Spherical Harmonic Wave Function =? 3D Wave Function

[drumercalzone]

Homework Statement

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.

Homework Equations

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

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?

 

[vela]

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.)