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So I am trying to do a problem from my book that says show the following equation satisfies the 3 dimensional wave equation. $$E(r,t)\space = \frac{A}{r}\ e^{i(kr-\omega t))} ~ $$

I am rather confused on how to attempt this. As the 3 dimensional wave equation has partial derivative with respect to x, y and z. Aren't there no x,y and z in the above equation? So don't I get 0 as the answer?

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3 Answers 3

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Hint :

Since this is a homework-exercise like problem we restrict our answers to hints. So, you must find the expressions \begin{equation} \dfrac{\partial^{2} E}{\partial t^{2}}\,, \quad \nabla^{2}E \tag{A}\label{A} \end{equation} and prove that there exists a positive constant $\;a\;$ such that \begin{equation} \dfrac{\partial^{2} E}{\partial t^{2}}\boldsymbol{=}a\cdot\nabla^{2}E \tag{B}\label{B} \end{equation} Fortunately enough, if a scalar function $\;\psi(r)\;$ of the spherical coordinates $\;r,\theta,\phi\;$ is independent of $\;\theta,\phi\;$ then \begin{equation} \nabla^{2}\psi\boldsymbol{=}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2\dfrac{\partial\psi}{\partial r}\right)\boldsymbol{\equiv}\dfrac{1}{r}\dfrac{\partial^{2}}{\partial r^{2}}\left(r\psi\right) \tag{C}\label{C} \end{equation}

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The three dimensional wave equation can be written in spherical coordinates. There, the transformation $(x,y,z)$ $\rightarrow$ $(r, \phi, \theta)$ will help you out.

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This equation is obviously in spherical coordinates, in which $r=\sqrt{x^2+y^2+z^2}$. You may either transform it and do partial derivative with respect to x, y and z in Cartesian coordinate, or do partial derivative with respect to r in spherical coordinate.

You may refer to this for a spherical wave.

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  • $\begingroup$ So if I replace r by $r=\sqrt{x^2+y^2+z^2}$ and use the 3 dimensional wave equation for cartesian coordinates. I should get the correct answer? $\endgroup$
    – Schnieder
    Commented Feb 6, 2019 at 6:20
  • $\begingroup$ Yes. However, I would prefer using spherical coordinate to do this problem. $\endgroup$
    – pinchun
    Commented Feb 6, 2019 at 7:13

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