Proving Kirchhoff's diffraction equation with Green's second identity

[gN]Berserker]

Hi Guys,

I assume you are familiar with the equations so i do not post them (please write if u want me to post them).

One of the steps to prove Kirchhoff's diffraction equation is to use Green's second identity.
This identity shows the relation between the solutions in the volume and boundary. The two solutions - are two scalar functions phi and psi that generate a vector field trough: A = phi*del(psi).
all till now is just definitions.

In order to get the Kirchhoff's diffraction equation a Green FUNCTION is plugged instead of psi into the identity. This green function is a solution to the Helmholtz equation with Dirac's delta as perturbation.

I do not understand:

1. the connection of Green's second identity to the topic - why it is used at all.
2. why the Green function is just plugged in, instead of one of the solutions.

I went trough few books: Born and wolf, Jackson, Goodman,,, but no one gives satisfactory explanation. actually all just ignore (2) and say something of the kind:
"We call the second solution function (psi) an auxiliary function that is also the Green function of Helmholtz equation."

What is the rational behind choosing one of the solutions as a Green function at all?
(I guess it makes sense to choose the Green function of Helmholtz equation as we speak about wave propagation).

Thanks!
Please write if you are not familiar with the equations, I will write them or link you.

 

[maajdl]

Try anything else that the solution for a Dirac's delta,
and you won't find the explicit solution that the Kirchhoff's equation provides.
The Dirac's delta is what pops the solution out of the Green identity.