Green Function and Boundary Conditions

1. Jul 14, 2008

robousy

Hey folks,

I'm trying to find the Green function for the equation
$$-\partial_\mu \partial^\mu \phi = K$$
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 $$G|_0= G|_a=0$$

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:

$$(-\partial_z^2-\lambda^2)g(z,z')=\delta(z-z')$$

where $$\lambda^2=\omega^2-k^2$$

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. Jul 14, 2008

smallphi

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.

3. Jul 15, 2008

robousy

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.

4. Jul 15, 2008

smallphi

The freeware version of 'Introduction to quantum fields in classical backgrounds' by S. Winitzki and V. Mukhanov:

http://homepages.physik.uni-muenchen.de/~Winitzki/T6/book.pdf

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.

5. Jul 15, 2008

robousy

Thats a very nice link. Great looking book. Thansk a lot smallphi!