S-Welle" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-et mw-list-item"><a href="https://et.wikipedia.org/wiki/S-laine" title="S-laine – Estonian" lang="et" hreflang="et" data-title="S-laine" data-language-autonym="Eesti" data-language-local-name="Estonian" class="interlanguage-link-target"><span>Eesti</span></a></li><li class="interlanguage-link interwiki-el mw-list-item"><a href="https://el.wikipedia.org/wiki/%CE%9A%CF%8D%CE%BC%CE%B1_S" title="Κύμα S – Greek" lang="el" hreflang="el" data-title="Κύμα S" data-language-autonym="Ελληνικά" data-language-local-name="Greek" class="interlanguage-link-target"><span>Ελληνικά</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Onda_S" title="Onda S – Spanish" lang="es" hreflang="es" data-title="Onda S" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fa mw-list-item"><a href="https://fa.wikipedia.org/wiki/%D9%85%D9%88%D8%AC_%D8%A7%D8%B3" title="موج اس – Persian" lang="fa" hreflang="fa" data-title="موج اس" data-language-autonym="فارسی" data-language-local-name="Persian" class="interlanguage-link-target"><span>فارسی</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/S%ED%8C%8C" title="S파 – Korean" lang="ko" hreflang="ko" data-title="S파" data-language-autonym="한국어" data-language-local-name="Korean" class="interlanguage-link-target"><span>한국어</span></a></li><li class="interlanguage-link interwiki-id mw-list-item"><a href="https://id.wikipedia.org/wiki/Gelombang-S" title="Gelombang-S – Indonesian" lang="id" hreflang="id" data-title="Gelombang-S" data-language-autonym="Bahasa Indonesia" data-language-local-name="Indonesian" class="interlanguage-link-target"><span>Bahasa Indonesia</span></a></li><li class="interlanguage-link interwiki-ht mw-list-item"><a href="https://ht.wikipedia.org/wiki/Onn_segond%C3%A8" title="Onn segondè – Haitian Creole" lang="ht" hreflang="ht" data-title="Onn segondè" data-language-autonym="Kreyòl ayisyen" data-language-local-name="Haitian Creole" class="interlanguage-link-target"><span>Kreyòl ayisyen</span></a></li><li class="interlanguage-link interwiki-ms mw-list-item"><a href="https://ms.wikipedia.org/wiki/Gelombang_S" title="Gelombang S – Malay" lang="ms" hreflang="ms" data-title="Gelombang S" 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The present address (URL) is a <a href="/wiki/Help:Permanent_link" title="Help:Permanent link">permanent link</a> to this revision, which may differ significantly from the <span class="plainlinks"><a class="external text" href="https://en.wikipedia.org/wiki/S_wave">current revision</a></span>.</b></p><div id="revision-info-plain" style="display: none;">Revision as of 20:13, 10 November 2024 by <a href="/wiki/User:Dawnseeker2000" class="mw-userlink" title="User:Dawnseeker2000" data-mw-revid="1256618779"><bdi>Dawnseeker2000</bdi></a> <span class="mw-usertoollinks">(<a href="/wiki/User_talk:Dawnseeker2000" class="mw-usertoollinks-talk" title="User talk:Dawnseeker2000">talk</a> | <a href="/wiki/Special:Contributions/Dawnseeker2000" class="mw-usertoollinks-contribs" title="Special:Contributions/Dawnseeker2000">contribs</a>)</span> <span class="comment">(minor formatting)</span></div></div><div id="mw-revision-nav">(<a href="/w/index.php?title=S_wave&diff=prev&oldid=1256618779" title="S wave">diff</a>) <a href="/w/index.php?title=S_wave&direction=prev&oldid=1256618779" title="S wave">← Previous revision</a> | <a href="/wiki/S_wave" title="S wave">Latest revision</a> (<a href="/w/index.php?title=S_wave&diff=cur&oldid=1256618779" title="S wave">diff</a>) | <a href="/w/index.php?title=S_wave&direction=next&oldid=1256618779" title="S wave">Newer revision →</a> (<a href="/w/index.php?title=S_wave&diff=next&oldid=1256618779" title="S wave">diff</a>)</div></div></div></div></div> <div id="mw-content-text" class="mw-body-content"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><div class="shortdescription nomobile noexcerpt noprint searchaux" style="display:none">Type of elastic body wave</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">For the lowest-energy electronic wavefunction in atomic physics, see <a href="/wiki/Atomic_orbital" title="Atomic orbital">atomic orbital</a>. For the S wave on an electrocardiogram, see <a href="/wiki/QRS_complex" title="QRS complex">QRS complex</a>.</div> <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 ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · 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.sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-night .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-list-title,html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle{background:transparent!important}html.skin-theme-clientpref-os .mw-parser-output .sidebar:not(.notheme) .sidebar-title-with-pretitle a{color:var(--color-progressive)!important}}@media print{body.ns-0 .mw-parser-output .sidebar{display:none!important}}</style><table class="sidebar sidebar-collapse nomobile nowraplinks"><tbody><tr><td class="sidebar-pretitle">Part of <a href="/wiki/Category:Earthquakes" title="Category:Earthquakes">a series</a> on</td></tr><tr><th class="sidebar-title-with-pretitle"><a href="/wiki/Earthquake" title="Earthquake">Earthquakes</a></th></tr><tr><td class="sidebar-image"><span typeof="mw:File"><a href="/wiki/File:Earthquake_-_The_Noun_Project.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Earthquake_-_The_Noun_Project.svg/80px-Earthquake_-_The_Noun_Project.svg.png" decoding="async" width="80" height="64" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Earthquake_-_The_Noun_Project.svg/120px-Earthquake_-_The_Noun_Project.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a0/Earthquake_-_The_Noun_Project.svg/160px-Earthquake_-_The_Noun_Project.svg.png 2x" data-file-width="512" data-file-height="410" /></a></span></td></tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Types_of_earthquake" title="Types of earthquake">Types</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Mainshock" title="Mainshock">Mainshock</a></li> <li><a href="/wiki/Foreshock" title="Foreshock">Foreshock</a></li> <li><a href="/wiki/Aftershock" title="Aftershock">Aftershock</a></li> <li><a href="/wiki/Blind_thrust_earthquake" title="Blind thrust earthquake">Blind thrust</a></li> <li><a href="/wiki/Doublet_earthquake" title="Doublet earthquake">Doublet</a></li> <li><a href="/wiki/Interplate_earthquake" title="Interplate earthquake">Interplate</a></li> <li><a href="/wiki/Intraplate_earthquake" title="Intraplate earthquake">Intraplate</a></li> <li><a href="/wiki/Megathrust_earthquake" title="Megathrust earthquake">Megathrust</a></li> <li><a href="/wiki/Remotely_triggered_earthquakes" title="Remotely triggered earthquakes">Remotely triggered</a></li> <li><a href="/wiki/Slow_earthquake" title="Slow earthquake">Slow</a></li> <li><a href="/wiki/Submarine_earthquake" title="Submarine earthquake">Submarine</a></li> <li><a href="/wiki/Supershear_earthquake" title="Supershear earthquake">Supershear</a></li> <li><a href="/wiki/Tsunami_earthquake" title="Tsunami earthquake">Tsunami</a></li> <li><a href="/wiki/Earthquake_swarm" title="Earthquake swarm">Earthquake swarm</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Causes</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Fault_(geology)" title="Fault (geology)">Fault movement</a></li> <li><a href="/wiki/Volcano_tectonic_earthquake" title="Volcano tectonic earthquake">Volcanism</a></li> <li><a href="/wiki/Induced_seismicity" title="Induced seismicity">Induced seismicity</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Characteristics</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Epicenter" title="Epicenter">Epicenter</a></li> <li><a href="/wiki/Epicentral_distance" title="Epicentral distance">Epicentral distance</a></li> <li><a href="/wiki/Hypocenter" title="Hypocenter">Hypocenter</a></li> <li><a href="/wiki/Shadow_zone" title="Shadow zone">Shadow zone</a></li> <li><a href="/wiki/Seismic_wave" title="Seismic wave">Seismic waves</a></li> <li><a href="/wiki/P_wave" title="P wave">P wave</a></li> <li><a class="mw-selflink selflink">S wave</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Measurement</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Seismometer" title="Seismometer">Seismometer</a></li> <li><a href="/wiki/Seismic_magnitude_scales" title="Seismic magnitude scales">Seismic magnitude scales</a></li> <li><a href="/wiki/Seismic_intensity_scales" title="Seismic intensity scales">Seismic intensity scales</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)"><a href="/wiki/Earthquake_prediction" title="Earthquake prediction">Prediction</a></div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><div style="display:inline-block; padding:0.2em 0.4em; line-height:1.2em; line-height:1.25em;"><a href="/wiki/Coordinating_Committee_for_Earthquake_Prediction" title="Coordinating Committee for Earthquake Prediction">Coordinating Committee for<br />Earthquake Prediction</a></div></li> <li><a href="/wiki/Earthquake_forecasting" title="Earthquake forecasting">Forecasting</a></li></ul></div></div></td> </tr><tr><td class="sidebar-content"> <div class="sidebar-list mw-collapsible mw-collapsed"><div class="sidebar-list-title" style="color: var(--color-base)">Other topics</div><div class="sidebar-list-content mw-collapsible-content hlist"> <ul><li><a href="/wiki/Shear_wave_splitting" title="Shear wave splitting">Shear wave splitting</a></li> <li><a href="/wiki/Adams%E2%80%93Williamson_equation" title="Adams–Williamson equation">Adams–Williamson equation</a></li> <li><a href="/wiki/Flinn%E2%80%93Engdahl_regions" class="mw-redirect" title="Flinn–Engdahl regions">Flinn–Engdahl regions</a></li> <li><a href="/wiki/Earthquake_engineering" title="Earthquake engineering">Earthquake engineering</a></li> <li><a href="/wiki/Seismite" title="Seismite">Seismite</a></li> <li><a href="/wiki/Seismology" title="Seismology">Seismology</a></li></ul></div></div></td> </tr><tr><td class="sidebar-below hlist"> <ul><li><a href="/wiki/Portal:Earth_sciences" title="Portal:Earth sciences">Earth Sciences Portal</a></li></ul> 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.thumbinner{width:100%!important;box-sizing:border-box;max-width:none!important;align-items:center}.mw-parser-output .tmulti .trow{justify-content:center}.mw-parser-output .tmulti .tsingle{float:none!important;max-width:100%!important;box-sizing:border-box;text-align:center}.mw-parser-output .tmulti .tsingle .thumbcaption{text-align:left}.mw-parser-output .tmulti .trow>.thumbcaption{text-align:center}}@media screen{html.skin-theme-clientpref-night .mw-parser-output .tmulti .multiimageinner img{background-color:white}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .tmulti .multiimageinner img{background-color:white}}</style><div class="thumb tmulti tright"><div class="thumbinner multiimageinner" style="width:392px;max-width:392px"><div class="trow"><div class="tsingle" style="width:194px;max-width:194px"><div class="thumbimage" style="height:145px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Onde_cisaillement_impulsion_1d_30_petit.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Onde_cisaillement_impulsion_1d_30_petit.gif/192px-Onde_cisaillement_impulsion_1d_30_petit.gif" decoding="async" width="192" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/6d/Onde_cisaillement_impulsion_1d_30_petit.gif/288px-Onde_cisaillement_impulsion_1d_30_petit.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/6/6d/Onde_cisaillement_impulsion_1d_30_petit.gif 2x" data-file-width="305" data-file-height="231" /></a></span></div><div class="thumbcaption">Plane shear wave</div></div><div class="tsingle" style="width:194px;max-width:194px"><div class="thumbimage" style="height:145px;overflow:hidden"><span typeof="mw:File"><a href="/wiki/File:Ondes_cisaillement_2d_20_petit.gif" class="mw-file-description"><img alt="" src="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ondes_cisaillement_2d_20_petit.gif/192px-Ondes_cisaillement_2d_20_petit.gif" decoding="async" width="192" height="145" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/31/Ondes_cisaillement_2d_20_petit.gif/288px-Ondes_cisaillement_2d_20_petit.gif 1.5x, //upload.wikimedia.org/wikipedia/commons/3/31/Ondes_cisaillement_2d_20_petit.gif 2x" data-file-width="305" data-file-height="231" /></a></span></div><div class="thumbcaption">Propagation of a spherical S wave in a 2d grid (empirical model)</div></div></div></div></div> <p>In <a href="/wiki/Seismology" title="Seismology">seismology</a> and other areas involving elastic waves, <b>S waves</b>, <b>secondary waves</b>, or <b>shear waves</b> (sometimes called <b>elastic S waves</b>) are a type of <a href="/wiki/Linear_elasticity#Elastic_wave" title="Linear elasticity">elastic wave</a> and are one of the two main types of elastic <a href="/wiki/Body_wave_(seismology)" class="mw-redirect" title="Body wave (seismology)">body waves</a>, so named because they move through the body of an object, unlike <a href="/wiki/Surface_wave" title="Surface wave">surface waves</a>.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> </p><p>S waves are <a href="/wiki/Transverse_wave" title="Transverse wave">transverse waves</a>, meaning that the direction of <a href="/wiki/Particle" title="Particle">particle</a> movement of an S wave is perpendicular to the direction of wave propagation, and the main restoring force comes from <a href="/wiki/Shear_stress" title="Shear stress">shear stress</a>.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup> Therefore, S waves cannot propagate in liquids<sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup> with zero (or very low) <a href="/wiki/Viscosity" title="Viscosity">viscosity</a>; however, they may propagate in liquids with high viscosity.<sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup> </p><p>The name <i>secondary wave</i> comes from the fact that they are the second type of wave to be detected by an earthquake <a href="/wiki/Seismograph" class="mw-redirect" title="Seismograph">seismograph</a>, after the <a href="/wiki/Compressional_wave" class="mw-redirect" title="Compressional wave">compressional</a> primary wave, or <a href="/wiki/P_wave" title="P wave">P wave</a>, because S waves travel more slowly in solids. Unlike P waves, S waves cannot travel through the molten <a href="/wiki/Earth%27s_outer_core" title="Earth's outer core">outer core</a> of the Earth, and this causes a <a href="/wiki/Shadow_zone" title="Shadow zone">shadow zone</a> for S waves opposite to their origin. They can still propagate through the solid <a href="/wiki/Earth%27s_inner_core" title="Earth's inner core">inner core</a>: when a P wave strikes the boundary of molten and solid cores at an oblique angle, S waves will form and propagate in the solid medium. When these S waves hit the boundary again at an oblique angle, they will in turn create P waves that propagate through the liquid medium. This property allows <a href="/wiki/Seismology" title="Seismology">seismologists</a> to determine some physical properties of the Earth's inner core.<sup id="cite_ref-UIC_6-0" class="reference"><a href="#cite_note-UIC-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="History">History</h2></div> <p>In 1830, the mathematician <a href="/wiki/Sim%C3%A9on_Denis_Poisson" title="Siméon Denis Poisson">Siméon Denis Poisson</a> presented to the <a href="/wiki/French_Academy_of_Sciences" title="French Academy of Sciences">French Academy of Sciences</a> an essay ("memoir") with a theory of the propagation of elastic waves in solids. In his memoir, he states that an earthquake would produce two different waves: one having a certain speed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>a</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle a}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ffd2487510aa438433a2579450ab2b3d557e5edc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.23ex; height:1.676ex;" alt="{\displaystyle a}"></span> and the other having a speed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\frac {a}{\sqrt {3}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>a</mi> <msqrt> <mn>3</mn> </msqrt> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {a}{\sqrt {3}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9d0faf67d0ab0a840626b9329ec62fa760102579" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:3.934ex; height:5.676ex;" alt="{\displaystyle {\frac {a}{\sqrt {3}}}}"></span>. At a sufficient distance from the source, when they can be considered <a href="/wiki/Plane_wave" title="Plane wave">plane waves</a> in the region of interest, the first kind consists of expansions and compressions in the direction perpendicular to the wavefront (that is, parallel to the wave's direction of motion); while the second consists of stretching motions occurring in directions parallel to the front (perpendicular to the direction of motion).<sup id="cite_ref-pois1831_7-0" class="reference"><a href="#cite_note-pois1831-7"><span class="cite-bracket">[</span>7<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="Theory">Theory</h2></div> <div class="mw-heading mw-heading3"><h3 id="Isotropic_medium">Isotropic medium</h3></div> <figure class="mw-default-size" typeof="mw:File/Thumb"><a href="/wiki/File:Speeds_of_seismic_waves.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Speeds_of_seismic_waves.svg/310px-Speeds_of_seismic_waves.svg.png" decoding="async" width="310" height="211" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Speeds_of_seismic_waves.svg/465px-Speeds_of_seismic_waves.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Speeds_of_seismic_waves.svg/620px-Speeds_of_seismic_waves.svg.png 2x" data-file-width="1321" data-file-height="901" /></a><figcaption>Velocity of seismic waves in the Earth versus depth. The negligible S-wave velocity in the outer core occurs because it is liquid, while in the solid inner core the S-wave velocity is non-zero.</figcaption></figure> <p>For the purpose of this explanation, a solid medium is considered <a href="/wiki/Isotropic" class="mw-redirect" title="Isotropic">isotropic</a> if its <a href="/wiki/Deformation_(mechanics)" class="mw-redirect" title="Deformation (mechanics)">strain (deformation)</a> in response to <a href="/wiki/Stress_(mechanics)" title="Stress (mechanics)">stress</a> is the same in all directions. Let <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {u}}=(u_{1},u_{2},u_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {u}}=(u_{1},u_{2},u_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/801e5df1f26005e2f849876a6c6df1f94bd98289" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.71ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {u}}=(u_{1},u_{2},u_{3})}"></span> be the displacement <a href="/wiki/Vector_(mathematics)" class="mw-redirect" title="Vector (mathematics)">vector</a> of a particle of such a medium from its "resting" position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/11b18a7b18c1c53b91f64f514ef7da05ae972183" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.659ex; height:2.843ex;" alt="{\displaystyle {\boldsymbol {x}}=(x_{1},x_{2},x_{3})}"></span> due elastic vibrations, understood to be a <a href="/wiki/Function_(mathematics)" title="Function (mathematics)">function</a> of the rest position <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {x}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">x</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {x}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/606b7680d510560a505937143775ea80fa958051" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.532ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {x}}}"></span> and time <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>t</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle t}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/65658b7b223af9e1acc877d848888ecdb4466560" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}"></span>. The deformation of the medium at that point can be described by the <a href="/wiki/Strain_tensor" class="mw-redirect" title="Strain tensor">strain tensor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {e}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">e</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {e}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e806bbc21b48eeda4b49efa21e9f9812e69eb61c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.288ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {e}}}"></span>, the 3×3 matrix whose elements are <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e_{ij}={\tfrac {1}{2}}\left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mfrac> <mn>1</mn> <mn>2</mn> </mfrac> </mstyle> </mrow> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle e_{ij}={\tfrac {1}{2}}\left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0ba641bc63f41b2acb9a647d6f6747a15e40358a" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:20.901ex; height:3.509ex;" alt="{\displaystyle e_{ij}={\tfrac {1}{2}}\left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)}"></span> </p><p>where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9b6be63a3ddb914c608027404afea96933df7c79" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.034ex; height:2.509ex;" alt="{\displaystyle \partial _{i}}"></span> denotes partial derivative with respect to position coordinate <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x_{i}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle x_{i}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e87000dd6142b81d041896a30fe58f0c3acb2158" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.129ex; height:2.009ex;" alt="{\displaystyle x_{i}}"></span>. The strain tensor is related to the 3×3 <a href="/wiki/Cauchy_stress_tensor" title="Cauchy stress tensor">stress tensor</a> <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\boldsymbol {\tau }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">τ</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\boldsymbol {\tau }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/beb1359149db88a28c86f0a3030894b71610a224" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.418ex; height:1.676ex;" alt="{\displaystyle {\boldsymbol {\tau }}}"></span> by the equation <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}e_{kk}+2\mu e_{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>λ</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mn>2</mn> <mi>μ</mi> <msub> <mi>e</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}e_{kk}+2\mu e_{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/058553f789b25b367297eddf692c66ef8355aef0" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:24.58ex; height:5.509ex;" alt="{\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}e_{kk}+2\mu e_{ij}}"></span> </p><p>Here <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \delta _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \delta _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fa75d04c11480d976e1396951e02cbb3c4f71568" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.51ex; height:3.009ex;" alt="{\displaystyle \delta _{ij}}"></span> is the <a href="/wiki/Kronecker_delta" title="Kronecker delta">Kronecker delta</a> (1 if <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i=j}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>i</mi> <mo>=</mo> <mi>j</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle i=j}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/706e0928b2bf0f24076b0c90bb20616ff2068343" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.859ex; height:2.509ex;" alt="{\displaystyle i=j}"></span>, 0 otherwise) and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><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></span><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 }"></span> and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> are the <a href="/wiki/Lam%C3%A9_parameters" title="Lamé parameters">Lamé parameters</a> (<span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> being the material's <a href="/wiki/Shear_modulus" title="Shear modulus">shear modulus</a>). It follows that <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}\partial _{k}u_{k}+\mu \left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <mo>=</mo> <mi>λ</mi> <msub> <mi>δ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>μ</mi> <mrow> <mo>(</mo> <mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}\partial _{k}u_{k}+\mu \left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ca6e4ddee23a0c4165e07aa19e63d58b336a55e2" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:36.153ex; height:5.509ex;" alt="{\displaystyle \tau _{ij}=\lambda \delta _{ij}\sum _{k}\partial _{k}u_{k}+\mu \left(\partial _{i}u_{j}+\partial _{j}u_{i}\right)}"></span> </p><p>From <a href="/wiki/Newton%27s_2nd_law" class="mw-redirect" title="Newton's 2nd law">Newton's law of inertia</a>, one also gets <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \partial _{t}^{2}u_{i}=\sum _{j}\partial _{j}\tau _{ij}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>τ</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> <mi>j</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \partial _{t}^{2}u_{i}=\sum _{j}\partial _{j}\tau _{ij}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/86677fbc56d8794e61cd2213767eeb8ea0a687cc" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:17.207ex; height:5.843ex;" alt="{\displaystyle \rho \partial _{t}^{2}u_{i}=\sum _{j}\partial _{j}\tau _{ij}}"></span> where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/1f7d439671d1289b6a816e6af7a304be40608d64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.202ex; height:2.176ex;" alt="{\displaystyle \rho }"></span> is the <a href="/wiki/Density" title="Density">density</a> (mass per unit volume) of the medium at that point, and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{t}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/ec987d5a79787a5086aa23e31e1d46663f276e7d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.06ex; height:2.509ex;" alt="{\displaystyle \partial _{t}}"></span> denotes partial derivative with respect to time. Combining the last two equations one gets the <i>seismic wave equation in homogeneous media</i> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \partial _{t}^{2}u_{i}=\lambda \partial _{i}\sum _{k}\partial _{k}u_{k}+\mu \sum _{j}{\bigl (}\partial _{i}\partial _{j}u_{j}+\partial _{j}\partial _{j}u_{i}{\bigr )}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mo>=</mo> <mi>λ</mi> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </munder> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>k</mi> </mrow> </msub> <mo>+</mo> <mi>μ</mi> <munder> <mo>∑</mo> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </munder> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-OPEN"> <mo maxsize="1.2em" minsize="1.2em">(</mo> </mrow> </mrow> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <mo>+</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>j</mi> </mrow> </msub> <msub> <mi>u</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>i</mi> </mrow> </msub> <mrow class="MJX-TeXAtom-ORD"> <mrow class="MJX-TeXAtom-CLOSE"> <mo maxsize="1.2em" minsize="1.2em">)</mo> </mrow> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \partial _{t}^{2}u_{i}=\lambda \partial _{i}\sum _{k}\partial _{k}u_{k}+\mu \sum _{j}{\bigl (}\partial _{i}\partial _{j}u_{j}+\partial _{j}\partial _{j}u_{i}{\bigr )}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a6d27cbef3a2f5211b6fb721f72ced6cf944f9f" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:47.264ex; height:5.843ex;" alt="{\displaystyle \rho \partial _{t}^{2}u_{i}=\lambda \partial _{i}\sum _{k}\partial _{k}u_{k}+\mu \sum _{j}{\bigl (}\partial _{i}\partial _{j}u_{j}+\partial _{j}\partial _{j}u_{i}{\bigr )}}"></span> </p><p>Using the <a href="/wiki/Nabla_operator" class="mw-redirect" title="Nabla operator">nabla operator</a> notation of <a href="/wiki/Vector_calculus" title="Vector calculus">vector calculus</a>, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla =(\partial _{1},\partial _{2},\partial _{3})}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>=</mo> <mo stretchy="false">(</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mo>,</mo> <msub> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>3</mn> </mrow> </msub> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla =(\partial _{1},\partial _{2},\partial _{3})}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0189fd825f7d2de767c0f9fa43726d3548e8f115" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.778ex; height:2.843ex;" alt="{\displaystyle \nabla =(\partial _{1},\partial _{2},\partial _{3})}"></span>, with some approximations, this equation can be written as <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho \partial _{t}^{2}{\boldsymbol {u}}=\left(\lambda +2\mu \right)\nabla \left(\nabla \cdot {\boldsymbol {u}}\right)-\mu \nabla \times \left(\nabla \times {\boldsymbol {u}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>ρ</mi> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>=</mo> <mrow> <mo>(</mo> <mrow> <mi>λ</mi> <mo>+</mo> <mn>2</mn> <mi>μ</mi> </mrow> <mo>)</mo> </mrow> <mi mathvariant="normal">∇</mi> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mrow> <mo>)</mo> </mrow> <mo>−</mo> <mi>μ</mi> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \rho \partial _{t}^{2}{\boldsymbol {u}}=\left(\lambda +2\mu \right)\nabla \left(\nabla \cdot {\boldsymbol {u}}\right)-\mu \nabla \times \left(\nabla \times {\boldsymbol {u}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/613f3db2a303682f4799488f7a667a933cd0a7e9" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:43.754ex; height:3.176ex;" alt="{\displaystyle \rho \partial _{t}^{2}{\boldsymbol {u}}=\left(\lambda +2\mu \right)\nabla \left(\nabla \cdot {\boldsymbol {u}}\right)-\mu \nabla \times \left(\nabla \times {\boldsymbol {u}}\right)}"></span> </p><p>Taking the <a href="/wiki/Curl_(mathematics)" title="Curl (mathematics)">curl</a> of this equation and applying vector identities, one gets <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \partial _{t}^{2}(\nabla \times {\boldsymbol {u}})={\frac {\mu }{\rho }}\nabla ^{2}\left(\nabla \times {\boldsymbol {u}}\right)}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msubsup> <mi mathvariant="normal">∂</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>t</mi> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msubsup> <mo stretchy="false">(</mo> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>μ</mi> <mi>ρ</mi> </mfrac> </mrow> <msup> <mi mathvariant="normal">∇</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow> <mo>(</mo> <mrow> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mrow> <mo>)</mo> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \partial _{t}^{2}(\nabla \times {\boldsymbol {u}})={\frac {\mu }{\rho }}\nabla ^{2}\left(\nabla \times {\boldsymbol {u}}\right)}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a418a1716e25b918fe411ee2827e14b6755779c" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.448ex; height:5.343ex;" alt="{\displaystyle \partial _{t}^{2}(\nabla \times {\boldsymbol {u}})={\frac {\mu }{\rho }}\nabla ^{2}\left(\nabla \times {\boldsymbol {u}}\right)}"></span> </p><p>This formula is the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a> applied to the vector quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \times {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>×</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \times {\boldsymbol {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/971412cd7f5dbe7d814b3bf693d116589c0f904b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.359ex; height:2.176ex;" alt="{\displaystyle \nabla \times {\boldsymbol {u}}}"></span>, which is the material's shear strain. Its solutions, the S waves, are <a href="/wiki/Linear_combination" title="Linear combination">linear combinations</a> of <a href="/wiki/Sinusoidal" class="mw-redirect" title="Sinusoidal">sinusoidal</a> <a href="/wiki/Plane_wave" title="Plane wave">plane waves</a> of various <a href="/wiki/Wavelength" title="Wavelength">wavelengths</a> and directions of propagation, but all with the same speed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \beta ={\sqrt {\mu /\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \beta ={\sqrt {\mu /\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7eb99c6cf4f141daa1b775e3def61c3887190883" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:10.52ex; height:3.343ex;" alt="{\textstyle \beta ={\sqrt {\mu /\rho }}}"></span>. Assuming that the medium of propagation is linear, elastic, isotropic, and homogeneous, this equation can be rewritten as <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu =\rho \beta ^{2}=\rho \omega ^{2}/k^{2}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo>=</mo> <mi>ρ</mi> <msup> <mi>β</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mi>ρ</mi> <msup> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <msup> <mi>k</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu =\rho \beta ^{2}=\rho \omega ^{2}/k^{2}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/6a8de7210f088a825cac578a07c99f61fe60f453" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.322ex; height:3.176ex;" alt="{\displaystyle \mu =\rho \beta ^{2}=\rho \omega ^{2}/k^{2}}"></span><sup id="cite_ref-:0_8-0" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> where <i>ω</i> is the angular frequency and <i><span class="texhtml mvar" style="font-style:italic;">k</span></i> is the wavenumber. Thus, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta =\omega /k}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> <mo>=</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta =\omega /k}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e652f7955337629629a49b613e67af9dc8cd808e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.25ex; height:2.843ex;" alt="{\displaystyle \beta =\omega /k}"></span>. </p><p>Taking the <a href="/wiki/Divergence" title="Divergence">divergence</a> of seismic wave equation in homogeneous media, instead of the curl, yields a wave equation describing propagation of the quantity <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \nabla \cdot {\boldsymbol {u}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">∇</mi> <mo>⋅</mo> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \nabla \cdot {\boldsymbol {u}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/eeeb5b1822fcb29f8955a291eb54ed1d0e206be9" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.198ex; height:2.176ex;" alt="{\displaystyle \nabla \cdot {\boldsymbol {u}}}"></span>, which is the material's compression strain. The solutions of this equation, the P waves, travel at the speed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\textstyle \alpha ={\sqrt {(\lambda +2\mu )/\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="false" scriptlevel="0"> <mi>α</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mo stretchy="false">(</mo> <mi>λ</mi> <mo>+</mo> <mn>2</mn> <mi>μ</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\textstyle \alpha ={\sqrt {(\lambda +2\mu )/\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/904a5ae11687294ce9f993e6b6be5777266cd5fd" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.843ex; height:3.343ex;" alt="{\textstyle \alpha ={\sqrt {(\lambda +2\mu )/\rho }}}"></span> that is more than twice the speed <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>β</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \beta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7ed48a5e36207156fb792fa79d29925d2f7901e8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.332ex; height:2.509ex;" alt="{\displaystyle \beta }"></span> of S waves. </p><p>The <a href="/wiki/Steady_state" title="Steady state">steady state</a> SH waves are defined by the <a href="/wiki/Helmholtz_equation" title="Helmholtz equation">Helmholtz equation</a><sup id="cite_ref-9" class="reference"><a href="#cite_note-9"><span class="cite-bracket">[</span>9<span class="cite-bracket">]</span></a></sup> <span class="mwe-math-element"><span class="mwe-math-mathml-display mwe-math-mathml-a11y" style="display: none;"><math display="block" xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \left(\nabla ^{2}+k^{2}\right){\boldsymbol {u}}=0}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow> <mo>(</mo> <mrow> <msup> <mi mathvariant="normal">∇</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> </mrow> <mo>)</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi mathvariant="bold-italic">u</mi> </mrow> <mo>=</mo> <mn>0</mn> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \left(\nabla ^{2}+k^{2}\right){\boldsymbol {u}}=0}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f4f7981fe6edaa0f39186a54cb2e6f69a2b1ebc8" class="mwe-math-fallback-image-display mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.457ex; height:3.343ex;" alt="{\displaystyle \left(\nabla ^{2}+k^{2}\right){\boldsymbol {u}}=0}"></span> where <span class="texhtml mvar" style="font-style:italic;">k</span> is the wave number. </p> <div class="mw-heading mw-heading3"><h3 id="S_waves_in_viscoelastic_materials">S waves in viscoelastic materials</h3></div> <p>Similar to in an elastic medium, in a <a href="/wiki/Viscoelasticity" title="Viscoelasticity">viscoelastic</a> material, the speed of a shear wave is described by a similar relationship <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\omega )=\omega /k(\omega )={\sqrt {\mu (\omega )/\rho }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi>ω</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>k</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>ρ</mi> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(\omega )=\omega /k(\omega )={\sqrt {\mu (\omega )/\rho }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b12c1d2d4694d4057f67a354aaa19b8c813afbc4" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:26.878ex; height:4.843ex;" alt="{\displaystyle c(\omega )=\omega /k(\omega )={\sqrt {\mu (\omega )/\rho }}}"></span>, however, here, <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/9fd47b2a39f7a7856952afec1f1db72c67af6161" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.402ex; height:2.176ex;" alt="{\displaystyle \mu }"></span> is a complex, frequency-dependent shear modulus and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c(\omega )}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>c</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle c(\omega )}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2b3ba17b314be72f43454f3e89f3f0396b71f4a3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.262ex; height:2.843ex;" alt="{\displaystyle c(\omega )}"></span> is the frequency dependent phase velocity.<sup id="cite_ref-:0_8-1" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> One common approach to describing the shear modulus in viscoelastic materials is through the <a href="/wiki/Kelvin%E2%80%93Voigt_material" title="Kelvin–Voigt material">Voigt Model</a> which states: <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu (\omega )=\mu _{0}+i\omega \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>μ</mi> <mo stretchy="false">(</mo> <mi>ω</mi> <mo stretchy="false">)</mo> <mo>=</mo> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> <mo>+</mo> <mi>i</mi> <mi>ω</mi> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu (\omega )=\mu _{0}+i\omega \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/aad52f0b79e5be807f0c92c49c011c4fc1fca318" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.469ex; height:2.843ex;" alt="{\displaystyle \mu (\omega )=\mu _{0}+i\omega \eta }"></span>, where <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>μ</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>0</mn> </mrow> </msub> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.456ex; height:2.176ex;" alt="{\displaystyle \mu _{0}}"></span> is the stiffness of the material and <span class="mwe-math-element"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>η</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \eta }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e4d701857cf5fbec133eebaf94deadf722537f64" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }"></span> is the viscosity.<sup id="cite_ref-:0_8-2" class="reference"><a href="#cite_note-:0-8"><span class="cite-bracket">[</span>8<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="S_wave_technology">S wave technology</h2></div> <div class="mw-heading mw-heading3"><h3 id="Magnetic_resonance_elastography">Magnetic resonance elastography</h3></div> <p><a href="/wiki/Magnetic_resonance_elastography" title="Magnetic resonance elastography">Magnetic resonance elastography</a> (MRE) is a method for studying the properties of biological materials in living organisms by propagating shear waves at desired frequencies throughout the desired organic tissue.<sup id="cite_ref-:1_10-0" class="reference"><a href="#cite_note-:1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> This method uses a vibrator to send the shear waves into the tissue and <a href="/wiki/Magnetic_resonance_imaging" title="Magnetic resonance imaging">magnetic resonance imaging</a> to view the response in the tissue.<sup id="cite_ref-11" class="reference"><a href="#cite_note-11"><span class="cite-bracket">[</span>11<span class="cite-bracket">]</span></a></sup> The measured wave speed and wavelengths are then measured to determine elastic properties such as the <a href="/wiki/Shear_modulus#:~:text=In_materials_science,_shear_modulus,Common_symbols" title="Shear modulus">shear modulus</a>. MRE has seen use in studies of a variety of human tissues including liver, brain, and bone tissues.<sup id="cite_ref-:1_10-1" class="reference"><a href="#cite_note-:1-10"><span class="cite-bracket">[</span>10<span class="cite-bracket">]</span></a></sup> </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div> <ul><li><a href="/wiki/Earthquake_Early_Warning_(Japan)" title="Earthquake Early Warning (Japan)">Earthquake Early Warning (Japan)</a></li> <li><a href="/wiki/Lamb_waves" title="Lamb waves">Lamb waves</a></li> <li><a href="/wiki/Longitudinal_wave" title="Longitudinal wave">Longitudinal wave</a></li> <li><a href="/wiki/Love_wave" title="Love wave">Love wave</a></li> <li><a href="/wiki/P_wave" title="P wave">P wave</a></li> <li><a href="/wiki/Rayleigh_wave" title="Rayleigh wave">Rayleigh wave</a></li> <li><a href="/wiki/Seismic_wave" title="Seismic wave">Seismic wave</a></li> <li><a href="/wiki/Shear_wave_splitting" title="Shear wave splitting">Shear wave splitting</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2></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"> <div class="mw-references-wrap mw-references-columns"><ol class="references"> <li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text"><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 class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.mtu.edu/geo/community/seismology/learn/seismology-study/">"Seismology | UPSeis | Michigan Tech"</a>. 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Retrieved <span class="nowrap">2019-12-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=Earth+Observatory+of+Singapore&rft.atitle=Why+can%27t+S-waves+travel+through+liquids%3F&rft_id=https%3A%2F%2Fwww.earthobservatory.sg%2Ffaq-on-earth-sciences%2Fwhy-cant-s-waves-travel-through-liquids&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></span> </li> <li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGreenwoodBamberger2002" class="citation journal cs1">Greenwood, Margaret Stautberg; Bamberger, Judith Ann (August 2002). "Measurement of viscosity and shear wave velocity of a liquid or slurry for on-line process control". <i>Ultrasonics</i>. <b>39</b> (9): 623–630. <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%2Fs0041-624x%2802%2900372-4">10.1016/s0041-624x(02)00372-4</a>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/12206629">12206629</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Ultrasonics&rft.atitle=Measurement+of+viscosity+and+shear+wave+velocity+of+a+liquid+or+slurry+for+on-line+process+control&rft.volume=39&rft.issue=9&rft.pages=623-630&rft.date=2002-08&rft_id=info%3Adoi%2F10.1016%2Fs0041-624x%2802%2900372-4&rft_id=info%3Apmid%2F12206629&rft.aulast=Greenwood&rft.aufirst=Margaret+Stautberg&rft.au=Bamberger%2C+Judith+Ann&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></span> </li> <li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://www.researchgate.net/post/Do_viscous_fluids_support_shear_waves_propagation">"Do viscous fluids support shear waves propagation?"</a>. <i>ResearchGate</i><span class="reference-accessdate">. Retrieved <span class="nowrap">2019-12-06</span></span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=unknown&rft.jtitle=ResearchGate&rft.atitle=Do+viscous+fluids+support+shear+waves+propagation%3F&rft_id=https%3A%2F%2Fwww.researchgate.net%2Fpost%2FDo_viscous_fluids_support_shear_waves_propagation&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span><sup class="noprint Inline-Template" style="white-space:nowrap;">[<i><a href="/wiki/Wikipedia:Reliable_sources" title="Wikipedia:Reliable sources"><span title="The material near this tag may rely on an unreliable source. (May 2024)">unreliable source?</span></a></i>]</sup></span> </li> <li id="cite_note-UIC-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-UIC_6-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://web.archive.org/web/20020507161231/http://tigger.uic.edu/~rdemar/geol107/lect16.htm">"Lecture 16 Seismographs and the earth's interior"</a>. University of Illinois at Chicago. 17 July 1997. Archived from <a rel="nofollow" class="external text" href="http://tigger.uic.edu/~rdemar/geol107/lect16.htm">the original</a> on 7 May 2002<span class="reference-accessdate">. Retrieved <span class="nowrap">8 June</span> 2010</span>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=Lecture+16+Seismographs+and+the+earth%27s+interior&rft.pub=University+of+Illinois+at+Chicago&rft.date=1997-07-17&rft_id=http%3A%2F%2Ftigger.uic.edu%2F~rdemar%2Fgeol107%2Flect16.htm&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></span> </li> <li id="cite_note-pois1831-7"><span class="mw-cite-backlink"><b><a href="#cite_ref-pois1831_7-0">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFPoisson1831" class="citation journal cs1 cs1-prop-foreign-lang-source">Poisson, S. D. (1831). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=NJ04Yqm2iFkC&pg=PA549">"Mémoire sur la propagation du mouvement dans les milieux élastiques"</a> [Memoir on the propagation of motion in elastic media]. <i>Mémoires de l'Académie des Sciences de l'Institut de France</i> (in French). <b>10</b>: 549–605.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=M%C3%A9moires+de+l%27Acad%C3%A9mie+des+Sciences+de+l%27Institut+de+France&rft.atitle=M%C3%A9moire+sur+la+propagation+du+mouvement+dans+les+milieux+%C3%A9lastiques&rft.volume=10&rft.pages=549-605&rft.date=1831&rft.aulast=Poisson&rft.aufirst=S.+D.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DNJ04Yqm2iFkC%26pg%3DPA549&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span> From p.595: "<i>On verra aisément que cet ébranlement donnera naissance à deux ondes sphériques qui se propageront uniformément, l'une avec une vitesse </i>a<i>, l'autre avec une vitesse </i>b<i> ou </i>a<i> / <span class="nowrap">√<span style="border-top:1px solid; padding:0 0.1em;">3</span></span></i>" ... (One will easily see that this quake will give birth to two spherical waves that will be propagated uniformly, one with a speed <i>a</i>, the other with a speed <i>b</i> or <i>a</i> /√3 ... ) From p.602: ... "<i>à une grande distance de l'ébranlement primitif, et lorsque les ondes mobiles sont devenues sensiblement planes dans chaque partie très-petite par rapport à leurs surfaces entières, il ne subsiste plus que des vitesses propres des molécules, normales ou parallèles à ces surfaces; les vitesses normal ayant lieu dans les ondes de la première espèce, où elles sont accompagnées de dilations qui leur sont proportionnelles, et les vitesses parallèles appartenant aux ondes de la seconde espèce, où elles ne sont accompagnées d'aucune dilatation ou condensation de volume, mais seulement de dilatations et de condensations linéaires.</i>" ( ... at a great distance from the original quake, and when the moving waves have become roughly planes in every tiny part in relation to their entire surface, there remain [in the elastic solid of the Earth] only the molecules' own speeds, normal or parallel to these surfaces; the normal speeds occur in waves of the first type, where they are accompanied by expansions that are proportional to them, and the parallel speeds belonging to waves of the second type, where they are not accompanied by any expansion or contraction of volume, but only by linear stretchings and squeezings.)</span> </li> <li id="cite_note-:0-8"><span class="mw-cite-backlink">^ <a href="#cite_ref-:0_8-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:0_8-1"><sup><i><b>b</b></i></sup></a> <a href="#cite_ref-:0_8-2"><sup><i><b>c</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFRouzeDengTrutnaPalmeri2018" class="citation journal cs1">Rouze; Deng; Trutna; Palmeri; Nightengale (May 2018). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5972540">"Characterization of Viscoelastic Materials Using Group Shear Wave Speeds"</a>. <i>Institute of Electrical and Electronics Engineers</i>. <b>65</b> (5): 780–794. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTUFFC.2018.2815505">10.1109/TUFFC.2018.2815505</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5972540">5972540</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/29733281">29733281</a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.jtitle=Institute+of+Electrical+and+Electronics+Engineers&rft.atitle=Characterization+of+Viscoelastic+Materials+Using+Group+Shear+Wave+Speeds&rft.volume=65&rft.issue=5&rft.pages=780-794&rft.date=2018-05&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5972540%23id-name%3DPMC&rft_id=info%3Apmid%2F29733281&rft_id=info%3Adoi%2F10.1109%2FTUFFC.2018.2815505&rft.au=Rouze&rft.au=Deng&rft.au=Trutna&rft.au=Palmeri&rft.au=Nightengale&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC5972540&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></span> </li> <li id="cite_note-9"><span class="mw-cite-backlink"><b><a href="#cite_ref-9">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFGraff2012" class="citation book cs1">Graff, Karl F. (2012-04-26). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=jorRAgAAQBAJ&pg=PP1"><i>Wave Motion in Elastic Solids</i></a>. Courier Corporation. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/978-0-486-13957-9" title="Special:BookSources/978-0-486-13957-9"><bdi>978-0-486-13957-9</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+Motion+in+Elastic+Solids&rft.pub=Courier+Corporation&rft.date=2012-04-26&rft.isbn=978-0-486-13957-9&rft.aulast=Graff&rft.aufirst=Karl+F.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DjorRAgAAQBAJ%26pg%3DPP1&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></span> </li> <li id="cite_note-:1-10"><span class="mw-cite-backlink">^ <a href="#cite_ref-:1_10-0"><sup><i><b>a</b></i></sup></a> <a href="#cite_ref-:1_10-1"><sup><i><b>b</b></i></sup></a></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFTwetenOkamotoSchmidtGarbow2015" class="citation journal cs1">Tweten, Dennis J.; Okamoto, Ruth J.; Schmidt, John L.; Garbow, Joel R.; Bayly, Philip V. (November 2015). <a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4663187">"Estimation of material parameters from slow and fast shear waves in an incompressible, transversely isotropic material"</a>. <i>Journal of Biomechanics</i>. <b>48</b> (15): 4002–4009. <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.jbiomech.2015.09.009">10.1016/j.jbiomech.2015.09.009</a>. <a href="/wiki/PMC_(identifier)" class="mw-redirect" title="PMC (identifier)">PMC</a> <span class="id-lock-free" title="Freely accessible"><a rel="nofollow" class="external text" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4663187">4663187</a></span>. <a href="/wiki/PMID_(identifier)" class="mw-redirect" title="PMID (identifier)">PMID</a> <a rel="nofollow" class="external text" href="https://pubmed.ncbi.nlm.nih.gov/26476762">26476762</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+Biomechanics&rft.atitle=Estimation+of+material+parameters+from+slow+and+fast+shear+waves+in+an+incompressible%2C+transversely+isotropic+material&rft.volume=48&rft.issue=15&rft.pages=4002-4009&rft.date=2015-11&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4663187%23id-name%3DPMC&rft_id=info%3Apmid%2F26476762&rft_id=info%3Adoi%2F10.1016%2Fj.jbiomech.2015.09.009&rft.aulast=Tweten&rft.aufirst=Dennis+J.&rft.au=Okamoto%2C+Ruth+J.&rft.au=Schmidt%2C+John+L.&rft.au=Garbow%2C+Joel+R.&rft.au=Bayly%2C+Philip+V.&rft_id=https%3A%2F%2Fwww.ncbi.nlm.nih.gov%2Fpmc%2Farticles%2FPMC4663187&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></span> </li> <li id="cite_note-11"><span class="mw-cite-backlink"><b><a href="#cite_ref-11">^</a></b></span> <span class="reference-text"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite class="citation web cs1"><a rel="nofollow" class="external text" href="https://medicine.utah.edu/radiology/research/labs/fus-lab/breast-mrgfus/mr-elastography">"MR Shear Wave Elastography"</a>. University of Utah Health. 10 November 2021.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=unknown&rft.btitle=MR+Shear+Wave+Elastography&rft.pub=University+of+Utah+Health&rft.date=2021-11-10&rft_id=https%3A%2F%2Fmedicine.utah.edu%2Fradiology%2Fresearch%2Flabs%2Ffus-lab%2Fbreast-mrgfus%2Fmr-elastography&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></span> </li> </ol></div></div> <div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div> <ul><li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFShearer1999" class="citation book cs1">Shearer, Peter (1999). <i>Introduction to Seismology</i> (1st ed.). Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-66023-8" title="Special:BookSources/0-521-66023-8"><bdi>0-521-66023-8</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Introduction+to+Seismology&rft.edition=1st&rft.pub=Cambridge+University+Press&rft.date=1999&rft.isbn=0-521-66023-8&rft.aulast=Shearer&rft.aufirst=Peter&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFAkiRichards2002" class="citation book cs1"><a href="/wiki/Keiiti_Aki" title="Keiiti Aki">Aki, Keiiti</a>; <a href="/wiki/Paul_G._Richards" title="Paul G. Richards">Richards, Paul G.</a> (2002). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=sRhawFG5_EcC"><i>Quantitative Seismology</i></a> (2nd ed.). University Science Books. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-935702-96-2" title="Special:BookSources/0-935702-96-2"><bdi>0-935702-96-2</bdi></a>.</cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=Quantitative+Seismology&rft.edition=2nd&rft.pub=University+Science+Books&rft.date=2002&rft.isbn=0-935702-96-2&rft.aulast=Aki&rft.aufirst=Keiiti&rft.au=Richards%2C+Paul+G.&rft_id=https%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3DsRhawFG5_EcC&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222"><cite id="CITEREFFowler1990" class="citation book cs1"><a href="/wiki/Mary_Fowler_(geologist)" class="mw-redirect" title="Mary Fowler (geologist)">Fowler, C. M. R.</a> (1990). <span class="id-lock-registration" title="Free registration required"><a rel="nofollow" class="external text" href="https://archive.org/details/solidearthintrod0000fowl"><i>The solid earth</i></a></span>. Cambridge, UK: Cambridge University Press. <a href="/wiki/ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <a href="/wiki/Special:BookSources/0-521-38590-3" title="Special:BookSources/0-521-38590-3"><bdi>0-521-38590-3</bdi></a>. <q>S-wave.</q></cite><span title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rft.genre=book&rft.btitle=The+solid+earth&rft.place=Cambridge%2C+UK&rft.pub=Cambridge+University+Press&rft.date=1990&rft.isbn=0-521-38590-3&rft.aulast=Fowler&rft.aufirst=C.+M.+R.&rft_id=https%3A%2F%2Farchive.org%2Fdetails%2Fsolidearthintrod0000fowl&rfr_id=info%3Asid%2Fen.wikipedia.org%3AS+wave" class="Z3988"></span></li></ul> <div class="navbox-styles"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><style data-mw-deduplicate="TemplateStyles:r1236075235">.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output 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aria-labelledby="Geotechnical_engineering" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1129693374"><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1239400231"><div class="navbar plainlinks hlist navbar-mini"><ul><li class="nv-view"><a href="/wiki/Template:Geotechnical_engineering" title="Template:Geotechnical engineering"><abbr title="View this template">v</abbr></a></li><li class="nv-talk"><a href="/wiki/Template_talk:Geotechnical_engineering" title="Template talk:Geotechnical engineering"><abbr title="Discuss this template">t</abbr></a></li><li class="nv-edit"><a href="/wiki/Special:EditPage/Template:Geotechnical_engineering" title="Special:EditPage/Template:Geotechnical engineering"><abbr title="Edit this template">e</abbr></a></li></ul></div><div id="Geotechnical_engineering" style="font-size:114%;margin:0 4em"><a href="/wiki/Geotechnical_engineering" title="Geotechnical engineering">Geotechnical engineering</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="/wiki/Offshore_geotechnical_engineering" title="Offshore geotechnical engineering">Offshore geotechnical engineering</a></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geotechnical_investigation" title="Geotechnical investigation">Investigation</a> <br />and<br />instrumentation</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Field (<i>in situ</i>)</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"><div class="div-col" style="column-width: 33em;"> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Core_sample.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Core_sample.svg/20px-Core_sample.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Core_sample.svg/30px-Core_sample.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5e/Core_sample.svg/40px-Core_sample.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Core_drill" title="Core drill">Core drill</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Cone_penetration_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Cone_penetration_test.svg/20px-Cone_penetration_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/28/Cone_penetration_test.svg/30px-Cone_penetration_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/28/Cone_penetration_test.svg/40px-Cone_penetration_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Cone_penetration_test" title="Cone penetration test">Cone penetration test</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Geo-electrical_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Geo-electrical_sounding.svg/20px-Geo-electrical_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/10/Geo-electrical_sounding.svg/30px-Geo-electrical_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/10/Geo-electrical_sounding.svg/40px-Geo-electrical_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Geo-electrical_sounding&action=edit&redlink=1" class="new" title="Geo-electrical sounding (page does not exist)">Geo-electrical sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:In_situ_permeameter_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/In_situ_permeameter_test.svg/20px-In_situ_permeameter_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/7b/In_situ_permeameter_test.svg/30px-In_situ_permeameter_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/7b/In_situ_permeameter_test.svg/40px-In_situ_permeameter_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Permeability_(Earth_sciences)" class="mw-redirect" title="Permeability (Earth sciences)">Permeability test</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Load_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Load_test.svg/20px-Load_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Load_test.svg/30px-Load_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/a2/Load_test.svg/40px-Load_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Load_test" class="mw-redirect" title="Load test">Load test</a> <ul><li><a href="/wiki/Static_load_testing" title="Static load testing">Static</a></li> <li><a href="/wiki/Dynamic_load_testing" title="Dynamic load testing">Dynamic</a></li> <li><a href="/wiki/Statnamic_load_test" title="Statnamic load test">Statnamic</a></li></ul></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Pore_pressure_measurement.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Pore_pressure_measurement.svg/20px-Pore_pressure_measurement.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Pore_pressure_measurement.svg/30px-Pore_pressure_measurement.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/fe/Pore_pressure_measurement.svg/40px-Pore_pressure_measurement.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> Pore pressure measurement <ul><li><a href="/wiki/Piezometer" class="mw-redirect" title="Piezometer">Piezometer</a></li> <li><a href="/wiki/Well#Classification" title="Well">Well</a></li></ul></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Ram_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Ram_sounding.svg/20px-Ram_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Ram_sounding.svg/30px-Ram_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Ram_sounding.svg/40px-Ram_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Ram_sounding&action=edit&redlink=1" class="new" title="Ram sounding (page does not exist)">Ram sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Rock_control_drilling.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Rock_control_drilling.svg/20px-Rock_control_drilling.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Rock_control_drilling.svg/30px-Rock_control_drilling.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/e/e4/Rock_control_drilling.svg/40px-Rock_control_drilling.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Rock_control_drilling&action=edit&redlink=1" class="new" title="Rock control drilling (page does not exist)">Rock control drilling</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Rotary_pressure_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Rotary_pressure_sounding.svg/20px-Rotary_pressure_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/8/81/Rotary_pressure_sounding.svg/30px-Rotary_pressure_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/8/81/Rotary_pressure_sounding.svg/40px-Rotary_pressure_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Rotary-pressure_sounding" title="Rotary-pressure sounding">Rotary-pressure sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Rotary_weight_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Rotary_weight_sounding.svg/20px-Rotary_weight_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Rotary_weight_sounding.svg/30px-Rotary_weight_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Rotary_weight_sounding.svg/40px-Rotary_weight_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Rotary_weight_sounding&action=edit&redlink=1" class="new" title="Rotary weight sounding (page does not exist)">Rotary weight sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Sample_series.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Sample_series.svg/20px-Sample_series.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/05/Sample_series.svg/30px-Sample_series.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/05/Sample_series.svg/40px-Sample_series.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Soil_test#Soil_testing" title="Soil test">Sample series</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Screw_plate_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Screw_plate_test.svg/20px-Screw_plate_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/75/Screw_plate_test.svg/30px-Screw_plate_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/75/Screw_plate_test.svg/40px-Screw_plate_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Screw_plate_test&action=edit&redlink=1" class="new" title="Screw plate test (page does not exist)">Screw plate test</a></li></ul> <ul><li><a href="/wiki/Deformation_monitoring" title="Deformation monitoring">Deformation monitoring</a> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Inclinometer.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Inclinometer.svg/20px-Inclinometer.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Inclinometer.svg/30px-Inclinometer.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/c/c6/Inclinometer.svg/40px-Inclinometer.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Inclinometer" title="Inclinometer">Inclinometer</a></li> <li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Settlement_recordings.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Settlement_recordings.svg/20px-Settlement_recordings.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/3/35/Settlement_recordings.svg/30px-Settlement_recordings.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/3/35/Settlement_recordings.svg/40px-Settlement_recordings.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Soil_consolidation" title="Soil consolidation">Settlement recordings</a></li></ul></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Shear_vane_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Shear_vane_test.svg/20px-Shear_vane_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Shear_vane_test.svg/30px-Shear_vane_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/a/ae/Shear_vane_test.svg/40px-Shear_vane_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Shear_vane_test" title="Shear vane test">Shear vane test</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Simple_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Simple_sounding.svg/20px-Simple_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Simple_sounding.svg/30px-Simple_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/d/d4/Simple_sounding.svg/40px-Simple_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/w/index.php?title=Simple_sounding&action=edit&redlink=1" class="new" title="Simple sounding (page does not exist)">Simple sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Standard_penetration_test.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Standard_penetration_test.svg/20px-Standard_penetration_test.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Standard_penetration_test.svg/30px-Standard_penetration_test.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/0/0d/Standard_penetration_test.svg/40px-Standard_penetration_test.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Standard_penetration_test" title="Standard penetration test">Standard penetration test</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Total_sounding.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Total_sounding.svg/20px-Total_sounding.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/6/60/Total_sounding.svg/30px-Total_sounding.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/6/60/Total_sounding.svg/40px-Total_sounding.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Total_sounding" title="Total sounding">Total sounding</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Trial_pit.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Trial_pit.svg/20px-Trial_pit.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/7/71/Trial_pit.svg/30px-Trial_pit.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/7/71/Trial_pit.svg/40px-Trial_pit.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Trial_pit" title="Trial pit">Trial pit</a></li></ul> <ul><li><span class="noviewer" typeof="mw:File"><a href="/wiki/File:Visible_rock.svg" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Visible_rock.svg/20px-Visible_rock.svg.png" decoding="async" width="20" height="20" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/14/Visible_rock.svg/30px-Visible_rock.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/14/Visible_rock.svg/40px-Visible_rock.svg.png 2x" data-file-width="512" data-file-height="512" /></a></span> <a href="/wiki/Bedrock" title="Bedrock">Visible bedrock</a></li></ul> <ul><li><a href="/wiki/Nuclear_densometer" class="mw-redirect" title="Nuclear densometer">Nuclear densometer test</a></li></ul> <ul><li><a href="/wiki/Exploration_geophysics" title="Exploration geophysics">Exploration geophysics</a></li></ul> <ul><li><a href="/wiki/Crosshole_sonic_logging" title="Crosshole sonic logging">Crosshole sonic logging</a></li></ul> <ul><li><a href="/wiki/Pile_integrity_test" title="Pile integrity test">Pile integrity test</a></li></ul> <ul><li><a href="/wiki/Wave_equation_analysis" title="Wave equation analysis">Wave equation analysis</a></li></ul></div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;"><a href="/wiki/Soil_test" title="Soil test">Laboratory <br />testing</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <li><a href="/wiki/Soil_classification" title="Soil classification">Soil classification</a></li> <li><a href="/wiki/Atterberg_limits" title="Atterberg limits">Atterberg limits</a></li> <li><a href="/wiki/California_bearing_ratio" title="California bearing ratio">California bearing ratio</a></li> <li><a href="/wiki/Direct_shear_test" title="Direct shear test">Direct shear test</a></li> <li><a href="/wiki/Hydrometer" title="Hydrometer">Hydrometer</a></li> <li><a href="/wiki/Proctor_compaction_test" title="Proctor compaction test">Proctor compaction test</a></li> <li><a href="/wiki/R-value_(soils)" title="R-value (soils)">R-value</a></li> <li><a href="/wiki/Sieve_analysis" title="Sieve analysis">Sieve analysis</a></li> <li><a href="/wiki/Triaxial_shear_test" title="Triaxial shear test">Triaxial shear test</a></li> <li><a href="/wiki/Oedometer_test" title="Oedometer test">Oedometer test</a></li> <li><a href="/wiki/Hydraulic_conductivity#Experimental_approach" title="Hydraulic conductivity">Hydraulic conductivity tests</a></li> <li><a href="/wiki/Water_content#Measurement" title="Water content">Water content tests</a></li> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Soil" title="Soil">Soil</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Types</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Clay" title="Clay">Clay</a></li> <li><a href="/wiki/Silt" title="Silt">Silt</a></li> <li><a href="/wiki/Sand" title="Sand">Sand</a></li> <li><a href="/wiki/Gravel" title="Gravel">Gravel</a></li> <li><a href="/wiki/Peat" title="Peat">Peat</a></li> <li><a href="/wiki/Loam" title="Loam">Loam</a></li> <li><a href="/wiki/Loess" title="Loess">Loess</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Properties</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Hydraulic_conductivity" title="Hydraulic conductivity">Hydraulic conductivity</a></li> <li><a href="/wiki/Water_content" title="Water content">Water content</a></li> <li><a href="/wiki/Void_ratio" title="Void ratio">Void ratio</a></li> <li><a href="/wiki/Bulk_density" title="Bulk density">Bulk density</a></li> <li><a href="/wiki/Thixotropy" title="Thixotropy">Thixotropy</a></li> <li><a href="/wiki/Reynolds%27_dilatancy" class="mw-redirect" title="Reynolds' dilatancy">Reynolds' dilatancy</a></li> <li><a href="/wiki/Angle_of_repose" title="Angle of repose">Angle of repose</a></li> <li><a href="/wiki/Friction#Angle_of_friction" title="Friction">Friction angle</a></li> <li><a href="/wiki/Cohesion_(geology)" title="Cohesion (geology)">Cohesion</a></li> <li><a href="/wiki/Porosity" title="Porosity">Porosity</a></li> <li><a href="/wiki/Permeability_(earth_sciences)" class="mw-redirect" title="Permeability (earth sciences)">Permeability</a></li> <li><a href="/wiki/Specific_storage" title="Specific storage">Specific storage</a></li> <li><a href="/wiki/Shear_strength_(soil)" title="Shear strength (soil)">Shear strength</a></li> <li><a href="/wiki/Soil_liquefaction" title="Soil liquefaction">Sensitivity</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Structures <br />(<a href="/wiki/Soil-structure_interaction" title="Soil-structure interaction">Interaction</a>)</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Natural features</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Topography" title="Topography">Topography</a></li> <li><a href="/wiki/Vegetation" title="Vegetation">Vegetation</a></li> <li><a href="/wiki/Terrain" title="Terrain">Terrain</a></li> <li><a href="/wiki/Topsoil" title="Topsoil">Topsoil</a></li> <li><a href="/wiki/Water_table" title="Water table">Water table</a></li> <li><a href="/wiki/Bedrock" title="Bedrock">Bedrock</a></li> <li><a href="/wiki/Subgrade" title="Subgrade">Subgrade</a></li> <li><a href="/wiki/Subsoil" title="Subsoil">Subsoil</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;"><a href="/wiki/Earthworks_(engineering)" title="Earthworks (engineering)">Earthworks</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li>Shoring structures <ul><li><a href="/wiki/Retaining_wall" title="Retaining wall">Retaining walls</a></li> <li><a href="/wiki/Gabion" title="Gabion">Gabion</a></li> <li><a href="/wiki/Ground_freezing" title="Ground freezing">Ground freezing</a></li> <li><a href="/wiki/Mechanically_stabilized_earth" title="Mechanically stabilized earth">Mechanically stabilized earth</a></li> <li><a href="/wiki/Pressure_grouting" title="Pressure grouting">Pressure grouting</a></li> <li><a href="/wiki/Slurry_wall" title="Slurry wall">Slurry wall</a></li> <li><a href="/wiki/Soil_nailing" title="Soil nailing">Soil nailing</a></li> <li><a href="/wiki/Tieback_(geotechnical)" title="Tieback (geotechnical)">Tieback</a></li></ul></li> <li><a href="/wiki/Land_development" title="Land development">Land development</a></li> <li><a href="/wiki/Landfill" title="Landfill">Landfill</a></li> <li><a href="/wiki/Digging" title="Digging">Excavation</a></li> <li><a href="/wiki/Trench" title="Trench">Trench</a></li> <li><a href="/wiki/Embankment_(earthworks)" title="Embankment (earthworks)">Embankment</a></li> <li><a href="/wiki/Cut_(earthworks)" title="Cut (earthworks)">Cut</a></li> <li><a href="/wiki/Causeway" title="Causeway">Causeway</a></li> <li><a href="/wiki/Terrace_(earthworks)" title="Terrace (earthworks)">Terracing</a></li> <li><a href="/wiki/Tunnel#Cut-and-cover" title="Tunnel">Cut-and-cover</a></li> <li><a href="/wiki/Cut_and_fill" title="Cut and fill">Cut and fill</a></li> <li><a href="/wiki/Fill_dirt" title="Fill dirt">Fill dirt</a></li> <li><a href="/wiki/Grading_(engineering)" class="mw-redirect" title="Grading (engineering)">Grading</a></li> <li><a href="/wiki/Land_reclamation" title="Land reclamation">Land reclamation</a></li> <li><a href="/wiki/Track_bed" title="Track bed">Track bed</a></li> <li><a href="/wiki/Erosion_control" title="Erosion control">Erosion control</a></li> <li><a href="/wiki/Earth_structure" title="Earth structure">Earth structure</a></li> <li><a href="/wiki/Expanded_clay_aggregate" title="Expanded clay aggregate">Expanded clay aggregate</a></li> <li><a href="/wiki/Crushed_stone" title="Crushed stone">Crushed stone</a></li> <li><a href="/wiki/Geosynthetics" title="Geosynthetics">Geosynthetics</a> <ul><li><a href="/wiki/Geotextile" title="Geotextile">Geotextile</a></li> <li><a href="/wiki/Geomembrane" title="Geomembrane">Geomembrane</a></li> <li><a href="/wiki/Geosynthetic_clay_liner" title="Geosynthetic clay liner">Geosynthetic clay liner</a></li> <li><a href="/wiki/Cellular_confinement" title="Cellular confinement">Cellular confinement</a></li></ul></li> <li><a href="/wiki/Infiltration_(hydrology)" title="Infiltration (hydrology)">Infiltration</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;"><a href="/wiki/Foundation_(engineering)" title="Foundation (engineering)">Foundations</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Shallow_foundation" title="Shallow foundation">Shallow</a></li> <li><a href="/wiki/Deep_foundation" title="Deep foundation">Deep</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Soil_mechanics" title="Soil mechanics">Mechanics</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Forces</th><td class="navbox-list-with-group navbox-list navbox-even" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Effective_stress" title="Effective stress">Effective stress</a></li> <li><a href="/wiki/Pore_water_pressure" title="Pore water pressure">Pore water pressure</a></li> <li><a href="/wiki/Lateral_earth_pressure" title="Lateral earth pressure">Lateral earth pressure</a></li> <li><a href="/wiki/Overburden_pressure" title="Overburden pressure">Overburden pressure</a></li> <li><a href="/wiki/Preconsolidation_pressure" title="Preconsolidation pressure">Preconsolidation pressure</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:10.0em;font-weight: normal;">Phenomena/<br />problems</th><td class="navbox-list-with-group navbox-list navbox-odd" style="padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Permafrost" title="Permafrost">Permafrost</a></li> <li><a href="/wiki/Frost_heaving" title="Frost heaving">Frost heaving</a></li> <li><a href="/wiki/Consolidation_(soil)" class="mw-redirect" title="Consolidation (soil)">Consolidation</a></li> <li><a href="/wiki/Soil_compaction" title="Soil compaction">Compaction</a></li> <li><a href="/wiki/Earthquake" title="Earthquake">Earthquake</a> <ul><li><a href="/wiki/Response_spectrum" title="Response spectrum">Response spectrum</a></li> <li><a href="/wiki/Seismic_hazard" title="Seismic hazard">Seismic hazard</a></li> <li><a class="mw-selflink selflink">Shear wave</a></li></ul></li> <li><a href="/wiki/Landslide" title="Landslide">Landslide</a> analysis <ul><li><a href="/wiki/Slope_stability_analysis" title="Slope stability analysis">Stability analysis</a></li> <li><a href="/wiki/Landslide_mitigation" title="Landslide mitigation">Mitigation</a></li> <li><a href="/wiki/Landslide_classification" title="Landslide classification">Classification</a></li> <li><a href="/wiki/Sliding_criterion_(geotechnical_engineering)" title="Sliding criterion (geotechnical engineering)">Sliding criterion</a></li> <li><a href="/wiki/Road#Slab_stabilization" title="Road">Slab stabilisation</a></li></ul></li> <li><a href="/wiki/Bearing_capacity" title="Bearing capacity">Bearing capacity</a> * <a href="/wiki/Stress_distribution_in_soil" title="Stress distribution in soil">Stress distribution in soil</a></li></ul> </div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Software" title="Software">Numerical analysis <br />software</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/SEEP2D" title="SEEP2D">SEEP2D</a></li> <li><a href="/w/index.php?title=STABL&action=edit&redlink=1" class="new" title="STABL (page does not exist)">STABL</a></li> <li><a href="/wiki/SVFlux" title="SVFlux">SVFlux</a></li> <li><a href="/wiki/SVSlope" title="SVSlope">SVSlope</a></li> <li><a href="/wiki/UTEXAS" title="UTEXAS">UTEXAS</a></li> <li><a href="/wiki/Plaxis" title="Plaxis">Plaxis</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Geoprofessions" title="Geoprofessions">Related fields</a></th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Geology" title="Geology">Geology</a></li> <li><a href="/wiki/Geochemistry" title="Geochemistry">Geochemistry</a></li> <li><a href="/wiki/Petrology" title="Petrology">Petrology</a></li> <li><a href="/wiki/Earthquake_engineering" title="Earthquake engineering">Earthquake engineering</a></li> <li><a href="/wiki/Geomorphology" title="Geomorphology">Geomorphology</a></li> <li><a href="/wiki/Soil_science" title="Soil science">Soil science</a></li> <li><a href="/wiki/Hydrology" title="Hydrology">Hydrology</a></li> <li><a href="/wiki/Hydrogeology" title="Hydrogeology">Hydrogeology</a></li> <li><a href="/wiki/Biogeography" title="Biogeography">Biogeography</a></li> <li><a href="/wiki/Earth_materials" title="Earth materials">Earth 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