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Spherical Harmonic Wave Function =? 3D Wave Function

by drumercalzone
Tags: harmonic, optics, spherical, wave
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drumercalzone
#1
Aug29-11, 01:02 PM
P: 13
1. The problem statement, all variables and given/known data
Prove that the spherical harmonic wave function [tex] \frac{1}{r}e^{i(kr-{\omega}t)} [/tex] is a solution of the three-dimensional wave equation, where [tex] r = (x^2+y^2+z^2)^{\frac{1}{2}} [/tex]. The proof is easier if spherical coordinates are used.

2. Relevant equations

Wave function: [tex] \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}[/tex]

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?
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vela
#2
Aug30-11, 09:27 PM
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You wrote the wave equation using Cartesian coordinates. More generally, you can write it as[tex]\nabla^2 U = \frac{1}{u^2}\frac{\partial^2 U}{\partial t^2}[/tex]
In your textbook, you can probably find how to write the Laplacian [itex]\nabla^2[/itex] using spherical coordinates. (Or just Google it.)


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