Green Function and Boundary Conditions

In summary, the problem is to find the Green function for the equation[tex]-\partial_\mu \partial^\mu \phi = K[/tex],where K is some source term. It can be solved using the Fourier transform, but the book I am using says that you need to impose boundary conditions so that each Fourier component or "mode" of K vanishes at z=0 and z=a.
  • #1
robousy
334
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 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:

[tex](-\partial_z^2-\lambda^2)g(z,z')=\delta(z-z')[/tex]

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 ??'.
 
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  • #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.
 
  • #3
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)[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
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.
 
Last edited by a moderator:
  • #5
Thats a very nice link. Great looking book. Thansk a lot smallphi!
 

Related to Green Function and Boundary Conditions

1. What is a Green function in physics?

A Green function, also known as a Green's function, is a mathematical function used to solve differential equations that describe physical systems. It represents the response of a system to a point source or impulse at a specific location.

2. How are Green functions related to boundary conditions?

Green functions are typically used to solve boundary value problems, where the behavior of a system is known at certain boundary points. The boundary conditions are used to determine the specific form of the Green function for the given system.

3. Can Green functions be used in any type of physical system?

Yes, Green functions have a wide range of applications in physics, including electromagnetics, quantum mechanics, fluid dynamics, and many others. They are a useful tool for solving a variety of problems in different fields of science and engineering.

4. How are Green functions calculated or derived?

The process of calculating or deriving a Green function depends on the specific problem at hand. In general, it involves solving a set of differential equations and applying boundary conditions to determine the form of the function. In some cases, analytic solutions may be possible, while in others numerical methods may be necessary.

5. What are some real-world applications of Green functions?

Green functions have many practical applications, including modeling the propagation of electromagnetic waves in materials, calculating the distribution of temperature in heat conduction problems, and predicting the behavior of particles in quantum mechanical systems. They are also commonly used in image and signal processing techniques.

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