GermanWikipedia - /page/entry.mu

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<title>Dispersionsrelation</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">Dispersionsrelation</h1>

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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="de" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="de" dir="ltr">In der <a href="Physik" title="Physik">Physik</a> beschreibt die Dispersionsrelation (lat. dispergere ‚verteilen', ‚ausbreiten', ‚zerstreuen') den Zusammenhang zwischen dem Ablauf eines physikalischen Prozesses (<a href="Frequenz" title="Frequenz">Frequenz</a>, <a href="Energie" title="Energie">Energie</a>) und den Eigenschaften der ihn beschreibenden Größen (<a href="Wellenzahl" title="Wellenzahl">Wellenzahl</a>, <a href="Brechungsindex" title="Brechungsindex">Brechungsindex</a>, <a href="Ausbreitungsgeschwindigkeit" class="mw-redirect" title="Ausbreitungsgeschwindigkeit">Ausbreitungsgeschwindigkeit</a>, <a href="Impuls_(Physik)" class="mw-redirect" title="Impuls (Physik)">Impuls</a>).

Mathematisch ist die Dispersionsrelation die Beziehung zwischen der <a href="Kreisfrequenz" title="Kreisfrequenz">Kreisfrequenz</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> und der <a href="Kreiswellenzahl" class="mw-redirect" title="Kreiswellenzahl">Kreiswellenzahl</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

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<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. Sie wird aus der linearen <a href="Wellengleichung" title="Wellengleichung">Wellengleichung</a> durch eine <a href="Fouriertransformation" class="mw-redirect" title="Fouriertransformation">Fouriertransformation</a> in Raum und Zeit gewonnen und hat die Form<a href="#cite_note-1">[1]</a>

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<annotation encoding="application/x-tex">{\displaystyle \omega =\Omega (k)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ede4a916610ecda6ba1eeb2530efe08aaa6eac3a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.243ex; height:2.843ex;" alt="{\displaystyle \omega =\Omega (k)}" loading="lazy">.</dd></dl>

Im einfachsten Fall sind Kreisfrequenz und Kreiswellenzahl stets proportional<a href="#cite_note-2">[2]</a>

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<mtext>Phase</mtext>

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<annotation encoding="application/x-tex">{\displaystyle \omega =v_{\text{Phase}}\cdot k}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/badf289b44cd961815154aaaa1d7e66e596c703f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.028ex; height:2.509ex;" alt="{\displaystyle \omega =v_{\text{Phase}}\cdot k}" loading="lazy">,</dd></dl>

mit der konstanten <a href="Phasengeschwindigkeit" title="Phasengeschwindigkeit">Phasengeschwindigkeit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}}">

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<mtext>Phase</mtext>

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<annotation encoding="application/x-tex">{\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b0d638ce457abd83c831df472925f2dbe602faa.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:10.973ex; height:4.843ex;" alt="{\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}}" loading="lazy">. In diesem Fall gibt es keine <a href="Dispersion_(Physik)" title="Dispersion (Physik)">Dispersion</a>.

Die Geschwindigkeit eines <a href="Wellenpaket" title="Wellenpaket">Wellenpakets</a> ist dagegen die <a href="Gruppengeschwindigkeit" title="Gruppengeschwindigkeit">Gruppengeschwindigkeit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} k}}}">

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<mtext>Gruppe</mtext>

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<annotation encoding="application/x-tex">{\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} k}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ada88c7f6ee6cf5d1603be4423453e933021d016.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:13.672ex; height:5.509ex;" alt="{\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} k}}}" loading="lazy"> oder im dreidimensionalen Fall<a href="#cite_note-3">[3]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\vec {v}}_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} {\vec {k}}}}}">

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<mtext>Gruppe</mtext>

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<annotation encoding="application/x-tex">{\displaystyle {\vec {v}}_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} {\vec {k}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c0776732085d8f5020c563de3655d61fdff20b0a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:13.719ex; height:6.176ex;" alt="{\displaystyle {\vec {v}}_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} {\vec {k}}}}}" loading="lazy">.

Ein Wellenpaket besteht aus Wellen verschiedener Frequenzen, die unterschiedliche Phasengeschwindigkeiten haben können. Daher läuft ein Wellenpaket im Allgemeinen auseinander. Wellenpakete, die aufgrund nichtlinearer Effekte trotz Dispersion nicht auseinanderlaufen, werden als <a href="Soliton" title="Soliton">Solitonen</a> bezeichnet<a href="#cite_note-4">[4]</a>.

<div class="mw-heading mw-heading2"><h2 id="Optik">Optik</h2></div>

Die Dispersionsrelation der <a href="Optik" title="Optik">Optik</a> als Ausbreitung <a href="Elektromagnetische_Welle" title="Elektromagnetische Welle">elektromagnetischer Wellen</a> in <a href="Nichtleiter" title="Nichtleiter">nichtleitenden</a> Medien<a href="#cite_note-5">[5]</a> lautet:

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<annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {1}{\sqrt {\mu \varepsilon }}}\,k=\Omega (k)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8a93bbcf117be0ac33908c53ac504b6750bbfdb1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:19.197ex; height:6.176ex;" alt="{\displaystyle \omega ={\frac {1}{\sqrt {\mu \varepsilon }}}\,k=\Omega (k)}" loading="lazy"></dd></dl>

mit der <a href="Permittivit%C3%A4t" title="Permittivität">Permittivität</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon }">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a30c89172e5b88edbd45d3e2772c7f5e562e5173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle \varepsilon }" loading="lazy"> und der <a href="Magnetische_Permeabilit%C3%A4t" title="Magnetische Permeabilität">Permeabilität</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

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<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" 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 }" loading="lazy">. Die Phasengeschwindigkeit von <a href="Licht" title="Licht">Licht</a> in einem <a href="Ausbreitungsmedium" title="Ausbreitungsmedium">Medium</a> beträgt

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<mtext>Phase</mtext>

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<annotation encoding="application/x-tex">{\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}={\frac {1}{\sqrt {\mu \varepsilon }}}={\frac {c}{n(\omega )}}=c_{\text{M}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3b097b0568e09a4d8a1ed162f2ac7dba35443d06.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:33.758ex; height:6.176ex;" alt="{\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}={\frac {1}{\sqrt {\mu \varepsilon }}}={\frac {c}{n(\omega )}}=c_{\text{M}}}" loading="lazy"></dd></dl>

mit der der <a href="Vakuumlichtgeschwindigkeit" class="mw-redirect" title="Vakuumlichtgeschwindigkeit">Vakuumlichtgeschwindigkeit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

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<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy">. Der (<a href="Komplexe_Zahl" title="Komplexe Zahl">komplexe</a>) <a href="Brechungsindex" title="Brechungsindex">Brechungsindex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n}">

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<annotation encoding="application/x-tex">{\displaystyle n}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a601995d55609f2d9f5e233e36fbe9ea26011b3b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.395ex; height:1.676ex;" alt="{\displaystyle n}" loading="lazy"> tritt in Abhängigkeit der Kreisfrequenz <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">

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<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"> auf:

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<annotation encoding="application/x-tex">{\displaystyle n(\omega )={\sqrt {{\frac {\mu }{\mu _{0}}}\,{\frac {\varepsilon }{\varepsilon _{0}}}}}\quad {\text{und }}c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6d8b2e5065c9d4ec0119fa804854966071921201.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:34.977ex; height:6.343ex;" alt="{\displaystyle n(\omega )={\sqrt {{\frac {\mu }{\mu _{0}}}\,{\frac {\varepsilon }{\varepsilon _{0}}}}}\quad {\text{und }}c={\frac {1}{\sqrt {\mu _{0}\varepsilon _{0}}}}}" loading="lazy"></dd></dl>

mit der <a href="Elektrische_Feldkonstante" title="Elektrische Feldkonstante"> elektrischen Feldkonstante</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \varepsilon _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \varepsilon _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/acb0a8377db20e42274444cb181d51b5532b5844.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.138ex; height:2.009ex;" alt="{\displaystyle \varepsilon _{0}}" loading="lazy"> und der <a href="Magnetische_Feldkonstante" title="Magnetische Feldkonstante">magnetischen Feldkonstante</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu _{0}}">

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<annotation encoding="application/x-tex">{\displaystyle \mu _{0}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fe2fd9b8decb38a3cd158e7b6c0c6e2d987fefcc.svg" 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}}" loading="lazy">. Die Gruppengeschwindigkeit<a href="#cite_note-6">[6]</a>

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<annotation encoding="application/x-tex">{\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} k}}={\frac {c}{n(\omega )+\omega {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d561bc6ba0d6af3ced6378ac9aee9621abdc467b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.505ex; width:29.315ex; height:7.009ex;" alt="{\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} k}}={\frac {c}{n(\omega )+\omega {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}}}" loading="lazy"></dd></dl>

kann je nach Vorzeichen von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}}">

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<annotation encoding="application/x-tex">{\displaystyle {\textstyle {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3cceb5fd090cc744f6b095dca3aff1ebc613c99c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:2.772ex; height:3.843ex;" alt="{\displaystyle {\textstyle {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}}" loading="lazy"> deutlich von der Phasengeschwindigkeit abweichen. Normale Dispersion liegt für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}>0}">

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<annotation encoding="application/x-tex">{\displaystyle {\textstyle {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}>0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9c9fba1bd9a1190977f7c8c3b5dc487c0a2cacbe.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.033ex; height:3.843ex;" alt="{\displaystyle {\textstyle {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}>0}" loading="lazy"> vor und <a href="Dispersion_(Physik)#Anomale_Dispersion" title="Dispersion (Physik)">anomale Dispersion</a> für <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\textstyle {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}<0}">

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<annotation encoding="application/x-tex">{\displaystyle {\textstyle {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}<0}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f94170a386002d6e7deab952c8fd30c448d5dfee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:7.033ex; height:3.843ex;" alt="{\displaystyle {\textstyle {\frac {\mathrm {d} n}{\mathrm {d} \omega }}}<0}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Teilchenphysik_und_Materiewellen">Teilchenphysik und Materiewellen</h2></div>

Da die Frequenz immer in Zusammenhang mit der Energie steht

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<annotation encoding="application/x-tex">{\displaystyle \omega ={\frac {E}{\hbar }}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f9da81c42488056ad3e56a99d73020e3dd3149f7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.156ex; height:5.343ex;" alt="{\displaystyle \omega ={\frac {E}{\hbar }}}" loading="lazy"></dd></dl>

und die <a href="Wellenzahl" title="Wellenzahl">Wellenzahl</a> (bzw. der <a href="Wellenvektor" title="Wellenvektor">Wellenvektor</a>) mit dem <a href="Impuls" title="Impuls">Impuls</a><a href="#cite_note-7">[7]</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle {\vec {k}}={\frac {\vec {p}}{\hbar }},}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e84a32d0af1fd0ca77c4a0af84d7f1476d82d902.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:7.118ex; height:5.676ex;" alt="{\displaystyle {\vec {k}}={\frac {\vec {p}}{\hbar }},}" loading="lazy"></dd></dl>

bezeichnet man die Energie-Impuls-Beziehungen der <a href="Teilchenphysik" title="Teilchenphysik">Teilchenphysik</a> auch als Dispersionsrelation (oder Dispersionsbeziehung) der <a href="Materiewelle" title="Materiewelle">Materiewelle</a>, z. B. bei <a href="Freies_Teilchen" title="Freies Teilchen">freien</a> <a href="Elektron" title="Elektron">Elektronen</a> im nicht-<a href="Spezielle_Relativit%C3%A4tstheorie" title="Spezielle Relativitätstheorie">relativistischen</a> Grenzfall:

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<msup>

</mrow>

</msup>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mtd>

</mtr>

<mtr>

<mtd>

</mtd>

<mtd>

</mtd>

<mtd>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msup>

</mrow>

</msup>

</mtd>

</mtr>

</mtable>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\begin{aligned}&&E&={\frac {p^{2}}{2m}}\\\Rightarrow &&\hbar \,\omega &={\frac {\hbar ^{2}k^{2}}{2m}}\\\Leftrightarrow &&\omega &={\frac {\hbar }{2m}}k^{2}=\Omega (k),\end{aligned}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0900481c5fec8b86c6560e42229f8c31aff413ad.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -8.005ex; width:28.707ex; height:17.176ex;" alt="{\displaystyle {\begin{aligned}&&E&={\frac {p^{2}}{2m}}\\\Rightarrow &&\hbar \,\omega &={\frac {\hbar ^{2}k^{2}}{2m}}\\\Leftrightarrow &&\omega &={\frac {\hbar }{2m}}k^{2}=\Omega (k),\end{aligned}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \hbar }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \hbar }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/de68de3a92517953436c93b5a76461d49160cc41.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.306ex; height:2.176ex;" alt="{\displaystyle \hbar }" loading="lazy"> die <a href="Reduzierte_Planck-Konstante" class="mw-redirect" title="Reduzierte Planck-Konstante">reduzierte Planck-Konstante</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0a07d98bb302f3856cbabc47b2b9016692e3f7bc.svg" 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}" loading="lazy"> die <a href="Masse_(Physik)" title="Masse (Physik)">Masse</a> des Teilchens bezeichnet<a href="#cite_note-8">[8]</a>. Die Phasengeschwindigkeit der Materiewelle eines freien Teilchens beträgt<a href="#cite_note-9">[9]</a>

<msub>

<mtext>Phase</mtext>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

<mrow>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}={\frac {\hbar k}{2m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0f54bdfe0961e9e9b56135f231cb9c9dac795bcb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:18.111ex; height:5.509ex;" alt="{\displaystyle v_{\text{Phase}}={\frac {\omega }{k}}={\frac {\hbar k}{2m}}}" loading="lazy"></dd></dl>

und die Gruppengeschwindigkeit<a href="#cite_note-10">[10]</a>:

<msub>

<mtext>Gruppe</mtext>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<msub>

<mtext>Phase</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} k}}={\frac {\hbar k}{m}}=2\cdot v_{\text{Phase}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bc852784a6e4340a7492f92b1f8ca574293be397.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:31.657ex; height:5.509ex;" alt="{\displaystyle v_{\text{Gruppe}}={\frac {\mathrm {d} \Omega }{\mathrm {d} k}}={\frac {\hbar k}{m}}=2\cdot v_{\text{Phase}}}" loading="lazy"></dd></dl>

Das Ergebnis erlaubt die klassische Beschreibung des freien Teilchens mit einer minimalen Ortsunschärfe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f3890eb866b6258d7a304fc34c70ee3fb3a81a70.svg" 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}" loading="lazy"> mit der Geschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\text{Gruppe}}=v=p/m}">

<msub>

<mtext>Gruppe</mtext>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{\text{Gruppe}}=v=p/m}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ff442ce22e3521028446bacb332915612a5d327e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:18.463ex; height:3.009ex;" alt="{\displaystyle v_{\text{Gruppe}}=v=p/m}" loading="lazy"> in den Fällen, in denen mit der Impulsunschärfe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta p=h\Delta v_{\text{Gruppe}}}">

<msub>

<mtext>Gruppe</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta p=h\Delta v_{\text{Gruppe}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b14ea227d2ebdc39cd298b763985c7abb9290866.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:16.245ex; height:2.843ex;" alt="{\displaystyle \Delta p=h\Delta v_{\text{Gruppe}}}" loading="lazy"> aus der <a href="Heisenbergsche_Unsch%C3%A4rferelation" title="Heisenbergsche Unschärferelation">Heisenbergsche Unschärferelation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x\cdot \Delta p\geq h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta x\cdot \Delta p\geq h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8d1de17f5cfd0e5733f79f1ac143b72d602798f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.487ex; height:2.509ex;" alt="{\displaystyle \Delta x\cdot \Delta p\geq h}" loading="lazy"> folgt, dass <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta x\geq h/m\Delta v_{\text{Gruppe}}}">

</mrow>

<msub>

<mtext>Gruppe</mtext>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta x\geq h/m\Delta v_{\text{Gruppe}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6b08a7d8aabdc94cc898919a06dd1f55ae2cb3e8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:19.608ex; height:3.009ex;" alt="{\displaystyle \Delta x\geq h/m\Delta v_{\text{Gruppe}}}" loading="lazy"> ist<a href="#cite_note-11">[11]</a>.

<div class="mw-heading mw-heading2"><h2 id="Festkörperphysik">Festkörperphysik</h2></div>

In der <a href="Festk%C3%B6rperphysik" title="Festkörperphysik">Festkörperphysik</a> wird die Dispersion als Zusammenhang zwischen Energie bzw. Kreisfrequenz und Wellenzahl eines <a href="Teilchen" title="Teilchen">Teilchens</a> oder <a href="Quasiteilchen" title="Quasiteilchen">Quasiteilchens</a> angegeben. In <a href="Festk%C3%B6rper" title="Festkörper">Festkörpern</a> wird dabei einerseits den <a href="Phonon" title="Phonon">Phononen</a> (Gitterschwingungen des <a href="Atomgitter" title="Atomgitter">Atomgitters</a>) eine <a href="Phonon#Dispersion" title="Phonon">Phononen-Dispersionsrelation</a> zugeordnet<a href="#cite_note-12">[12]</a>, andererseits kann den Elektronen eine Elektronen-Dispersionsrelation zugeordnet werden, die mit Hilfe der <a href="Bandstruktur" title="Bandstruktur">Bandstruktur</a> beschrieben wird<a href="#cite_note-13">[13]</a><a href="#cite_note-14">[14]</a>.

<div class="mw-heading mw-heading2"><h2 id="Beispiele_von_Dispersionsrelationen">Beispiele von Dispersionsrelationen</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Elektronen_im_Festkörper_(Bandstruktur)">Elektronen im Festkörper (Bandstruktur)</h3></div>

In einem Kristall mit periodischer Gitterstruktur kann die Dispersionsrelation für Elektronen durch eine Bandstruktur beschrieben werden<a href="#cite_note-15">[15]</a>:

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E(\mathbf {k} )=E_{0}+{\frac {\hbar ^{2}k^{2}}{2m^{*}}}=\hbar \omega \quad {\text{bzw. }}\omega =\omega _{0}+{\frac {\hbar k^{2}}{2m^{*}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aad76706f28d387d4e659ebb64535b1b56a03e38.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:47.564ex; height:5.676ex;" alt="{\displaystyle E(\mathbf {k} )=E_{0}+{\frac {\hbar ^{2}k^{2}}{2m^{*}}}=\hbar \omega \quad {\text{bzw. }}\omega =\omega _{0}+{\frac {\hbar k^{2}}{2m^{*}}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m^{*}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m^{*}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7f2650f1055b63acf24bf275e50b4f59c3e53685.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.095ex; height:2.343ex;" alt="{\displaystyle m^{*}}" loading="lazy"> die <a href="Effektive_Masse" title="Effektive Masse">effektive Masse</a> des Elektrons ist.

<div class="mw-heading mw-heading3"><h3 id="Plasmonen_(Kollektive_Schwingungen_in_einem_Plasma)">Plasmonen (Kollektive Schwingungen in einem Plasma)</h3></div>

Für <a href="Plasmonen" class="mw-redirect" title="Plasmonen">Plasmonen</a> als longitudinale Plasmaschwingungen ist die Dispersionsrelation<a href="#cite_note-16">[16]</a>:

<msqrt>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {\omega _{p}^{2}+c^{2}k^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3070af6dd6097c79145b009552ed6329f3d3a734.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.54ex; height:4.843ex;" alt="{\displaystyle \omega ={\sqrt {\omega _{p}^{2}+c^{2}k^{2}}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/86a67b81c2de995bd608d5b2df50cd8cd7d92455.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.007ex; height:1.676ex;" alt="{\displaystyle c}" loading="lazy"> die Lichtgeschwindigkeit und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega _{p}={\sqrt {4\pi n_{e}e^{2}/m_{e}}}}">

<msub>

</mrow>

</msub>

<msqrt>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega _{p}={\sqrt {4\pi n_{e}e^{2}/m_{e}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/042acfd90a1612f8ab5a53163848d02f44e81487.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:19.154ex; height:4.843ex;" alt="{\displaystyle \omega _{p}={\sqrt {4\pi n_{e}e^{2}/m_{e}}}}" loading="lazy"> die <a href="Plasmafrequenz" class="mw-redirect" title="Plasmafrequenz">Plasmafrequenz</a> ist, gebildet aus der <a href="Elektronendichte" title="Elektronendichte">Elektronendichte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle n_{e}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle n_{e}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5e21303faca6e2167f4ecbf2a75aed982e817035.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.393ex; height:2.009ex;" alt="{\displaystyle n_{e}}" loading="lazy">, der <a href="Elektronenladung" class="mw-redirect" title="Elektronenladung">Elektronenladung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle e}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle e}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cd253103f0876afc68ebead27a5aa9867d927467.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.083ex; height:1.676ex;" alt="{\displaystyle e}" loading="lazy"> und der Elektronenmasse <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m_{e}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m_{e}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8303b668e94e02d8f3db8c5b3ebd069ca5da9ba5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.039ex; height:2.009ex;" alt="{\displaystyle m_{e}}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Schwerewellen_auf_einem_Meer_endlicher_Tiefe">Schwerewellen auf einem Meer endlicher Tiefe</h3></div>

Für Meereswellen als <a href="Schwerewellen" class="mw-redirect" title="Schwerewellen">Schwerewellen</a> lautet die Dispersionsrelation nach<a href="#cite_note-17">[17]</a>

<msqrt>

</msqrt>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<mrow>

<msup>

</mrow>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =\Omega (k)={\sqrt {gk\tanh(kh)}}\qquad {\text{mit }}\tanh(kh)={\frac {1}{2}}{\frac {{\text{e}}^{kh}-{\text{e}}^{-kh}}{{\text{e}}^{kh}+{\text{e}}^{-kh}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/96360067392b8ff199043a7c0a9fdd5bfdc62757.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:60.691ex; height:6.176ex;" alt="{\displaystyle \omega =\Omega (k)={\sqrt {gk\tanh(kh)}}\qquad {\text{mit }}\tanh(kh)={\frac {1}{2}}{\frac {{\text{e}}^{kh}-{\text{e}}^{-kh}}{{\text{e}}^{kh}+{\text{e}}^{-kh}}}}" loading="lazy"></dd></dl>

mit der <a href="Schwerebeschleunigung" class="mw-redirect" title="Schwerebeschleunigung">Schwerebeschleunigung</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> und der Wassertiefe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy">. Die Funktion

<div class="mw-heading mw-heading3"><h3 id="Tiefwasserwellen">Tiefwasserwellen</h3></div>

Für Tiefwasserwellen gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kh\gg 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle kh\gg 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/47db8b6fe83cc743f2f921caa6cf73150606c1b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.327ex; height:2.176ex;" alt="{\displaystyle kh\gg 1}" loading="lazy"> und damit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tanh(kh)\approx 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tanh(kh)\approx 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fac275a789edea4b8a033a0991e13fe2319161d8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.273ex; height:2.843ex;" alt="{\displaystyle \tanh(kh)\approx 1}" loading="lazy">. Für diese Wellen nähert sich die Dispersionsrelation zu<a href="#cite_note-Kuhlmann145-18">[18]</a>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {gk}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a8fb2c35d92792f8e7426d42edf12bb71ca7e5e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:9.195ex; height:3.509ex;" alt="{\displaystyle \omega ={\sqrt {gk}}}" loading="lazy"></dd></dl>

mit der Schwerebeschleunigung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> und der Wellenzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.

<div class="mw-heading mw-heading3"><h3 id="Seichtwasserwellen">Seichtwasserwellen</h3></div>

Für Seichtwasserwellen der Tiefe <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b26be3e694314bc90c3215047e4a2010c6ee184a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.339ex; height:2.176ex;" alt="{\displaystyle h}" loading="lazy"> gilt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kh\ll 1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle kh\ll 1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/8939dd5c178d7ca22fbbe6279182b87b19b21d6a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.327ex; height:2.176ex;" alt="{\displaystyle kh\ll 1}" loading="lazy"> und damit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tanh(kh)\approx (kh)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \tanh(kh)\approx (kh)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d2887559b6e75bebad34231810d6ac7d1fc30945.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.47ex; height:2.843ex;" alt="{\displaystyle \tanh(kh)\approx (kh)}" loading="lazy">. Für diese Oberflächenwellen beträgt die Dispersionsrelation<a href="#cite_note-Kuhlmann145-18">[18]</a>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {gk\tanh(kh)}}\approx {\sqrt {gk\,kh}}={\sqrt {gh}}\,k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/495f92faae8990c15abb76528f7d9de8d08d3aca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.756ex; height:4.843ex;" alt="{\displaystyle \omega ={\sqrt {gk\tanh(kh)}}\approx {\sqrt {gk\,kh}}={\sqrt {gh}}\,k}" loading="lazy"></dd></dl>

mit der Schwerebeschleunigung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle g}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle g}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d3556280e66fe2c0d0140df20935a6f057381d77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.116ex; height:2.009ex;" alt="{\displaystyle g}" loading="lazy"> und der Wellenzahl <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. Damit sind sowohl die Phasengeschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{\Omega }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{\Omega }}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7ebada76ce471e11072140f8022ef0396d999b31.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.546ex; height:2.009ex;" alt="{\displaystyle v_{\Omega }}" loading="lazy"> und die Gruppengeschwindigkeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v_{g}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{g}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/156cf81ae5c302f373a5f109ca5259f636bbb9c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.149ex; height:2.343ex;" alt="{\displaystyle v_{g}}" loading="lazy"> konstant:

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{\Omega }={\frac {\omega }{k}}={\sqrt {gh}}\quad {\text{und}}\quad v_{g}={\frac {{\text{d}}\omega }{{\text{d}}k}}={\sqrt {gh}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/72e98058279e9015233d0daf1d0cf306f38a758b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:41.026ex; height:5.509ex;" alt="{\displaystyle v_{\Omega }={\frac {\omega }{k}}={\sqrt {gh}}\quad {\text{und}}\quad v_{g}={\frac {{\text{d}}\omega }{{\text{d}}k}}={\sqrt {gh}}}" loading="lazy"></dd></dl>

Im Gegensatz zu Sturmwellen, bei denen die Wasserschichten ab einer Tiefe von etwa 200 m unbewegt bleiben, wird bei einem <a href="Tsunami" title="Tsunami">Tsunami</a> das gesamte Wasservolumen vom Meeresboden bis zur Oberfläche in Bewegung gesetzt. Auf dem offenen Meer können Tsunamis Wellenlängen von 100 bis 300 km erreichen, in seltenen Fällen sogar bis zu 500 km. Die Wellenzahlen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=2\pi /\lambda }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=2\pi /\lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/eecfda2366cb3a64ef6d4747e2460284155a1960.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.322ex; height:2.843ex;" alt="{\displaystyle k=2\pi /\lambda }" loading="lazy"> reichen dabei von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{6,3}}\cdot 10^{-5}{\text{ m}}^{-1}}">

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\text{6,3}}\cdot 10^{-5}{\text{ m}}^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/867ec6f90b62497129c603194dab89599f0d223f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:14.158ex; height:3.009ex;" alt="{\displaystyle {\text{6,3}}\cdot 10^{-5}{\text{ m}}^{-1}}" loading="lazy"> bis <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 10^{-5}{\text{ m}}^{-1}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 10^{-5}{\text{ m}}^{-1}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0c0c72d97f4e815caab40433e953d6d41fe2d22b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.507ex; height:2.676ex;" alt="{\displaystyle 10^{-5}{\text{ m}}^{-1}}" loading="lazy">. Selbst in Ozeanen mit einer Tiefe von etwa <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle h=5.000\,{\text{m}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle h=5.000\,{\text{m}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/afd4fb5097e3c5b4bf992e210db3012efe6bb4ac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.057ex; height:2.176ex;" alt="{\displaystyle h=5.000\,{\text{m}}}" loading="lazy"> ist das Produkt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kh}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle kh}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/34bb259c0161d2a55bbb60cc055ec2c34f64add2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.55ex; height:2.176ex;" alt="{\displaystyle kh}" loading="lazy"> maximal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle kh\sim {\text{0,3}}<1}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle kh\sim {\text{0,3}}<1}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/d5bd38f7fb7e3750f6bd5c10848c0f8c01c64b6d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.881ex; height:2.509ex;" alt="{\displaystyle kh\sim {\text{0,3}}<1}" loading="lazy">. Tsunamis sind also vom Verhalten her Seichtwasserwellen, die Geschwindigkeiten von

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

</mrow>

</mrow>

<mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

<msqrt>

</msqrt>

</mrow>

<msqrt>

</mrow>

</msqrt>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v_{g}={\frac {{\text{d}}\omega }{{\text{d}}k}}={\sqrt {gh}}={\sqrt {{\text{9,81}}\cdot 5.000}}\,{\text{m/s}}=220\,{\text{m/s}}=800\,{\text{km/h}}.}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ab58c492472cf04817abb5c9df5cc2d3ed5f45e0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:61.859ex; height:5.509ex;" alt="{\displaystyle v_{g}={\frac {{\text{d}}\omega }{{\text{d}}k}}={\sqrt {gh}}={\sqrt {{\text{9,81}}\cdot 5.000}}\,{\text{m/s}}=220\,{\text{m/s}}=800\,{\text{km/h}}.}" loading="lazy"></dd></dl>

erreichen – fast die Geschwindigkeit eines Jumbo-Jets!

<div class="mw-heading mw-heading3"><h3 id="Schwerewellen_unter_dem_Einfluss_von_Oberflächenspannung">Schwerewellen unter dem Einfluss von Oberflächenspannung</h3></div>

Berücksichtigt man bei den Meereswellen zusätzlich die <a href="Kapillarwelle" title="Kapillarwelle">Kapillarwellen</a>, so erweitert sich die Dispersionsrelation zu<a href="#cite_note-19">[19]</a>

<msqrt>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =\Omega (k)={\sqrt {gk+{\frac {\sigma k^{3}}{\rho }}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa4be121db2b2d73a2279bca6d5adc8a86e35e75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:24.264ex; height:7.676ex;" alt="{\displaystyle \omega =\Omega (k)={\sqrt {gk+{\frac {\sigma k^{3}}{\rho }}}}}" loading="lazy"></dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> die <a href="Oberfl%C3%A4chenspannung" title="Oberflächenspannung">Oberflächenspannung</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" 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 }" loading="lazy"> die <a href="Dichte" title="Dichte">Dichte</a> des Wassers.

<div class="mw-heading mw-heading3"><h3 id="Schwerewellen_unter_dem_Einfluss_von_Oberflächenspannung_und_Tiefe">Schwerewellen unter dem Einfluss von Oberflächenspannung und Tiefe</h3></div>

Betrachtet man Meereswellen als Schwerewellen unter dem Einfluss von Oberflächenspannung und Tiefe, so ergänzt sich die Dispersionsrelation zu<a href="#cite_note-20">[20]</a>

<msqrt>

<mrow>

<mrow>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega ={\sqrt {\left(gk+{\frac {\sigma k^{3}}{\rho }}\right)\tanh(kh)}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/020f3dce9eb065863fbabe04918382c3fd061868.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.287ex; height:7.509ex;" alt="{\displaystyle \omega ={\sqrt {\left(gk+{\frac {\sigma k^{3}}{\rho }}\right)\tanh(kh)}}}" loading="lazy"></dd></dl>

Dabei ist <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> die Oberflächenspannung und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" 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 }" loading="lazy"> die Dichte des Wassers.

<div class="mw-heading mw-heading3"><h3 id="Elastische_Wellen_in_isotropen_Medien">Elastische Wellen in isotropen Medien</h3></div>

In isotropen Festkörpern existieren zwei Arten von elastischen Wellen mit ihren entsprechenden Dispersionsrelationen:

<ul><li><a href="Longitudinalwelle" title="Longitudinalwelle">Longitudinalwellen</a> (P-Wellen)<a href="#cite_note-21">[21]</a>:</li></ul>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

<mfrac>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

<msqrt>

<mfrac>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =c_{L}k,\quad c_{L}={\sqrt {\frac {\lambda +2\mu }{\rho }}}={\sqrt {\frac {K+{\frac {4}{3}}\mu }{\rho }}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/908df855e1810e071745718b6234e753908cc5eb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:41.071ex; height:8.509ex;" alt="{\displaystyle \omega =c_{L}k,\quad c_{L}={\sqrt {\frac {\lambda +2\mu }{\rho }}}={\sqrt {\frac {K+{\frac {4}{3}}\mu }{\rho }}}}" loading="lazy"></dd></dl>

<ul><li><a href="Transversalwelle" title="Transversalwelle">Transversalwellen</a> (S-Wellen)<a href="#cite_note-22">[22]</a>:</li></ul>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =c_{T}k,\quad c_{T}={\sqrt {\frac {\mu }{\rho }}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fa4933003dcc51d6bbb6d6aebbd1c20db38a9173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:21.564ex; height:6.343ex;" alt="{\displaystyle \omega =c_{T}k,\quad c_{T}={\sqrt {\frac {\mu }{\rho }}}}" loading="lazy"></dd></dl>

Hierbei sind <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" 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 }" loading="lazy"> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" 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 }" loading="lazy"> die <a href="Lam%C3%A9-Konstanten" title="Lamé-Konstanten">Lamé-Konstanten</a> und <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K=\lambda +{\textstyle {\frac {2}{3}}}\mu }">

<mfrac>

</mfrac>

</mrow>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle K=\lambda +{\textstyle {\frac {2}{3}}}\mu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/79928816f5fd796819d118d0ca73f896614e2202.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.338ex; width:12.42ex; height:3.676ex;" alt="{\displaystyle K=\lambda +{\textstyle {\frac {2}{3}}}\mu }" loading="lazy"> der <a href="Kompressionsmodul" title="Kompressionsmodul">Kompressionsmodul</a>.

<div class="mw-heading mw-heading3"><h3 id="Biegewellen_in_Stäben">Biegewellen in Stäben</h3></div>

Für Schwingungen transversal in Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> für <a href="Biegewelle" title="Biegewelle">Biegewellen</a>, die sich in einem dünnen Stab in Richtung <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle z}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/bf368e72c009decd9b6686ee84a375632e11de98.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.088ex; height:1.676ex;" alt="{\displaystyle z}" loading="lazy"> ausbreiten, gilt<a href="#cite_note-23">[23]</a>:

<msup>

</mrow>

</msup>

</mrow>

<msqrt>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<mrow>

</mrow>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =\beta \ k^{2},\quad {\text{mit}}\quad \beta ={\sqrt {\frac {EI_{y}}{\rho A}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/027708f66252adfe3155f645e133a76c90c20302.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:29.327ex; height:7.676ex;" alt="{\displaystyle \omega =\beta \ k^{2},\quad {\text{mit}}\quad \beta ={\sqrt {\frac {EI_{y}}{\rho A}}}}" loading="lazy"></dd></dl>

wobei:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle E}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4232c9de2ee3eec0a9c0a19b15ab92daa6223f9b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.776ex; height:2.176ex;" alt="{\displaystyle E}" loading="lazy"> der <a href="Elastizit%C3%A4tsmodul" title="Elastizitätsmodul">Elastizitätsmodul</a> ist,</dd>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{y}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/6f3eedd0871fd35e1fd8992c53812cd60d1b07c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:2.072ex; height:2.843ex;" alt="{\displaystyle I_{y}}" loading="lazy"> das axiale <a href="Fl%C3%A4chentr%C3%A4gheitsmoment" title="Flächenträgheitsmoment">Flächenträgheitsmoment</a> ist,</dd>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle A}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/7daff47fa58cdfd29dc333def748ff5fa4c923e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.743ex; height:2.176ex;" alt="{\displaystyle A}" loading="lazy"> die Querschnittsfläche ist.</dd></dl>

Die Frequenz oder Dispersionsrelation ist also proportional zum Quadrat des Wellenvektors. In einem unbegrenzten Medium hängt sie linear vom Wellenvektor ab.

<div class="mw-heading mw-heading3"><h3 id="Querwellen_in_Drähten_(Saiten)">Querwellen in Drähten (Saiten)</h3></div>

Die Dispersionsrelation für Torsionswellen in einem zylindrischen Stab lautet<a href="#cite_note-24">[24]</a>:

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =k{\sqrt {\frac {\sigma }{\rho }}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3a868074499b913b7ab75a419546e416b531ad43.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:10.245ex; height:6.176ex;" alt="{\displaystyle \omega =k{\sqrt {\frac {\sigma }{\rho }}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \sigma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \sigma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/59f59b7c3e6fdb1d0365a494b81fb9a696138c36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle \sigma }" loading="lazy"> die Spannung des Drahtes ist.

<div class="mw-heading mw-heading3"><h3 id="Torsionsschwingungen_eines_Stabes">Torsionsschwingungen eines Stabes</h3></div>

Torsionswellen in einem zylindrischen Stab folgen der Dispersionsrelation<a href="#cite_note-25">[25]</a>:

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

<msqrt>

<mfrac>

</mfrac>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =k{\sqrt {\frac {G}{\rho }}}=k{\sqrt {\frac {\mu }{\rho }}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/aa1613c402a842e3785d28b0ab3a1bd39b7e476c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.171ex; width:19.613ex; height:7.509ex;" alt="{\displaystyle \omega =k{\sqrt {\frac {G}{\rho }}}=k{\sqrt {\frac {\mu }{\rho }}}}" loading="lazy"></dd></dl>

wobei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> der <a href="Schubmodul" title="Schubmodul">Schubmodul</a> ist, bzw. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mu }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mu }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9fd47b2a39f7a7856952afec1f1db72c67af6161.svg" 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 }" loading="lazy"> die zweite <a href="Lam%C3%A9-Konstante" class="mw-redirect" title="Lamé-Konstante">Lamé-Konstante</a>.

<div class="mw-heading mw-heading3"><h3 id="Hörbare_Schallwellen">Hörbare Schallwellen</h3></div>

Hörbare Schallwellen sind ein weiteres Beispiel für dispersionsfreie Wellen, da die Dispersionsrelation linear vom Wellenvektor abhängt<a href="#cite_note-26">[26]</a>:

<msqrt>

<mfrac>

</mfrac>

</mrow>

</msqrt>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \omega =k{\sqrt {\gamma {\frac {p}{\rho }}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/73052e425d945c83d5c7e1b018fddb8dbf034be2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:11.38ex; height:6.343ex;" alt="{\displaystyle \omega =k{\sqrt {\gamma {\frac {p}{\rho }}}}}" loading="lazy"></dd></dl>

bei einem Gasdruck <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"> mit der Dichte <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \rho }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \rho }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1f7d439671d1289b6a816e6af7a304be40608d64.svg" 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 }" loading="lazy"> und dem <a href="Adiabatenexponent" class="mw-redirect" title="Adiabatenexponent">Adiabatenexponent</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \gamma }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \gamma }</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/a223c880b0ce3da8f64ee33c4f0010beee400b1a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.262ex; height:2.176ex;" alt="{\displaystyle \gamma }" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Literatur">Literatur</h2></div>

<ul><li><a href="Dieter_Meschede" title="Dieter Meschede">Dieter Meschede</a>: <cite style="font-style:italic">Optik, Licht und Laser</cite>. Springer-Verlag, 2015, ISBN 3-663-10954-2, S. 29 f.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Dieter+Meschede&rft.btitle=Optik%2C+Licht+und+Laser&rft.date=2015&rft.genre=book&rft.isbn=3663109542&rft.pages=29f&rft.pub=Springer-Verlag" style="display:none"> </li>

<li><cite style="font-style:italic">Dispersionsrelation</cite>. In: Ulrich Kilian u. Christine Weber (Hrsg.): <cite style="font-style:italic">Lexikon der Physik</cite>. Spektrum Akademischer Verlag, 2003, ISBN 978-3-86025-296-3 (<a rel="nofollow" class="external text" href="https://www.spektrum.de/lexikon/physik/dispersionsrelation/3190">spektrum.de</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.atitle=Dispersionsrelation&rft.btitle=Lexikon+der+Physik&rft.date=2003&rft.genre=book&rft.isbn=9783860252963&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </li>

<li><cite style="font-style:italic">Dispersionsrelation</cite>. In: <a href="Harry_Paul_(Physiker)" title="Harry Paul (Physiker)">Harry Paul</a> (Hrsg.): <cite style="font-style:italic">Lexikon der Optik</cite>. Spektrum Akademischer Verlag, 1999, ISBN 978-3-8274-0382-7 (<a rel="nofollow" class="external text" href="https://www.spektrum.de/lexikon/optik/dispersionsrelation/663">spektrum.de</a>).<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.atitle=Dispersionsrelation&rft.btitle=Lexikon+der+Optik&rft.date=1999&rft.genre=book&rft.isbn=9783827403827&rft.pub=Spektrum+Akademischer+Verlag" style="display:none"> </li></ul>

<div class="mw-heading mw-heading2"><h2 id="Einzelnachweise">Einzelnachweise</h2></div>

<li id="cite_note-1"><a href="#cite_ref-1">↑</a> Kip S. Thorne, Roger D. Blandford: <cite style="font-style:italic">Modern Classical Physics - Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics</cite>. 1. Auflage. Princeton University Press, Princeton 2017, ISBN 0-691-15902-5, S. 353.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Kip+S.+Thorne%2C+Roger+D.+Blandford&rft.btitle=Modern+Classical+Physics+-+Optics%2C+Fluids%2C+Plasmas%2C+Elasticity%2C+Relativity%2C+and+Statistical+Physics&rft.date=2017&rft.edition=1.&rft.genre=book&rft.isbn=0691159025&rft.pages=353&rft.place=Princeton&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-2"><a href="#cite_ref-2">↑</a> Kip S. Thorne, Roger D. Blandford: <cite style="font-style:italic">Modern Classical Physics - Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics</cite>. 1. Auflage. Princeton University Press, Princeton 2017, ISBN 0-691-15902-5, S. 352.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Kip+S.+Thorne%2C+Roger+D.+Blandford&rft.btitle=Modern+Classical+Physics+-+Optics%2C+Fluids%2C+Plasmas%2C+Elasticity%2C+Relativity%2C+and+Statistical+Physics&rft.date=2017&rft.edition=1.&rft.genre=book&rft.isbn=0691159025&rft.pages=352&rft.place=Princeton&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-3"><a href="#cite_ref-3">↑</a> Kip S. Thorne, Roger D. Blandford: <cite style="font-style:italic">Modern Classical Physics - Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics</cite>. 1. Auflage. Princeton University Press, Princeton 2017, ISBN 0-691-15902-5, S. 355.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Kip+S.+Thorne%2C+Roger+D.+Blandford&rft.btitle=Modern+Classical+Physics+-+Optics%2C+Fluids%2C+Plasmas%2C+Elasticity%2C+Relativity%2C+and+Statistical+Physics&rft.date=2017&rft.edition=1.&rft.genre=book&rft.isbn=0691159025&rft.pages=355&rft.place=Princeton&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-4"><a href="#cite_ref-4">↑</a> David J. Barber, R. Loudon: <cite style="font-style:italic">An Introduction to the Properties of Condensed Matter</cite>. 1. Auflage. Cambridge University Press, Cambridge 1989, ISBN 978-0-521-26907-0, S. 217.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=David+J.+Barber%2C+R.+Loudon&rft.btitle=An+Introduction+to+the+Properties+of+Condensed+Matter&rft.date=1989&rft.edition=1.&rft.genre=book&rft.isbn=9780521269070&rft.pages=217&rft.place=Cambridge&rft.pub=Cambridge+University+Press" style="display:none">

</li>

<li id="cite_note-5"><a href="#cite_ref-5">↑</a> John David Jackson: <cite style="font-style:italic">Klassische Elektrodynamik</cite>. 5. Auflage. de Gruyter, Berlin 1981, ISBN 3-11-008074-5, S. 342.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=John+David+Jackson&rft.btitle=Klassische+Elektrodynamik&rft.date=1981&rft.edition=5.&rft.genre=book&rft.isbn=3110080745&rft.pages=342&rft.place=Berlin&rft.pub=de+Gruyter" style="display:none">

</li>

<li id="cite_note-6"><a href="#cite_ref-6">↑</a> John David Jackson: <cite style="font-style:italic">Klassische Elektrodynamik</cite>. 5. Auflage. de Gruyter, Berlin 1981, ISBN 3-11-008074-5, S. 376.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=John+David+Jackson&rft.btitle=Klassische+Elektrodynamik&rft.date=1981&rft.edition=5.&rft.genre=book&rft.isbn=3110080745&rft.pages=376&rft.place=Berlin&rft.pub=de+Gruyter" style="display:none">

</li>

<li id="cite_note-7"><a href="#cite_ref-7">↑</a> Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë: <cite style="font-style:italic">Quantum Mechanics, Volume 1 : Basic Concepts, Tools, and Applications.</cite> John Wiley & Sons, Kassel 2021, ISBN 0-471-16432-1, S. 11.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Claude+Cohen-Tannoudji%2C+Bernard+Diu%2C+Franck+Lalo%C3%AB&rft.btitle=Quantum+Mechanics%2C+Volume+1+%3A+Basic+Concepts%2C+Tools%2C+and+Applications.&rft.date=2021&rft.genre=book&rft.isbn=0471164321&rft.pages=11&rft.place=Kassel&rft.pub=John+Wiley+%26+Sons" style="display:none">

</li>

<li id="cite_note-8"><a href="#cite_ref-8">↑</a> Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë: <cite style="font-style:italic">Quantum Mechanics, Volume 1 : Basic Concepts, Tools, and Applications.</cite> John Wiley & Sons, Kassel 2021, ISBN 0-471-16432-1, S. 22.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Claude+Cohen-Tannoudji%2C+Bernard+Diu%2C+Franck+Lalo%C3%AB&rft.btitle=Quantum+Mechanics%2C+Volume+1+%3A+Basic+Concepts%2C+Tools%2C+and+Applications.&rft.date=2021&rft.genre=book&rft.isbn=0471164321&rft.pages=22&rft.place=Kassel&rft.pub=John+Wiley+%26+Sons" style="display:none">

</li>

<li id="cite_note-9"><a href="#cite_ref-9">↑</a> Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë: <cite style="font-style:italic">Quantum Mechanics, Volume 1 : Basic Concepts, Tools, and Applications.</cite> John Wiley & Sons, Kassel 2021, ISBN 0-471-16432-1, S. 29.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Claude+Cohen-Tannoudji%2C+Bernard+Diu%2C+Franck+Lalo%C3%AB&rft.btitle=Quantum+Mechanics%2C+Volume+1+%3A+Basic+Concepts%2C+Tools%2C+and+Applications.&rft.date=2021&rft.genre=book&rft.isbn=0471164321&rft.pages=29&rft.place=Kassel&rft.pub=John+Wiley+%26+Sons" style="display:none">

</li>

<li id="cite_note-10"><a href="#cite_ref-10">↑</a> Claude Cohen-Tannoudji, Bernard Diu, Franck Laloë: <cite style="font-style:italic">Quantum Mechanics, Volume 1 : Basic Concepts, Tools, and Applications.</cite> John Wiley & Sons, Kassel 2021, ISBN 0-471-16432-1, S. 30.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Claude+Cohen-Tannoudji%2C+Bernard+Diu%2C+Franck+Lalo%C3%AB&rft.btitle=Quantum+Mechanics%2C+Volume+1+%3A+Basic+Concepts%2C+Tools%2C+and+Applications.&rft.date=2021&rft.genre=book&rft.isbn=0471164321&rft.pages=30&rft.place=Kassel&rft.pub=John+Wiley+%26+Sons" style="display:none">

</li>

<li id="cite_note-11"><a href="#cite_ref-11">↑</a> Werner Heisenberg: <cite style="font-style:italic">Physikalische Prinzipien der Quantentheorie -</cite>. B. I.-Wissenschaftsverlag, Mannheim 1958, ISBN 3-411-00001-5, S. 10.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Werner+Heisenberg&rft.btitle=Physikalische+Prinzipien+der+Quantentheorie+-&rft.date=1958&rft.genre=book&rft.isbn=3411000015&rft.pages=10&rft.place=Mannheim&rft.pub=B.+I.-Wissenschaftsverlag" style="display:none">

</li>

<li id="cite_note-12"><a href="#cite_ref-12">↑</a> Neil W. Ashcroft, N. David Mermin: <cite style="font-style:italic">Solid State Physics -</cite>. 1. Auflage. Holt, Rinehart and Winston, Philadelphia 1976, ISBN 0-03-049346-3, S. 432.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Neil+W.+Ashcroft%2C+N.+David+Mermin&rft.btitle=Solid+State+Physics+-&rft.date=1976&rft.edition=1.&rft.genre=book&rft.isbn=0030493463&rft.pages=432&rft.place=Philadelphia&rft.pub=Holt%2C+Rinehart+and+Winston" style="display:none">

</li>

<li id="cite_note-13"><a href="#cite_ref-13">↑</a> Neil W. Ashcroft, N. David Mermin: <cite style="font-style:italic">Solid State Physics -</cite>. 1. Auflage. Holt, Rinehart and Winston, Philadelphia 1976, ISBN 0-03-049346-3, S. 140.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Neil+W.+Ashcroft%2C+N.+David+Mermin&rft.btitle=Solid+State+Physics+-&rft.date=1976&rft.edition=1.&rft.genre=book&rft.isbn=0030493463&rft.pages=140&rft.place=Philadelphia&rft.pub=Holt%2C+Rinehart+and+Winston" style="display:none">

</li>

<li id="cite_note-14"><a href="#cite_ref-14">↑</a> Neil W. Ashcroft, N. David Mermin: <cite style="font-style:italic">Solid State Physics -</cite>. 1. Auflage. Holt, Rinehart and Winston, Philadelphia 1976, ISBN 0-03-049346-3, S. 158.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Neil+W.+Ashcroft%2C+N.+David+Mermin&rft.btitle=Solid+State+Physics+-&rft.date=1976&rft.edition=1.&rft.genre=book&rft.isbn=0030493463&rft.pages=158&rft.place=Philadelphia&rft.pub=Holt%2C+Rinehart+and+Winston" style="display:none">

</li>

<li id="cite_note-15"><a href="#cite_ref-15">↑</a> Neil W. Ashcroft, N. David Mermin: <cite style="font-style:italic">Solid State Physics -</cite>. 1. Auflage. Holt, Rinehart and Winston, Philadelphia 1976, ISBN 0-03-049346-3, S. 214.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Neil+W.+Ashcroft%2C+N.+David+Mermin&rft.btitle=Solid+State+Physics+-&rft.date=1976&rft.edition=1.&rft.genre=book&rft.isbn=0030493463&rft.pages=214&rft.place=Philadelphia&rft.pub=Holt%2C+Rinehart+and+Winston" style="display:none">

</li>

<li id="cite_note-16"><a href="#cite_ref-16">↑</a> Frank S. jr., Crawford: <cite style="font-style:italic">Berkeley Physik Kurs 3 - Schwingungen und Wellen</cite>. 1. Auflage. Vieweg + Teubner, Braunschweig 1974, ISBN 978-3-322-90778-3, S. 53.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Frank+S.+jr.%2C+Crawford&rft.btitle=Berkeley+Physik+Kurs+3+-+Schwingungen+und+Wellen&rft.date=1974&rft.edition=1.&rft.genre=book&rft.isbn=9783322907783&rft.pages=53&rft.place=Braunschweig&rft.pub=Vieweg+%2B+Teubner" style="display:none">

</li>

<li id="cite_note-17"><a href="#cite_ref-17">↑</a> David J. Barber, R. Loudon: <cite style="font-style:italic">An Introduction to the Properties of Condensed Matter</cite>. 1. Auflage. Cambridge University Press, Cambridge 1989, ISBN 978-0-521-26907-0, S. 216.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=David+J.+Barber%2C+R.+Loudon&rft.btitle=An+Introduction+to+the+Properties+of+Condensed+Matter&rft.date=1989&rft.edition=1.&rft.genre=book&rft.isbn=9780521269070&rft.pages=216&rft.place=Cambridge&rft.pub=Cambridge+University+Press" style="display:none">

</li>

<li id="cite_note-Kuhlmann145-18">↑ <a href="#cite_ref-Kuhlmann145_18-0">a</a> <a href="#cite_ref-Kuhlmann145_18-1">b</a> Hendrik C. Kuhlmann: <cite style="font-style:italic">Strömungsmechanik -</cite>. Pearson Studium, München 2007, ISBN 978-3-8273-7230-7, S. 145.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Hendrik+C.+Kuhlmann&rft.btitle=Str%C3%B6mungsmechanik+-&rft.date=2007&rft.genre=book&rft.isbn=9783827372307&rft.pages=145&rft.place=M%C3%BCnchen&rft.pub=Pearson+Studium" style="display:none">

</li>

<li id="cite_note-19"><a href="#cite_ref-19">↑</a> Hendrik C. Kuhlmann: <cite style="font-style:italic">Strömungsmechanik -</cite>. Pearson Studium, München 2007, ISBN 978-3-8273-7230-7, S. 147.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Hendrik+C.+Kuhlmann&rft.btitle=Str%C3%B6mungsmechanik+-&rft.date=2007&rft.genre=book&rft.isbn=9783827372307&rft.pages=147&rft.place=M%C3%BCnchen&rft.pub=Pearson+Studium" style="display:none">

</li>

<li id="cite_note-20"><a href="#cite_ref-20">↑</a> Hendrik C. Kuhlmann: <cite style="font-style:italic">Strömungsmechanik -</cite>. Pearson Studium, München 2007, ISBN 978-3-8273-7230-7, S. 148.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Hendrik+C.+Kuhlmann&rft.btitle=Str%C3%B6mungsmechanik+-&rft.date=2007&rft.genre=book&rft.isbn=9783827372307&rft.pages=148&rft.place=M%C3%BCnchen&rft.pub=Pearson+Studium" style="display:none">

</li>

<li id="cite_note-21"><a href="#cite_ref-21">↑</a> Kip S. Thorne, Roger D. Blandford: <cite style="font-style:italic">Modern Classical Physics - Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics</cite>. 1. Auflage. Princeton University Press, Princeton 2017, ISBN 0-691-15902-5, S. 637.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Kip+S.+Thorne%2C+Roger+D.+Blandford&rft.btitle=Modern+Classical+Physics+-+Optics%2C+Fluids%2C+Plasmas%2C+Elasticity%2C+Relativity%2C+and+Statistical+Physics&rft.date=2017&rft.edition=1.&rft.genre=book&rft.isbn=0691159025&rft.pages=637&rft.place=Princeton&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-22"><a href="#cite_ref-22">↑</a> Kip S. Thorne, Roger D. Blandford: <cite style="font-style:italic">Modern Classical Physics - Optics, Fluids, Plasmas, Elasticity, Relativity, and Statistical Physics</cite>. 1. Auflage. Princeton University Press, Princeton 2017, ISBN 0-691-15902-5, S. 639.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Kip+S.+Thorne%2C+Roger+D.+Blandford&rft.btitle=Modern+Classical+Physics+-+Optics%2C+Fluids%2C+Plasmas%2C+Elasticity%2C+Relativity%2C+and+Statistical+Physics&rft.date=2017&rft.edition=1.&rft.genre=book&rft.isbn=0691159025&rft.pages=639&rft.place=Princeton&rft.pub=Princeton+University+Press" style="display:none">

</li>

<li id="cite_note-23"><a href="#cite_ref-23">↑</a> L. D. Landau, E. M. Lifschitz: <cite style="font-style:italic">Lehrbuch der theoretischen Physik, Band 7, Elastizitätstheorie -</cite>. 4. Auflage. Akademie Verlag, Berlin 1975, S. 127.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=L.+D.+Landau%2C+E.+M.+Lifschitz&rft.btitle=Lehrbuch+der+theoretischen+Physik%2C+Band+7%2C+Elastizit%C3%A4tstheorie+-&rft.date=1975&rft.edition=4.&rft.genre=book&rft.pages=127&rft.place=Berlin&rft.pub=Akademie+Verlag" style="display:none">

</li>

<li id="cite_note-24"><a href="#cite_ref-24">↑</a> Anton Hammer, Hildegard Hammer, Karl Hammer: <cite style="font-style:italic">Taschenbuch der Physik</cite>. 9. Auflage. Lindauer, München 2004, ISBN 3-87488-094-X, S. 83.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Anton+Hammer%2C+Hildegard+Hammer%2C+Karl+Hammer&rft.btitle=Taschenbuch+der+Physik&rft.date=2004&rft.edition=9.&rft.genre=book&rft.isbn=387488094X&rft.pages=83&rft.place=M%C3%BCnchen&rft.pub=Lindauer" style="display:none">

</li>

<li id="cite_note-25"><a href="#cite_ref-25">↑</a> L. D. Landau, E. M. Lifschitz: <cite style="font-style:italic">Lehrbuch der theoretischen Physik, Band 7, Elastizitätstheorie -</cite>. 4. Auflage. Akademie Verlag, Berlin 1975, S. 130.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=L.+D.+Landau%2C+E.+M.+Lifschitz&rft.btitle=Lehrbuch+der+theoretischen+Physik%2C+Band+7%2C+Elastizit%C3%A4tstheorie+-&rft.date=1975&rft.edition=4.&rft.genre=book&rft.pages=130&rft.place=Berlin&rft.pub=Akademie+Verlag" style="display:none">

</li>

<li id="cite_note-26"><a href="#cite_ref-26">↑</a> Frank S. jr., Crawford: <cite style="font-style:italic">Berkeley Physik Kurs 3 - Schwingungen und Wellen</cite>. 1. Auflage. Vieweg + Teubner, Braunschweig 1974, ISBN 978-3-322-90778-3, S. 160.<span class="Z3988" title="ctx_ver=Z39.88-2004&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Abook&rfr_id=info:sid/de.wikipedia.org:Dispersionsrelation&rft.au=Frank+S.+jr.%2C+Crawford&rft.btitle=Berkeley+Physik+Kurs+3+-+Schwingungen+und+Wellen&rft.date=1974&rft.edition=1.&rft.genre=book&rft.isbn=9783322907783&rft.pages=160&rft.place=Braunschweig&rft.pub=Vieweg+%2B+Teubner" style="display:none">

</li>

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