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Green Function and Boundary Conditions

  1. Jul 14, 2008 #1
    Hey folks,

    I'm trying to find the Green function for the equation
    [tex]-\partial_\mu \partial^\mu \phi = K[/tex]
    where K is some source term. Its a 2D problem with the wave confined to a rectangular cavity where the cavity is located at z = 0 and z=a.
    This tells me that [tex]G|_0= G|_a=0[/tex]

    I've pretty much solved this problem for the case of NO boundary conditions, eg an infinte wave, I'm just stuck on where to put in info on the BC's.

    I am confident I have derived the following expression correctly as it matches with a book I am using:


    where [tex]\lambda^2=\omega^2-k^2 [/tex]

    So really this is the problem I need to solve - where g is the reduced Greens function.
    I can solve this by taking the FT and using contour integrals...pretty standard, but this is for an infinite wave. My question is 'where and how do I impose boundary conditions for g(0,z')=g(a,z')=0 ??'.
  2. jcsd
  3. Jul 14, 2008 #2
    My rough guess is that you have to demand that each Fourier component or "mode" of g(z) vanishes at z=0 and z=a. That means you have to use sines (because they are all zero at z=0) with specific allowed frequencies so that they are zero also at z=a.
  4. Jul 15, 2008 #3
    Hi smallphi, yes the solution is sines - I'm just hoping to find out how to put in the BCs. Eg - whats the next step in solving [tex](-\partial_z^2-\lambda^2)g(z,zsingle-quote)=\delta(z-zsingle-quote)[tex].

    Like I said I've done this by Fourier transforming then solving using contour method. But I'm not sure at what stage to use the BC's.
  5. Jul 15, 2008 #4
    The freeware version of 'Introduction to quantum fields in classical backgrounds' by S. Winitzki and V. Mukhanov:


    contain an appendix A2: Green's functions, boundary conditions and contours,

    which may shed some light on how to incorporate the boundary conditions in the contour method.
  6. Jul 15, 2008 #5
    Thats a very nice link. Great looking book. Thansk a lot smallphi!
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