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String vibration - Wikipedia

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mw-list-item"><a href="https://bn.wikipedia.org/wiki/%E0%A6%B8%E0%A7%8D%E0%A6%9F%E0%A7%8D%E0%A6%B0%E0%A6%BF%E0%A6%82_%E0%A6%95%E0%A6%AE%E0%A7%8D%E0%A6%AA%E0%A6%A8" title="স্ট্রিং কম্পন – Bangla" lang="bn" hreflang="bn" data-title="স্ট্রিং কম্পন" data-language-autonym="বাংলা" data-language-local-name="Bangla" class="interlanguage-link-target"><span>বাংলা</span></a></li><li class="interlanguage-link interwiki-da mw-list-item"><a href="https://da.wikipedia.org/wiki/Snorb%C3%B8lger" title="Snorbølger – Danish" lang="da" hreflang="da" data-title="Snorbølger" data-language-autonym="Dansk" data-language-local-name="Danish" class="interlanguage-link-target"><span>Dansk</span></a></li><li class="interlanguage-link interwiki-de mw-list-item"><a href="https://de.wikipedia.org/wiki/Saitenschwingung" title="Saitenschwingung – German" lang="de" hreflang="de" data-title="Saitenschwingung" data-language-autonym="Deutsch" data-language-local-name="German" class="interlanguage-link-target"><span>Deutsch</span></a></li><li class="interlanguage-link interwiki-es mw-list-item"><a href="https://es.wikipedia.org/wiki/Cuerda_vibrante" title="Cuerda vibrante – Spanish" lang="es" hreflang="es" data-title="Cuerda vibrante" data-language-autonym="Español" data-language-local-name="Spanish" class="interlanguage-link-target"><span>Español</span></a></li><li class="interlanguage-link interwiki-fr mw-list-item"><a href="https://fr.wikipedia.org/wiki/Onde_sur_une_corde_vibrante" title="Onde sur une corde vibrante – French" lang="fr" hreflang="fr" data-title="Onde sur une corde vibrante" data-language-autonym="Français" data-language-local-name="French" class="interlanguage-link-target"><span>Français</span></a></li><li class="interlanguage-link interwiki-ko mw-list-item"><a href="https://ko.wikipedia.org/wiki/%EB%81%88%EC%9D%98_%EC%A7%84%EB%8F%99" title="끈의 진동 – Korean" lang="ko" hreflang="ko" data-title="끈의 진동" data-language-autonym="한국어" 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data-language-autonym="Magyar" data-language-local-name="Hungarian" class="interlanguage-link-target"><span>Magyar</span></a></li><li class="interlanguage-link interwiki-mk mw-list-item"><a href="https://mk.wikipedia.org/wiki/%D0%96%D0%B8%D1%87%D0%B5%D0%BD%D0%BE_%D1%82%D1%80%D0%B5%D0%BF%D0%B5%D1%80%D0%B5%D1%9A%D0%B5" title="Жичено треперење – Macedonian" lang="mk" hreflang="mk" data-title="Жичено треперење" data-language-autonym="Македонски" data-language-local-name="Macedonian" class="interlanguage-link-target"><span>Македонски</span></a></li><li class="interlanguage-link interwiki-no mw-list-item"><a href="https://no.wikipedia.org/wiki/Svingende_streng" title="Svingende streng – Norwegian Bokmål" lang="nb" hreflang="nb" data-title="Svingende streng" data-language-autonym="Norsk bokmål" data-language-local-name="Norwegian Bokmål" class="interlanguage-link-target"><span>Norsk bokmål</span></a></li><li class="interlanguage-link interwiki-simple mw-list-item"><a href="https://simple.wikipedia.org/wiki/Vibrating_string" title="Vibrating string – Simple English" lang="en-simple" hreflang="en-simple" data-title="Vibrating string" data-language-autonym="Simple English" data-language-local-name="Simple English" class="interlanguage-link-target"><span>Simple English</span></a></li><li class="interlanguage-link interwiki-tr mw-list-item"><a href="https://tr.wikipedia.org/wiki/Dize_titre%C5%9Fimi" title="Dize titreşimi – Turkish" lang="tr" hreflang="tr" data-title="Dize titreşimi" data-language-autonym="Türkçe" data-language-local-name="Turkish" class="interlanguage-link-target"><span>Türkçe</span></a></li><li class="interlanguage-link interwiki-uk mw-list-item"><a href="https://uk.wikipedia.org/wiki/%D0%92%D1%96%D0%B1%D1%80%D0%B0%D1%86%D1%96%D1%8F_%D1%81%D1%82%D1%80%D1%83%D0%BD%D0%B8" title="Вібрація струни – Ukrainian" lang="uk" hreflang="uk" data-title="Вібрація струни" data-language-autonym="Українська" data-language-local-name="Ukrainian" 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<div class="mw-indicators"> </div> <div id="siteSub" class="noprint">From Wikipedia, the free encyclopedia</div> </div> <div id="contentSub"><div id="mw-content-subtitle"><span class="mw-redirectedfrom">(Redirected from <a href="/w/index.php?title=Vibrating_string&amp;redirect=no" class="mw-redirect" title="Vibrating string">Vibrating string</a>)</span></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">A wave</div> <figure typeof="mw:File/Thumb"><a href="/wiki/File:Standing_waves_on_a_string.gif" class="mw-file-description"><img src="//upload.wikimedia.org/wikipedia/commons/thumb/5/5c/Standing_waves_on_a_string.gif/250px-Standing_waves_on_a_string.gif" decoding="async" width="250" height="235" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/5/5c/Standing_waves_on_a_string.gif 1.5x" data-file-width="360" data-file-height="339" /></a><figcaption>Vibration, <a href="/wiki/Standing_wave" title="Standing wave">standing waves</a> in a string. The <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental</a> and the first 5 <a href="/wiki/Overtone" title="Overtone">overtones</a> in the <a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">harmonic series</a>.</figcaption></figure> <p>A <a href="/wiki/Vibration" title="Vibration">vibration</a> in a <a href="/wiki/Strings_(music)" class="mw-redirect" title="Strings (music)">string</a> is a <a href="/wiki/Wave" title="Wave">wave</a>. <a href="/wiki/Acoustic_resonance#Resonance_of_a_string" title="Acoustic resonance">Resonance</a> causes a <b>vibrating string</b> to produce a <a href="/wiki/Sound" title="Sound">sound</a> with constant <a href="/wiki/Frequency" title="Frequency">frequency</a>, i.e. constant <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitch</a>. If the length or tension of the string is correctly adjusted, the sound produced is a <a href="/wiki/Musical_tone" title="Musical tone">musical tone</a>. Vibrating strings are the basis of <a href="/wiki/String_instrument" title="String instrument">string instruments</a> such as <a href="/wiki/Guitar" title="Guitar">guitars</a>, <a href="/wiki/Cello" title="Cello">cellos</a>, and <a href="/wiki/Piano" title="Piano">pianos</a>. For an homogenous string, the motion is given by the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>. </p> <meta property="mw:PageProp/toc" /> <div class="mw-heading mw-heading2"><h2 id="Wave">Wave</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=String_vibration&amp;action=edit&amp;section=1" title="Edit section: Wave"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>The velocity of propagation of a wave in a string (<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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" /></span>) is proportional to the <a href="/wiki/Square_root" title="Square root">square root</a> of the force of tension of the string (<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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span>) and inversely proportional to the square root of the linear density (<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>&#x3bc;<!-- 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>) of the string: </p><p><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 v={\sqrt {T \over \mu }}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>T</mi> <mi>&#x3bc;<!-- μ --></mi> </mfrac> </msqrt> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\sqrt {T \over \mu }}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/bbb26a33c562b55d2fa78450063cc83d6f75bd12" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:9.669ex; height:7.676ex;" alt="{\displaystyle v={\sqrt {T \over \mu }}.}" /></span> </p><p>This relationship was discovered by <a href="/wiki/Vincenzo_Galilei" title="Vincenzo Galilei">Vincenzo Galilei</a> in the late 1500s. <sup class="noprint Inline-Template Template-Fact" style="white-space:nowrap;">&#91;<i><a href="/wiki/Wikipedia:Citation_needed" title="Wikipedia:Citation needed"><span title="This claim needs references to reliable sources. (November 2017)">citation needed</span></a></i>&#93;</sup> </p> <div class="mw-heading mw-heading3"><h3 id="Derivation">Derivation</h3><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=String_vibration&amp;action=edit&amp;section=2" title="Edit section: Derivation"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <figure class="mw-default-size mw-halign-right" typeof="mw:File"><a href="/wiki/File:StringParameters.svg" class="mw-file-description" title="Illustration for a vibrating string"><img alt="Illustration for a vibrating string" src="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/StringParameters.svg/380px-StringParameters.svg.png" decoding="async" width="380" height="304" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/1/15/StringParameters.svg/570px-StringParameters.svg.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/1/15/StringParameters.svg/760px-StringParameters.svg.png 2x" data-file-width="380" data-file-height="304" /></a><figcaption>Illustration for a vibrating string</figcaption></figure> <p>Source:<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">&#91;</span>1<span class="cite-bracket">&#93;</span></a></sup> </p><p>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 \Delta x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x394;<!-- Δ --></mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}" /></span> be the <a href="/wiki/Length" title="Length">length</a> of a piece of string, <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 m}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>m</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle m}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0a07d98bb302f3856cbabc47b2b9016692e3f7bc" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.04ex; height:1.676ex;" alt="{\displaystyle m}" /></span> its <a href="/wiki/Mass" title="Mass">mass</a>, 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>&#x3bc;<!-- 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> its <a href="/wiki/Linear_density" title="Linear density">linear density</a>. If angles <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x3b1;<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x3b2;<!-- 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> are small, then the horizontal components of <a href="/wiki/Tension_(mechanics)" class="mw-redirect" title="Tension (mechanics)">tension</a> on either side can both be approximated by a constant <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span>, for which the net horizontal force is zero. Accordingly, using the <a href="/wiki/Small-angle_approximation" title="Small-angle approximation">small angle approximation</a>, the horizontal tensions acting on both sides of the string segment are given by </p> <dl><dd><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_{1x}=T_{1}\cos(\alpha )\approx T.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b1;<!-- α --></mi> <mo stretchy="false">)</mo> <mo>&#x2248;<!-- ≈ --></mo> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{1x}=T_{1}\cos(\alpha )\approx T.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4c42f5eadba613e29a1fbed7fe16c1b0481b932a" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.039ex; height:2.843ex;" alt="{\displaystyle T_{1x}=T_{1}\cos(\alpha )\approx T.}" /></span></dd> <dd><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_{2x}=T_{2}\cos(\beta )\approx T.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>x</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b2;<!-- β --></mi> <mo stretchy="false">)</mo> <mo>&#x2248;<!-- ≈ --></mo> <mi>T</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle T_{2x}=T_{2}\cos(\beta )\approx T.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/d268cfe6ae13b625b663135000e43d77d149eb27" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.883ex; height:2.843ex;" alt="{\displaystyle T_{2x}=T_{2}\cos(\beta )\approx T.}" /></span></dd></dl> <p>From Newton's second law for the vertical component, the mass (which is the product of its linear density and length) of this piece times its acceleration, <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>, will be equal to the net force on the piece: </p> <dl><dd><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 \Sigma F_{y}=T_{1y}-T_{2y}=-T_{2}\sin(\beta )+T_{1}\sin(\alpha )=\Delta ma\approx \mu \Delta x{\frac {\partial ^{2}y}{\partial t^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x3a3;<!-- Σ --></mi> <msub> <mi>F</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>y</mi> </mrow> </msub> <mo>=</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> <mi>y</mi> </mrow> </msub> <mo>&#x2212;<!-- − --></mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> <mi>y</mi> </mrow> </msub> <mo>=</mo> <mo>&#x2212;<!-- − --></mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b2;<!-- β --></mi> <mo stretchy="false">)</mo> <mo>+</mo> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b1;<!-- α --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mi mathvariant="normal">&#x394;<!-- Δ --></mi> <mi>m</mi> <mi>a</mi> <mo>&#x2248;<!-- ≈ --></mo> <mi>&#x3bc;<!-- μ --></mi> <mi mathvariant="normal">&#x394;<!-- Δ --></mi> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Sigma F_{y}=T_{1y}-T_{2y}=-T_{2}\sin(\beta )+T_{1}\sin(\alpha )=\Delta ma\approx \mu \Delta x{\frac {\partial ^{2}y}{\partial t^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/5fd1fdc3917af5f1632c80f3a00ff218c7170c8b" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:63.219ex; height:6.009ex;" alt="{\displaystyle \Sigma F_{y}=T_{1y}-T_{2y}=-T_{2}\sin(\beta )+T_{1}\sin(\alpha )=\Delta ma\approx \mu \Delta x{\frac {\partial ^{2}y}{\partial t^{2}}}.}" /></span></dd></dl> <p>Dividing this expression by <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> and substituting the first and second equations obtains (we can choose either the first or the second equation for <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span>, so we conveniently choose each one with the matching angle <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>&#x3b2;<!-- 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> 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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x3b1;<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></span>) </p> <dl><dd><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 {T_{2}\sin(\beta )}{T_{2}\cos(\beta )}}+{\frac {T_{1}\sin(\alpha )}{T_{1}\cos(\alpha )}}=-\tan(\beta )+\tan(\alpha )={\frac {\mu \Delta x}{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mo>&#x2212;<!-- − --></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>sin</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b2;<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msub> <mi>cos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b2;<!-- β --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>+</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>sin</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b1;<!-- α --></mi> <mo stretchy="false">)</mo> </mrow> <mrow> <msub> <mi>T</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>1</mn> </mrow> </msub> <mi>cos</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b1;<!-- α --></mi> <mo stretchy="false">)</mo> </mrow> </mfrac> </mrow> <mo>=</mo> <mo>&#x2212;<!-- − --></mo> <mi>tan</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b2;<!-- β --></mi> <mo stretchy="false">)</mo> <mo>+</mo> <mi>tan</mi> <mo>&#x2061;<!-- ⁡ --></mo> <mo stretchy="false">(</mo> <mi>&#x3b1;<!-- α --></mi> <mo stretchy="false">)</mo> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>&#x3bc;<!-- μ --></mi> <mi mathvariant="normal">&#x394;<!-- Δ --></mi> <mi>x</mi> </mrow> <mi>T</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle -{\frac {T_{2}\sin(\beta )}{T_{2}\cos(\beta )}}+{\frac {T_{1}\sin(\alpha )}{T_{1}\cos(\alpha )}}=-\tan(\beta )+\tan(\alpha )={\frac {\mu \Delta x}{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/46124409db8b7a3f073a7c631ed272a1874a8374" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:59.508ex; height:6.509ex;" alt="{\displaystyle -{\frac {T_{2}\sin(\beta )}{T_{2}\cos(\beta )}}+{\frac {T_{1}\sin(\alpha )}{T_{1}\cos(\alpha )}}=-\tan(\beta )+\tan(\alpha )={\frac {\mu \Delta x}{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}" /></span></dd></dl> <p>According to the small-angle approximation, the tangents of the angles at the ends of the string piece are equal to the slopes at the ends, with an additional minus sign due to the definition of <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 \alpha }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x3b1;<!-- α --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \alpha }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b79333175c8b3f0840bfb4ec41b8072c83ea88d3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.488ex; height:1.676ex;" alt="{\displaystyle \alpha }" /></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 \beta }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x3b2;<!-- 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>. Using this fact and rearranging provides </p> <dl><dd><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 {1}{\Delta x}}\left(\left.{\frac {\partial y}{\partial x}}\right|^{x+\Delta x}-\left.{\frac {\partial y}{\partial x}}\right|^{x}\right)={\frac {\mu }{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mi mathvariant="normal">&#x394;<!-- Δ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mrow> <mo>(</mo> <mrow> <msup> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> <mo>+</mo> <mi mathvariant="normal">&#x394;<!-- Δ --></mi> <mi>x</mi> </mrow> </msup> <mo>&#x2212;<!-- − --></mo> <msup> <mrow> <mo fence="true" stretchy="true" symmetric="true"></mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mi>x</mi> </mrow> </mfrac> </mrow> <mo>|</mo> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mi>x</mi> </mrow> </msup> </mrow> <mo>)</mo> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>&#x3bc;<!-- μ --></mi> <mi>T</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{\Delta x}}\left(\left.{\frac {\partial y}{\partial x}}\right|^{x+\Delta x}-\left.{\frac {\partial y}{\partial x}}\right|^{x}\right)={\frac {\mu }{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/851f3fb7217d34cf73c6c7f49f0d063c517ecb06" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:35.811ex; height:7.509ex;" alt="{\displaystyle {\frac {1}{\Delta x}}\left(\left.{\frac {\partial y}{\partial x}}\right|^{x+\Delta x}-\left.{\frac {\partial y}{\partial x}}\right|^{x}\right)={\frac {\mu }{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}" /></span></dd></dl> <p>In the limit that <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x394;<!-- Δ --></mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}" /></span> approaches zero, the left hand side is the definition of the second derivative of <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 y}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>y</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle y}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/b8a6208ec717213d4317e666f1ae872e00620a0d" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.155ex; height:2.009ex;" alt="{\displaystyle y}" /></span>, </p> <dl><dd><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 {\partial ^{2}y}{\partial x^{2}}}={\frac {\mu }{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>x</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>&#x3bc;<!-- μ --></mi> <mi>T</mi> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <msup> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> <mi>y</mi> </mrow> <mrow> <mi mathvariant="normal">&#x2202;<!-- ∂ --></mi> <msup> <mi>t</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mrow> </mfrac> </mrow> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {\partial ^{2}y}{\partial x^{2}}}={\frac {\mu }{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/2a0afffe8a860a4f2cfd2f99a3f62e37a6999ae3" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:15.145ex; height:6.009ex;" alt="{\displaystyle {\frac {\partial ^{2}y}{\partial x^{2}}}={\frac {\mu }{T}}{\frac {\partial ^{2}y}{\partial t^{2}}}.}" /></span></dd></dl> <p>this equation is known as the <a href="/wiki/Wave_equation" title="Wave equation">wave equation</a>, and the coefficient of the second time derivative term is equal to <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 {1}{v^{2}}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <msup> <mi>v</mi> <mrow class="MJX-TeXAtom-ORD"> <mn>2</mn> </mrow> </msup> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle {\frac {1}{v^{2}}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/0d7042e7d28b92ca06498a09d475d5ce177c4d57" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:3.018ex; height:5.509ex;" alt="{\displaystyle {\frac {1}{v^{2}}}}" /></span>; thus </p> <dl><dd><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 v={\sqrt {T \over \mu }},}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>T</mi> <mi>&#x3bc;<!-- μ --></mi> </mfrac> </msqrt> </mrow> <mo>,</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\sqrt {T \over \mu }},}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/808a903e1dcaf490ff627a2cc9d1e519249ca9a2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:9.669ex; height:7.676ex;" alt="{\displaystyle v={\sqrt {T \over \mu }},}" /></span></dd></dl> <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 v}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/e07b00e7fc0847fbd16391c778d65bc25c452597" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.128ex; height:1.676ex;" alt="{\displaystyle v}" /></span> is the <a href="/wiki/Speed" title="Speed">speed</a> of propagation of the wave in the string. However, this derivation is only valid for small amplitude vibrations; for those of large amplitude, <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 x}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi mathvariant="normal">&#x394;<!-- Δ --></mi> <mi>x</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f3890eb866b6258d7a304fc34c70ee3fb3a81a70" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.266ex; height:2.176ex;" alt="{\displaystyle \Delta x}" /></span> is not a good approximation for the length of the string piece, the horizontal component of tension is not necessarily constant. The horizontal tensions are not well approximated by <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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span>. </p> <div class="mw-heading mw-heading2"><h2 id="Frequency_of_the_wave">Frequency of the wave</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=String_vibration&amp;action=edit&amp;section=3" title="Edit section: Frequency of the wave"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>Once the speed of propagation is known, the <a href="/wiki/Frequency" title="Frequency">frequency</a> of the <a href="/wiki/Sound" title="Sound">sound</a> produced by the string can be calculated. The <a href="/wiki/Speed" title="Speed">speed</a> of propagation of a wave is equal to the <a href="/wiki/Wavelength" title="Wavelength">wavelength</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 \lambda }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x3bb;<!-- 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> divided by the <a href="/wiki/Wave_period" class="mw-redirect" title="Wave period">period</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 \tau }"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>&#x3c4;<!-- τ --></mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \tau }</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/38a7dcde9730ef0853809fefc18d88771f95206c" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.202ex; height:1.676ex;" alt="{\displaystyle \tau }" /></span>, or multiplied by the <a href="/wiki/Frequency" title="Frequency">frequency</a> <span style="white-space:nowrap;"><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 f}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/132e57acb643253e7810ee9702d9581f159a1c61" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" /></span>:</span> </p> <dl><dd><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 v={\frac {\lambda }{\tau }}=\lambda f.}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>v</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>&#x3bb;<!-- λ --></mi> <mi>&#x3c4;<!-- τ --></mi> </mfrac> </mrow> <mo>=</mo> <mi>&#x3bb;<!-- λ --></mi> <mi>f</mi> <mo>.</mo> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle v={\frac {\lambda }{\tau }}=\lambda f.}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/f177dee03384bafcdf2f367e2c8729d9d3d400b2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:12.797ex; height:5.343ex;" alt="{\displaystyle v={\frac {\lambda }{\tau }}=\lambda f.}" /></span></dd></dl> <p>If the length of the string is <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" /></span>, the <a href="/wiki/Fundamental_frequency" title="Fundamental frequency">fundamental harmonic</a> is the one produced by the vibration whose <a href="/wiki/Node_(physics)" title="Node (physics)">nodes</a> are the two ends of the string, so <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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" /></span> is half of the wavelength of the fundamental harmonic. Hence one obtains <a href="/wiki/Mersenne%27s_laws" title="Mersenne&#39;s laws">Mersenne's laws</a>: </p> <dl><dd><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 f={\frac {v}{2L}}={1 \over 2L}{\sqrt {T \over \mu }}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>f</mi> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>v</mi> <mrow> <mn>2</mn> <mi>L</mi> </mrow> </mfrac> </mrow> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mn>1</mn> <mrow> <mn>2</mn> <mi>L</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>T</mi> <mi>&#x3bc;<!-- μ --></mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f={\frac {v}{2L}}={1 \over 2L}{\sqrt {T \over \mu }}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/28c5262b65611c01d276cd922b72ac2e0fb9cf29" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:19.435ex; height:7.676ex;" alt="{\displaystyle f={\frac {v}{2L}}={1 \over 2L}{\sqrt {T \over \mu }}}" /></span></dd></dl> <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 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/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.636ex; height:2.176ex;" alt="{\displaystyle T}" /></span> is the <a href="/wiki/Tension_(mechanics)" class="mw-redirect" title="Tension (mechanics)">tension</a> (in Newtons), <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>&#x3bc;<!-- 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 the <a href="/wiki/Linear_density" title="Linear density">linear density</a> (that is, the <a href="/wiki/Mass" title="Mass">mass</a> per unit length), 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 L}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <mi>L</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle L}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" /></span> is the <a href="/wiki/Length" title="Length">length</a> of the vibrating part of the string. Therefore: </p> <ul><li>the shorter the string, the higher the frequency of the fundamental</li> <li>the higher the tension, the higher the frequency of the fundamental</li> <li>the lighter the string, the higher the frequency of the fundamental</li></ul> <p>Moreover, if we take the nth harmonic as having a wavelength given by <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 _{n}=2L/n}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>&#x3bb;<!-- λ --></mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mn>2</mn> <mi>L</mi> <mrow class="MJX-TeXAtom-ORD"> <mo>/</mo> </mrow> <mi>n</mi> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle \lambda _{n}=2L/n}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/4fce2e56e3fb2361ada2c413c48be842a6b8625e" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.975ex; height:2.843ex;" alt="{\displaystyle \lambda _{n}=2L/n}" /></span>, then we easily get an expression for the frequency of the nth harmonic: </p> <dl><dd><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 f_{n}={\frac {nv}{2L}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mrow> <mi>n</mi> <mi>v</mi> </mrow> <mrow> <mn>2</mn> <mi>L</mi> </mrow> </mfrac> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}={\frac {nv}{2L}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/192a23dc880483a5c6108186d93ee2670a3d50d2" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:9.038ex; height:4.676ex;" alt="{\displaystyle f_{n}={\frac {nv}{2L}}}" /></span></dd></dl> <p>And for a string under a tension T with linear density <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>&#x3bc;<!-- 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>, then </p> <dl><dd><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 f_{n}={\frac {n}{2L}}{\sqrt {\frac {T}{\mu }}}}"> <semantics> <mrow class="MJX-TeXAtom-ORD"> <mstyle displaystyle="true" scriptlevel="0"> <msub> <mi>f</mi> <mrow class="MJX-TeXAtom-ORD"> <mi>n</mi> </mrow> </msub> <mo>=</mo> <mrow class="MJX-TeXAtom-ORD"> <mfrac> <mi>n</mi> <mrow> <mn>2</mn> <mi>L</mi> </mrow> </mfrac> </mrow> <mrow class="MJX-TeXAtom-ORD"> <msqrt> <mfrac> <mi>T</mi> <mi>&#x3bc;<!-- μ --></mi> </mfrac> </msqrt> </mrow> </mstyle> </mrow> <annotation encoding="application/x-tex">{\displaystyle f_{n}={\frac {n}{2L}}{\sqrt {\frac {T}{\mu }}}}</annotation> </semantics> </math></span><img src="https://wikimedia.org/api/rest_v1/media/math/render/svg/7e738f3975fc8ac6028f328bc3e64bc93789a827" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.338ex; width:13.834ex; height:7.676ex;" alt="{\displaystyle f_{n}={\frac {n}{2L}}{\sqrt {\frac {T}{\mu }}}}" /></span></dd></dl> <div class="mw-heading mw-heading2"><h2 id="Observing_string_vibrations">Observing string vibrations</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=String_vibration&amp;action=edit&amp;section=4" title="Edit section: Observing string vibrations"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <p>One can see the <a href="/wiki/Waveforms" class="mw-redirect" title="Waveforms">waveforms</a> on a vibrating string if the frequency is low enough and the vibrating string is held in front of a <a href="/wiki/CRT_screen" class="mw-redirect" title="CRT screen">CRT screen</a> such as one of a <a href="/wiki/Television" title="Television">television</a> or a <a href="/wiki/Computer" title="Computer">computer</a> (<i>not</i> of an analog oscilloscope). This effect is called the <a href="/wiki/Stroboscopic_effect" title="Stroboscopic effect">stroboscopic effect</a>, and the rate at which the string seems to vibrate is the difference between the frequency of the string and the <a href="/wiki/Refresh_rate" title="Refresh rate">refresh rate</a> of the screen. The same can happen with a <a href="/wiki/Fluorescent_lamp" title="Fluorescent lamp">fluorescent lamp</a>, at a rate that is the difference between the frequency of the string and the frequency of the <a href="/wiki/Alternating_current" title="Alternating current">alternating current</a>. (If the refresh rate of the screen equals the frequency of the string or an integer multiple thereof, the string will appear still but deformed.) In daylight and other non-oscillating light sources, this effect does not occur and the string appears still but thicker, and lighter or blurred, due to <a href="/wiki/Persistence_of_vision" title="Persistence of vision">persistence of vision</a>. </p><p>A similar but more controllable effect can be obtained using a <a href="/wiki/Stroboscope" title="Stroboscope">stroboscope</a>. This device allows matching the frequency of the <a href="/wiki/Xenon_flash_lamp" class="mw-redirect" title="Xenon flash lamp">xenon flash lamp</a> to the frequency of vibration of the string. In a dark room, this clearly shows the waveform. Otherwise, one can use <a href="/wiki/Bend_(guitar)" class="mw-redirect" title="Bend (guitar)">bending</a> or, perhaps more easily, by adjusting the machine heads, to obtain the same, or a multiple, of the AC frequency to achieve the same effect. For example, in the case of a guitar, the 6th (lowest pitched) string pressed to the third fret gives a G at 97.999&#160;Hz. A slight adjustment can alter it to 100&#160;Hz, exactly one octave above the alternating current frequency in Europe and most countries in Africa and Asia, 50&#160;Hz. In most countries of the Americas—where the AC frequency is 60&#160;Hz—altering A# on the fifth string, first fret from 116.54&#160;Hz to 120&#160;Hz produces a similar effect. </p> <div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=String_vibration&amp;action=edit&amp;section=5" title="Edit section: See also"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><a href="/wiki/Fret" title="Fret">Fretted instruments</a></li> <li><a href="/wiki/Musical_acoustics" title="Musical acoustics">Musical acoustics</a></li> <li><a href="/wiki/Vibrations_of_a_circular_membrane" class="mw-redirect" title="Vibrations of a circular membrane">Vibrations of a circular drum</a></li> <li><a href="/wiki/Melde%27s_experiment" title="Melde&#39;s experiment">Melde's experiment</a></li> <li><a href="/wiki/3rd_bridge" title="3rd bridge">3rd bridge</a> (harmonic resonance based on equal string divisions)</li> <li><a href="/wiki/String_resonance" class="mw-redirect" title="String resonance">String resonance</a></li> <li><a href="/wiki/Reflection_phase_change" title="Reflection phase change">Reflection phase change</a></li></ul> <div class="mw-heading mw-heading2"><h2 id="References">References</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=String_vibration&amp;action=edit&amp;section=6" title="Edit section: References"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li><style data-mw-deduplicate="TemplateStyles:r1238218222">.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("//upload.wikimedia.org/wikipedia/commons/6/65/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("//upload.wikimedia.org/wikipedia/commons/d/d6/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("//upload.wikimedia.org/wikipedia/commons/a/aa/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("//upload.wikimedia.org/wikipedia/commons/4/4c/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}</style><cite id="CITEREFMoltenoN._B._Tufillaro2004" class="citation journal cs1">Molteno, T. C. A.; N. B. Tufillaro (September 2004). "An experimental investigation into the dynamics of a string". <i>American Journal of Physics</i>. <b>72</b> (9): <span class="nowrap">1157–</span>1169. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/2004AmJPh..72.1157M">2004AmJPh..72.1157M</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.1764557">10.1119/1.1764557</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=American+Journal+of+Physics&amp;rft.atitle=An+experimental+investigation+into+the+dynamics+of+a+string&amp;rft.volume=72&amp;rft.issue=9&amp;rft.pages=%3Cspan+class%3D%22nowrap%22%3E1157-%3C%2Fspan%3E1169&amp;rft.date=2004-09&amp;rft_id=info%3Adoi%2F10.1119%2F1.1764557&amp;rft_id=info%3Abibcode%2F2004AmJPh..72.1157M&amp;rft.aulast=Molteno&amp;rft.aufirst=T.+C.+A.&amp;rft.au=N.+B.+Tufillaro&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AString+vibration" class="Z3988"></span></li> <li><link rel="mw-deduplicated-inline-style" href="mw-data:TemplateStyles:r1238218222" /><cite id="CITEREFTufillaro1989" class="citation journal cs1">Tufillaro, N. B. (1989). "Nonlinear and chaotic string vibrations". <i>American Journal of Physics</i>. <b>57</b> (5): 408. <a href="/wiki/Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1989AmJPh..57..408T">1989AmJPh..57..408T</a>. <a href="/wiki/Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1119%2F1.16011">10.1119/1.16011</a>.</cite><span title="ctx_ver=Z39.88-2004&amp;rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&amp;rft.genre=article&amp;rft.jtitle=American+Journal+of+Physics&amp;rft.atitle=Nonlinear+and+chaotic+string+vibrations&amp;rft.volume=57&amp;rft.issue=5&amp;rft.pages=408&amp;rft.date=1989&amp;rft_id=info%3Adoi%2F10.1119%2F1.16011&amp;rft_id=info%3Abibcode%2F1989AmJPh..57..408T&amp;rft.aulast=Tufillaro&amp;rft.aufirst=N.+B.&amp;rfr_id=info%3Asid%2Fen.wikipedia.org%3AString+vibration" class="Z3988"></span></li></ul> <dl><dt>Specific</dt></dl> <div class="mw-references-wrap"><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"><a rel="nofollow" class="external text" href="http://www.animations.physics.unsw.edu.au/jw/wave_equation_speed.htm">The wave equation and wave speed</a></span> </li> </ol></div> <div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2><span class="mw-editsection"><span class="mw-editsection-bracket">[</span><a href="/w/index.php?title=String_vibration&amp;action=edit&amp;section=7" title="Edit section: External links"><span>edit</span></a><span class="mw-editsection-bracket">]</span></span></div> <ul><li>"<a rel="nofollow" class="external text" href="http://demonstrations.wolfram.com/TheVibratingString/">The Vibrating String</a>" by <a href="/wiki/Alain_Goriely" title="Alain Goriely">Alain Goriely</a> and Mark Robertson-Tessi, <a 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title="Soundproofing">Soundproofing</a></li> <li><a class="mw-selflink selflink">String vibration</a></li> <li><a href="/wiki/Sympathetic_resonance" title="Sympathetic resonance">Sympathetic resonance</a></li></ul> </div></td><td class="noviewer navbox-image" rowspan="5" style="width:1px;padding:0 0 0 2px"><div><div class="plainlist"><ul><li><span typeof="mw:File"><a href="/wiki/File:Spectrogram_showing_shared_partials.png" class="mw-file-description"><img alt="A spectrogram of a violin playing a note and then a perfect fifth above it. The shared partials are highlighted by the white dashes." src="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Spectrogram_showing_shared_partials.png/80px-Spectrogram_showing_shared_partials.png" decoding="async" width="80" height="67" class="mw-file-element" srcset="//upload.wikimedia.org/wikipedia/commons/thumb/2/26/Spectrogram_showing_shared_partials.png/120px-Spectrogram_showing_shared_partials.png 1.5x, //upload.wikimedia.org/wikipedia/commons/thumb/2/26/Spectrogram_showing_shared_partials.png/160px-Spectrogram_showing_shared_partials.png 2x" data-file-width="300" data-file-height="250" /></a></span></li><li><span style="font-size:85%;"><a href="/wiki/Spectrogram" title="Spectrogram">Spectrogram</a></span></li></ul></div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Psychoacoustics" title="Psychoacoustics">Psychoacoustics</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/Pitch_(music)" title="Pitch (music)">Pitch</a> <ul><li><a href="/wiki/Bark_scale" title="Bark scale">Bark scale</a></li> <li><a href="/wiki/Mel_scale" title="Mel scale">Mel scale</a></li></ul></li> <li><a href="/wiki/Equal-loudness_contour" title="Equal-loudness contour">Equal-loudness contour</a> <ul><li><a href="/wiki/Fletcher%E2%80%93Munson_curves" class="mw-redirect" title="Fletcher–Munson curves">Fletcher–Munson curves</a></li></ul></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="/wiki/Audio_frequency" title="Audio frequency">Audio frequency</a> and <a href="/wiki/Pitch_(music)" title="Pitch (music)">pitch</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/Beat_(acoustics)" title="Beat (acoustics)">Beat</a></li> <li><a href="/wiki/Formant" title="Formant">Formant</a></li> <li><a href="/wiki/Fundamental_frequency" title="Fundamental frequency">Fundamental frequency</a></li> <li><a href="/wiki/Frequency_spectrum" class="mw-redirect" title="Frequency spectrum">Frequency spectrum</a> <ul><li><a href="/wiki/Harmonic_spectrum" title="Harmonic spectrum">harmonic spectrum</a></li></ul></li> <li><a href="/wiki/Harmonic" title="Harmonic">Harmonic</a> <ul><li><a href="/wiki/Harmonic_series_(music)" title="Harmonic series (music)">Series</a></li> <li><a href="/wiki/Inharmonicity" title="Inharmonicity">Inharmonicity</a></li> <li><a href="/wiki/Missing_fundamental" title="Missing fundamental">Missing fundamental</a></li></ul></li> <li><a href="/wiki/Combination_tone" title="Combination tone">Combination tone</a></li> <li><a href="/wiki/Mersenne%27s_laws" title="Mersenne&#39;s laws">Mersenne's laws</a></li> <li><a href="/wiki/Overtone" title="Overtone">Overtone</a></li> <li><a href="/wiki/Acoustic_resonance" title="Acoustic resonance">Resonance</a></li> <li><a href="/wiki/Standing_wave" title="Standing wave">Standing wave</a> <ul><li><a href="/wiki/Node_(physics)" title="Node (physics)">Node</a></li></ul></li> <li><a href="/wiki/Undertone_series" title="Undertone series">Subharmonic</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Acousticians</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/John_Backus_(acoustician)" title="John Backus (acoustician)">John Backus</a></li> <li><a href="/wiki/Jens_Blauert" title="Jens Blauert">Jens Blauert</a></li> <li><a href="/wiki/Ernst_Chladni" title="Ernst Chladni">Ernst Chladni</a></li> <li><a href="/wiki/Hermann_von_Helmholtz" title="Hermann von Helmholtz">Hermann von Helmholtz</a></li> <li><a href="/wiki/Carleen_Hutchins" title="Carleen Hutchins">Carleen Hutchins</a></li> <li><a href="/wiki/Franz_Melde" title="Franz Melde">Franz Melde</a></li> <li><a href="/wiki/Marin_Mersenne" title="Marin Mersenne">Marin Mersenne</a></li> <li><a href="/wiki/Werner_Meyer-Eppler" title="Werner Meyer-Eppler">Werner Meyer-Eppler</a></li> <li><a href="/wiki/John_William_Strutt,_3rd_Baron_Rayleigh" title="John William Strutt, 3rd Baron Rayleigh">Lord Rayleigh</a></li> <li><a href="/wiki/Joseph_Sauveur" title="Joseph Sauveur">Joseph Sauveur</a></li> <li><a href="/wiki/D._Van_Holliday" title="D. Van Holliday">D. Van Holliday</a></li> <li><a href="/wiki/Thomas_Young_(scientist)" title="Thomas Young (scientist)">Thomas Young</a></li></ul> </div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</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/Echo" title="Echo">Echo</a></li> <li><a href="/wiki/Infrasound" title="Infrasound">Infrasound</a></li> <li><a href="/wiki/Sound" title="Sound">Sound</a></li> <li><a href="/wiki/Ultrasound" title="Ultrasound">Ultrasound</a></li> <li><a href="/wiki/Musical_acoustics" title="Musical acoustics">Musical acoustics</a> <ul><li><a href="/wiki/Piano_acoustics" title="Piano acoustics">Piano</a></li> <li><a href="/wiki/Violin_acoustics" title="Violin acoustics">Violin</a></li></ul></li></ul> </div></td></tr><tr><td class="navbox-abovebelow hlist" colspan="3"><div> <ul><li><span class="noviewer" typeof="mw:File"><span 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title="String (music)">strings</a>, <a href="/wiki/Piano_wire" title="Piano wire">wires</a>, and <a href="/wiki/String_instrument" title="String instrument">instruments</a></div></th></tr><tr><td class="navbox-abovebelow" colspan="2"><div><a href="/wiki/List_of_string_instruments" title="List of string instruments">List</a> (<a href="/wiki/List_of_chordophones_by_Hornbostel%E2%80%93Sachs_number" title="List of chordophones by Hornbostel–Sachs number"> Hornbostel–Sachs numbers</a>)</div></td></tr><tr><td colspan="2" class="navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"> <ul><li><a href="/wiki/Bow_(music)" title="Bow (music)">Bow</a></li> <li><a href="/wiki/Bridge_(instrument)" title="Bridge (instrument)">Bridge</a> <ul><li><a href="/wiki/3rd_bridge" title="3rd bridge">Third bridge</a></li></ul></li> <li><a href="/wiki/Chordophone" class="mw-redirect" title="Chordophone">Chordophone</a></li> <li><a href="/wiki/Course_(music)" title="Course 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