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I've been wrestling with this question for quite some time now, and the result was like 20 leaves of paper packed with scribbling...anyway, here's the question:

I need to find the solution to the following source excited Helmholtz equation:

$$(\nabla^2 + k^2)G_1(r,a) = -\delta(r-a)$$ With the boundary condition: $$\left. G_1(r,a)\right|_{z=0} = 0$$

Now, the solution is given by the integral: \begin{multline} G_1(r,a)=\iiint G(r,r') \delta(r'-a) dV' \\ + \iint [G(r,r')\nabla'G_1(r',a)-G_1(r',a)\nabla'G(r,r')]\cdot \hat{n} dS' \end{multline} (The surface integral is on a closed surface, for some reason MathJaX doesn't understand what \oiint is) The second term in the surface integral drops since it's $0$ on all the boundaries of the volume that interests us (which is $z \geq 0$).

HOWEVER, I do not know what the gradient of $G_1$ is on $z=0$, and I understand that I'm free to chose it however I see fit as long as it only affects the $z<0$ region.

Even simplifying the problem to a one dimensional problem I still can't solve it.

I do know what the solution is, because I've been thought many years ago in introduction to Electromagnetics that when encountering a boundary over which the electrical potential is $0$, I need to mirror all my sources and give them a negative sign, and if I plug that solution back into the equation is does satisfy it and the boundary condition.

But how THAT solution is supposed to be inferred from the mathematical problem is beyond me. Any help/hints/insights would be most welcome.

(Sorry for the paucity of tags, I wanted to use "Helmholtz" and "Green" but I'm still a newbie so I can't create new tags...)

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What is $G(r,r')$? What are the domains of integration? –  Willie Wong Mar 17 '11 at 19:51
1  
$G(r,r')$ is the solution to the Helmholtz equation in free space: $G(r,r')=\frac{e^{-jk|r-r'|}}{4\pi|r-r'|}$. And the domains of the integration are over the $z\geq0$ "half" volume. –  Dany Zatuchna Mar 17 '11 at 19:54

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