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<title>D’Alembert-Operator</title>

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<h1 id="firstHeading" class="firstHeading mw-first-heading">D’Alembert-Operator</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">Der <a href="Jean_Baptiste_le_Rond_d%E2%80%99Alembert" class="mw-redirect" title="Jean Baptiste le Rond d’Alembert">D’Alembert</a>-Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/029b77f09ebeaf7528fc831fe57848be51f2240b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }" loading="lazy"> ist ein <a href="Differentialoperator" title="Differentialoperator">Differentialoperator</a> zweiter Ordnung, der auf Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(t,x_{1},\dots ,x_{d-1})}">

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<annotation encoding="application/x-tex">{\displaystyle f(t,x_{1},\dots ,x_{d-1})}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/3ae4fcbf3dff86504e142bda54609df6b66f5fd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:17.046ex; height:2.843ex;" alt="{\displaystyle f(t,x_{1},\dots ,x_{d-1})}" loading="lazy"> der <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e85ff03cbe0c7341af6b982e47e9f90d235c66ab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.216ex; height:2.176ex;" alt="{\displaystyle d}" loading="lazy">-dimensionalen Raumzeit wirkt (z. B. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d=4}">

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<annotation encoding="application/x-tex">{\displaystyle d=4}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/07c9d2f1986f62d1fd4cec8f910ecda996b1880b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:5.477ex; height:2.176ex;" alt="{\displaystyle d=4}" loading="lazy">).

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<annotation encoding="application/x-tex">{\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\sum _{i=1}^{d-1}{\frac {\partial ^{2}}{\partial x_{i}{}^{2}}}\ .}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/570df00ef9cda0171021ef63bd1d64274a880efc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:25ex; height:7.343ex;" alt="{\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\sum _{i=1}^{d-1}{\frac {\partial ^{2}}{\partial x_{i}{}^{2}}}\ .}" loading="lazy"></dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/029b77f09ebeaf7528fc831fe57848be51f2240b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }" loading="lazy"> (gesprochen Box)

ähnelt dem des <a href="Laplace-Operator" title="Laplace-Operator">Laplace-Operators</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> und es gilt die Beziehung

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<annotation encoding="application/x-tex">{\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\Delta _{x}\ .}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b5a9fcae221d1a6efe801db7749470f087e9cd33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.171ex; width:19.028ex; height:6.009ex;" alt="{\displaystyle \Box ={\frac {1}{c^{2}}}{\frac {\partial ^{2}}{\partial t^{2}}}-\Delta _{x}\ .}" loading="lazy"></dd></dl>

Der D'Alembert-Operator ist der Differentialoperator der <a href="Wellengleichung" title="Wellengleichung">Wellengleichung</a> und der <a href="Klein-Gordon-Gleichung" title="Klein-Gordon-Gleichung">Klein-Gordon-Gleichung</a> und heißt auch Wellenoperator oder Quabla-Operator.

In der Physik wird auch die Konvention verwendet, dass die Zeit-Koordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> in der obig angegebenen Gleichung mit der Geschwindigkeit <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"> zusammengefasst wird. Diese Zusammenfassung lässt sich wiederum als Wegstrecke interpretieren. Dabei wäre die Koordinate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \tau =ct}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/2476a7ab60af6fadfba49ed5ae89c1a992d8c6c7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.147ex; height:2.009ex;" alt="{\displaystyle \tau =ct}" loading="lazy"> die Strecke, die von der Welle in der Zeit <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/65658b7b223af9e1acc877d848888ecdb4466560.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.84ex; height:2.009ex;" alt="{\displaystyle t}" loading="lazy"> mit der Geschwindigkeit

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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"> durchlaufen wird.

<div class="mw-heading mw-heading2"><h2 id="Lorentzinvarianz_des_D'Alembert-Operators">Lorentzinvarianz des D'Alembert-Operators</h2></div>

Die Koeffizienten der zweiten Ableitungen im Wellenoperator sind die Komponenten der (inversen) <a href="Minkowski-Raum" title="Minkowski-Raum"> Raumzeitmetrik</a>

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<annotation encoding="application/x-tex">{\displaystyle \Box =\eta ^{mn}\partial _{x^{m}}\partial _{x^{n}}\ ,\ \eta ={\text{diag}}(1,-1,\dots ,-1)\ .}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0b700a1d65d0ba807eeffc246bdf349cd439f130.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:42.936ex; height:2.843ex;" alt="{\displaystyle \Box =\eta ^{mn}\partial _{x^{m}}\partial _{x^{n}}\ ,\ \eta ={\text{diag}}(1,-1,\dots ,-1)\ .}" loading="lazy"></dd></dl>

In der ebenso verbreiteten Konvention, das Negative dieser <a href="Quadratische_Form" title="Quadratische Form">quadratischen Form</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\text{diag}}(-1,+1,\dots ,+1)}">

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<annotation encoding="application/x-tex">{\displaystyle {\text{diag}}(-1,+1,\dots ,+1)}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/48f2658b2d58841f31c29b33a88a346555a75efc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.197ex; height:2.843ex;" alt="{\displaystyle {\text{diag}}(-1,+1,\dots ,+1)}" loading="lazy">, als Raumzeitmetrik zu bezeichnen,

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/029b77f09ebeaf7528fc831fe57848be51f2240b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }" loading="lazy"> für das Negative des hier definierten D'Alembert-Operators.

So wie die Raumzeitmetrik <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \eta }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e4d701857cf5fbec133eebaf94deadf722537f64.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:1.169ex; height:2.176ex;" alt="{\displaystyle \eta }" loading="lazy"> ist der D'Alembert-Operator <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Box }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/029b77f09ebeaf7528fc831fe57848be51f2240b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.808ex; height:2.176ex;" alt="{\displaystyle \Box }" loading="lazy"> invariant unter Translationen und <a href="Lorentztransformation" class="mw-redirect" title="Lorentztransformation">Lorentztransformationen</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Lambda }">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/0ac0a4a98a414e3480335f9ba652d12571ec6733.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.613ex; height:2.176ex;" alt="{\displaystyle \Lambda }" loading="lazy">. Angewendet auf Lorentzverkettete Funktionen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f\circ \Lambda ^{-1}}">

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<annotation encoding="application/x-tex">{\displaystyle f\circ \Lambda ^{-1}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cc9962ae46de7dc6b54e05e1d9afd8a3d157659a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.419ex; height:3.009ex;" alt="{\displaystyle f\circ \Lambda ^{-1}}" loading="lazy"> ergibt er dasselbe, wie die

Lorentzverkettete abgeleitete Funktion

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<annotation encoding="application/x-tex">{\displaystyle (\Box f)\circ \Lambda ^{-1}=\Box \,(f\circ \Lambda ^{-1})\ .}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/766ada4ac5a1f653a808fd9ec253d9b4f32556f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:26.786ex; height:3.176ex;" alt="{\displaystyle (\Box f)\circ \Lambda ^{-1}=\Box \,(f\circ \Lambda ^{-1})\ .}" loading="lazy"></dd></dl>

<div class="mw-heading mw-heading2"><h2 id="Greensche_Funktion">Greensche Funktion</h2></div>

Eine <a href="Greensche_Funktion" title="Greensche Funktion">Greensche Funktion</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(t,t',x,x')}">

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<annotation encoding="application/x-tex">{\displaystyle G(t,t',x,x')}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/4b4606bbb0455f455c8a9ba6c218fc71076e1f86.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.446ex; height:3.009ex;" alt="{\displaystyle G(t,t',x,x')}" loading="lazy"> des D'Alembert-Operators erfüllt als dessen Rechtsinverses die Definitionsgleichung

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<annotation encoding="application/x-tex">{\displaystyle \square (t,\mathbf {x} )G(t-t^{\prime },\mathbf {x} -\mathbf {x} ^{\prime })=\delta (t-t^{\prime })\delta (\mathbf {x} -\mathbf {x} ^{\prime })}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/1053e6d5caa21c9d92439f7b9623af9c1ace7aeb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:43.489ex; height:3.009ex;" alt="{\displaystyle \square (t,\mathbf {x} )G(t-t^{\prime },\mathbf {x} -\mathbf {x} ^{\prime })=\delta (t-t^{\prime })\delta (\mathbf {x} -\mathbf {x} ^{\prime })}" loading="lazy">.</dd></dl>

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

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c5321cfa797202b3e1f8620663ff43c4660ea03a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.049ex; height:2.343ex;" alt="{\displaystyle \delta }" loading="lazy"> die Diracsche <a href="Delta-Distribution" title="Delta-Distribution">Delta-Distribution</a>. Da es sich um einen nicht explizit zeit- und ortsabhängigen Operator handelt, hängt <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

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</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"> nur von den Differenzen <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (t-t')}">

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<annotation encoding="application/x-tex">{\displaystyle (t-t')}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c81ad862750efdc78530cd1d18e956664895e570.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.014ex; height:3.009ex;" alt="{\displaystyle (t-t')}" loading="lazy"> sowie <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (x-x')}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/e1dee28ea6aae8fe367cd1fa61aac00f3f59b388.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.994ex; height:3.009ex;" alt="{\displaystyle (x-x')}" loading="lazy"> ab, weshalb wir <a href="Ohne_Beschr%C3%A4nkung_der_Allgemeinheit" title="Ohne Beschränkung der Allgemeinheit">ohne Beschränkung der Allgemeinheit</a> die gestrichenen Koordinaten null setzen können. Für die <a href="Fouriertransformierte" class="mw-redirect" title="Fouriertransformierte">Fouriertransformierte</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(\omega ,k)}">

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

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74fbb16d434f31b521ebf67910eb12003a83eac0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.327ex; height:2.843ex;" alt="{\displaystyle G(\omega ,k)}" loading="lazy">

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<annotation encoding="application/x-tex">{\displaystyle G(t,\mathbf {x} )={\frac {1}{(2\pi )^{3}}}\iint d\omega \ d^{3}\mathbf {k} \;\mathrm {e} ^{\mathrm {i} (\omega t-\mathbf {kx} )}\ G(\omega ,\mathbf {k} )}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c36f1594de8d88fa269a95738939f28042af1bbf.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:43.977ex; height:6.009ex;" alt="{\displaystyle G(t,\mathbf {x} )={\frac {1}{(2\pi )^{3}}}\iint d\omega \ d^{3}\mathbf {k} \;\mathrm {e} ^{\mathrm {i} (\omega t-\mathbf {kx} )}\ G(\omega ,\mathbf {k} )}" loading="lazy"></dd></dl>

ergibt sich dann folgende <a href="Algebraische_Gleichung" title="Algebraische Gleichung">algebraische Gleichung</a>:

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<annotation encoding="application/x-tex">{\displaystyle G(\omega ,\mathbf {k} )={\frac {1}{-(\omega /c)^{2}+k^{2}}}}</annotation>

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</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/413d7755659b18f28cd99ed8a960fd9eb731096e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:24.854ex; height:6.009ex;" alt="{\displaystyle G(\omega ,\mathbf {k} )={\frac {1}{-(\omega /c)^{2}+k^{2}}}}" loading="lazy"></dd></dl>

Die <a href="Polstelle" title="Polstelle">Polstellen</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(\omega ,k)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74fbb16d434f31b521ebf67910eb12003a83eac0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.327ex; height:2.843ex;" alt="{\displaystyle G(\omega ,k)}" loading="lazy"> liegen genau dort, wo die <a href="Dispersionsrelation" title="Dispersionsrelation">Dispersionsrelation</a> für elektromagnetische Wellen im <a href="Vakuum" title="Vakuum">Vakuum</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega ^{2}=c^{2}k^{2}}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/cad04ebcefd21ed39fa92dd42701f58207e8ce4a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.925ex; height:2.676ex;" alt="{\displaystyle \omega ^{2}=c^{2}k^{2}}" loading="lazy">) erfüllt ist. Die Lösungen der homogenen Wellengleichung fallen also genau mit den Polen der Greenschen Funktion zusammen, was ein für <a href="%C3%9Cbertragungsfunktion" title="Übertragungsfunktion">Antwortfunktionen</a> typisches <a href="Resonanz" title="Resonanz">Resonanzverhalten</a> ist.

Um die Rücktransformation durchführen zu können, betrachten wir die <a href="Analytische_Fortsetzung" title="Analytische Fortsetzung">analytische Fortsetzung</a> von <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G(\omega ,k)}">

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/74fbb16d434f31b521ebf67910eb12003a83eac0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.327ex; height:2.843ex;" alt="{\displaystyle G(\omega ,k)}" loading="lazy"> für komplexe Frequenzen. Mit Hilfe des <a href="Residuenkalk%C3%BCl" class="mw-redirect" title="Residuenkalkül">Residuenkalküls</a> kann man die Pole bei <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle |\omega |=ck}">

</mrow>

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ba7f45ee9034fa0c5ffa91a6d26c8dfd2c356542.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.056ex; height:2.843ex;" alt="{\displaystyle |\omega |=ck}" loading="lazy"> „umschiffen“, wobei verschiedene Pfade verschiedenen Randbedingungen entsprechen. Man unterscheidet:

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/23fa05c4739ac98a7d9ab6eae782092f399fe08e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.527ex; height:2.843ex;" alt="{\displaystyle G(\omega ,\mathbf {k} )}" loading="lazy"></th>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/fd5451cfffcea5ef5888a78c4d1f3a47a02e504e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.839ex; height:2.843ex;" alt="{\displaystyle G(t,x)}" loading="lazy">

</th></tr>

<tr>

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

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/ea64c8eaee51f08eecca685b15f7fed130db38fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.338ex; height:2.509ex;" alt="{\displaystyle G^{+}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{-(\omega /c+\mathrm {i} \epsilon )^{2}+k^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/c8aab190c718f2c581781835bacab8a0b0522776.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.66ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{-(\omega /c+\mathrm {i} \epsilon )^{2}+k^{2}}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi x}}\delta \left(t-{\frac {x}{c}}\right)\Theta (t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/af61377baa42565b4a8cb827c7c14ba03fca564e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:19.561ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4\pi x}}\delta \left(t-{\frac {x}{c}}\right)\Theta (t)}" loading="lazy">

</td></tr>

<tr>

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

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/b17f1aad85ad42f94562d4202ac0e9f1b924b631.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.338ex; height:2.509ex;" alt="{\displaystyle G^{-}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mrow>

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{-(\omega /c-\mathrm {i} \epsilon )^{2}+k^{2}}}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/9907dfe7572c1f8f1a5575f70c9a027d3ebd39b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.671ex; width:18.66ex; height:6.009ex;" alt="{\displaystyle {\frac {1}{-(\omega /c-\mathrm {i} \epsilon )^{2}+k^{2}}}}" loading="lazy">

</td>

<mfrac>

<mrow>

</mrow>

</mfrac>

</mrow>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\frac {1}{4\pi x}}\delta \left(t+{\frac {x}{c}}\right)\Theta (-t)}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/926f69fb8824cfe8cb70db719a1e5b1906da87e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:21.37ex; height:5.176ex;" alt="{\displaystyle {\frac {1}{4\pi x}}\delta \left(t+{\frac {x}{c}}\right)\Theta (-t)}" loading="lazy">

</td></tr></tbody></table>

Die Greensche Funktion im Frequenzraum ist dabei im <a href="Grenzwert_(Funktion)" title="Grenzwert (Funktion)">Grenzwert</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \epsilon \to 0^{+}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \epsilon \to 0^{+}}</annotation>

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/56202e05acacec248c9d731e1d2f3fb0aaf4b304.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.231ex; height:2.509ex;" alt="{\displaystyle \epsilon \to 0^{+}}" loading="lazy"> zu verstehen, was den verschiedenen Pfaden um die Pole im Integral entspricht.

Der Faktor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle t-x/c}">

</mrow>

</mstyle>

</mrow>

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

</semantics>

</math><img src="./_assets_/eb734a37dd21ce173a46342d1cc64c92/5327359cdccb3ba3ac1b221d3f37df06dba0d640.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.179ex; height:2.843ex;" alt="{\displaystyle t-x/c}" loading="lazy"> entspricht dem Ausbreitungsgesetz einer <a href="Kugelwelle" title="Kugelwelle">Kugelwelle</a>.

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

<ul><li><a href="Torsten_Flie%C3%9Fbach" title="Torsten Fließbach">Torsten Fließbach</a>: Elektrodynamik. Lehrbuch zur theoretischen Physik II. 6. Auflage. Springer Spektrum Akademischer Verlag, Berlin / Heidelberg 2012, ISBN 978-3-8274-3035-9.</li></ul></div><div><div class="zim-footer">

Dieser Artikel wurde von <a class="external text" title="Zuletzt bearbeitet am 2024-12-01" href="https://de.wikipedia.org/wiki/?title=D%E2%80%99Alembert-Operator&oldid=250851586">Wikipedia</a> herausgegeben. Der Text ist unter <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.de">Creative Commons Attribution-Share Alike 4.0</a> verfügbar, sofern nicht anders angegeben. Für die Mediendateien können zusätzliche Bedingungen gelten.

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