Green's Function for Helmholtz Eqn in Cube

[Herr Malus]

Homework Statement

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

Homework Equations

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

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]\nabla²[/itex]u+k²u=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.

 

[susskind_leon]

I guess you could do that. You can always use Fourier transform to obtain the Green's function without BC and add a particular solution of the homogeneous Helmholtz equation that satisfies the BC instead, which I imagine would be a less awkward approach.

 

[Herr Malus]

So using the Fourier transform would just give you the green's function for helmholtz in 3d space which is something like an exponential divided by 4∏r followed by simply any function satisfying the BC? So in a 1-d case:
Helmholtz Green's function + sinkx for x<x' or
Helmholtz Green's function +sink(x-L) for x>x'

Thanks for the help.