# Show the following equation satisfies the wave equation [closed]

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?

• $r = r(x,y,z)$. Commented Feb 6, 2019 at 6:53
• Welcome to Physics! Please note that this site is not a place to obtain solutions to worked problems. Please see this Meta post on asking homework-like questions and this Meta post for "check my work problems". Commented Feb 6, 2019 at 17:34
• Either use $r=\sqrt{x^2+y^2+z^2}$ or find the wave equation in spherical coordinates. Commented Feb 7, 2019 at 0:55

Since this is a homework-exercise like problem we restrict our answers to hints. So, you must find the expressions $$$$\dfrac{\partial^{2} E}{\partial t^{2}}\,, \quad \nabla^{2}E \tag{A}\label{A}$$$$ and prove that there exists a positive constant $$\;a\;$$ such that $$$$\dfrac{\partial^{2} E}{\partial t^{2}}\boldsymbol{=}a\cdot\nabla^{2}E \tag{B}\label{B}$$$$ Fortunately enough, if a scalar function $$\;\psi(r)\;$$ of the spherical coordinates $$\;r,\theta,\phi\;$$ is independent of $$\;\theta,\phi\;$$ then $$$$\nabla^{2}\psi\boldsymbol{=}\dfrac{1}{r^2}\dfrac{\partial}{\partial r}\left(r^2\dfrac{\partial\psi}{\partial r}\right)\boldsymbol{\equiv}\dfrac{1}{r}\dfrac{\partial^{2}}{\partial r^{2}}\left(r\psi\right) \tag{C}\label{C}$$$$
The three dimensional wave equation can be written in spherical coordinates. There, the transformation $$(x,y,z)$$ $$\rightarrow$$ $$(r, \phi, \theta)$$ will help you out.
This equation is obviously in spherical coordinates, in which $$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.
• 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? Commented Feb 6, 2019 at 6:20