Green Function and Boundary Conditions

[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 infinite 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 ??'.

 

[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.

 

[robousy]

Hi smallphi, yes the solution is sines - I'm just hoping to find out how to put in the BCs. Eg - what's the next step in solving [tex](-\partial_z^2-\lambda^2)g(z,zsingle-quote)=\delta(z-zsingle-quote)$$.<br /> <br /> Like I said I've done this by Fourier transforming then solving using contour method. But I'm not sure at what stage to use the BC's.$$[/tex]

 

[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.

 

[robousy]

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