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bdforbes
#56
Aug2-09, 12:29 AM
P: 152
I'm now trying to determine the fundamental solution to the Helmholtz wave equation. My starting point is

[tex](\nabla^2+k^2)G(\vec{r})=-4\pi\delta(\vec{r})[/tex].

Treating this as a distributional statement, it is equivalent to

[tex] \int G(\vec{r})(\nabla^2+k^2)e^{-2\pi i\vec{\xi}\cdot\vec{r}}d^3\vec{r}=-4\pi[/tex]

This leads to the desired result after some contour integration.

But if I want to be a little more rigorous and not rely on distributional techniques, I run into trouble. Integrating both sides over punctured [itex]\textbf{R}^3[/itex]:

[tex]
\begin{align*}
0&=\lim_{\epsilon\to 0}\int_{B^c_\epsilon}e^{-2\pi i\vec{\xi}\cdot\vec{r}}(\nabla^2+k^2)G(\vec{r})d^3\vec{r}\\
&=\lim_{\epsilon\to 0}\int_{B^c_\epsilon}\left[\nabla^2\left(G(\vec{r})e^{-2\pi i\vec{\xi}\cdot\vec{r}}\right)-G(\vec{r})\nabla^2e^{-2\pi i\vec{\xi}\cdot\vec{r}}-2\nabla G(\vec{r})\cdot\nabla e^{-2\pi i\vec{\xi}\cdot\vec{r}}\right]d^3\vec{r}+k^2\tilde{G}(\vec{\xi})\\
&=\left[4\pi^2\xi^2+k^2\right]\tilde{G}(\vec{\xi})+\lim_{\epsilon\to 0}\left\{
\int_{\partial B^c_\epsilon}\nabla\left(G(\vec{r})e^{-2\pi i\vec{\xi}\cdot\vec{r}}\right)\cdot d\vec{A}+4\pi i\vec{\xi}\cdot\int_{B^c_\epsilon}e^{-2\pi i\vec{\xi}\cdot\vec{r}}\nabla G(\vec{r})d^3\vec{r}\right\}\\
&=\left[k^2-4\pi^2\xi^2\right]\tilde{G}(\vec{\xi})+\lim_{\epsilon\to 0}\left\{\int_{\partial B^c_\epsilon}\nabla\left(G(\vec{r})e^{-2\pi i\vec{\xi}\cdot\vec{r}}\right)\cdot d\vec{A}+4\pi i\vec{\xi}\cdot\int_{\partial B^c_\epsilon}G(\vec{r})e^{-2\pi i\vec{\xi}\cdot\vec{r}}d\vec{A}\right\}
\end{align*}
[/tex]

The problem is that I want the remaining integrals to evaluate to a constant, but it appears that it will depend on

[tex]\lim_{\epsilon\to 0}G(\epsilon)[/tex]

and

[tex]\lim_{\epsilon\to 0}\left.\frac{\partial G}{\partial r}\eval\right|_{r=\epsilon}[/tex]

I can solve the integrals by assuming spherical symmetry of G(r), but I just end up with functions of [itex]\xi[/itex], [itex]\epsilon[/itex] and G, and it is unclear if there is even convergence in the [itex]\epsilon\to 0[/itex] limit.

How can I proceed from this point?

EDIT:

I just had a thought. We are dealing with a second order PDE, so there are two degrees of freedom to the solution. Could we choose G(0) and G'(0) in such a way as to get reduce the remaining integrals to a normalization constant?