Spherical Harmonic Wave Function =? 3D Wave Function

1. Aug 29, 2011

drumercalzone

1. The problem statement, all variables and given/known data
Prove that the spherical harmonic wave function $$\frac{1}{r}e^{i(kr-{\omega}t)}$$ is a solution of the three-dimensional wave equation, where $$r = (x^2+y^2+z^2)^{\frac{1}{2}}$$. The proof is easier if spherical coordinates are used.

2. Relevant equations

Wave function: $$\frac{{\partial}^2U}{\partial x^2} + \frac{{\partial}^2U}{\partial y^2} + \frac{{\partial}^2U}{\partial z^2} = \frac{1}{u^2} \frac{{\partial}^2U}{\partial t^2}$$

3. The attempt at a solution

I really just don't even know where to start. Do I first convert the x,y,z into polar coordinates? or do I just substitue what's above in for r? But then what's up with imaginary part?

2. Aug 30, 2011

vela

Staff Emeritus
You wrote the wave equation using Cartesian coordinates. More generally, you can write it as$$\nabla^2 U = \frac{1}{u^2}\frac{\partial^2 U}{\partial t^2}$$
In your textbook, you can probably find how to write the Laplacian $\nabla^2$ using spherical coordinates. (Or just Google it.)