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Green's Function for Helmholtz Eqn in Cube

  1. Nov 28, 2011 #1
    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]\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.
     
  2. jcsd
  3. Nov 29, 2011 #2
    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.
     
  4. Nov 29, 2011 #3
    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.
     
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