(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Find the Green's Function for the Helmholtz Eqn in the cube 0≤x,y,z≤L by solving the equation:

[itex]\nabla[/itex]^{2}u+k^{2}u=δ(x-x')

with u=0 on the surface of the cube

This is problem 9-4 in Mathews and Walker Mathematical Methods of Physics

2. Relevant equations

Sines, they have the properties we're looking for.

3. The attempt at a solution

So, if the solution is in one dimension we obviously have

G=asinkx x<x' and

G=bsink(x-L) for x>x'

And these are obtained by solving [itex]\nabla^{2}[/itex]u+k^{2}u=0 and looking for a and b based on the matching condition at x'. I'm wondering if I can do the same thing for this three dimensional case, using separation of variables and solving for k^2 as the eigenvalue. Then enforcing orthogonality on the functions I get. My problem with this is just that it seems very awkward to have eight sets of equation for the Green's function, based on whether x<x' while y>y' and z<z' for example.

Any help appreciated.

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# Green's Function for Helmholtz Eqn in Cube

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