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Green's Function for the Helmholtz Equation

If we fourier transform the wave equation, or alternatively attempt to find solutions with a specified harmonic behavior in time $e^{-i\omega t}$, we convert it into the following spatial form:

\begin{displaymath}
\left(\nabla^2+ k^2\right) \phi(\mbox{\boldmath$x$}) = -\frac{\rho_\omega}{\epsilon_0}
\end{displaymath} (11.41)

(for example, from the wave equation above, where $\rho(\mbox{\boldmath$x$},t) =
\rho_\omega(\mbox{\boldmath$x$})e^{-i\omega t}$, $\phi(\mbox{\boldmath$x$},t) =
\phi_\omega(\mbox{\boldmath$x$})e^{-i\omega t}$, and $k^2 c^2 = \omega^2$ by assumption). This is called the inhomogeneous Helmholtz equation (IHE).

The Green's function therefore has to solve the PDE:

\begin{displaymath}
\left(\nabla^2+ k^2\right) G(\mbox{\boldmath$x$},\mbox{\bol...
...h$x$}_0) = \delta(\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0)
\end{displaymath} (11.42)

Once again, the Green's function satisfies the homogeneous Helmholtz equation (HHE). Furthermore, clearly the Poisson equation is the $k \to 0$ limit of the Helmholtz equation. It is straightforward to show that there are several functions that are good candidates for $G$. They are:
$\displaystyle G_0(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0)$ $\textstyle =$ $\displaystyle \frac{-\cos(k\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert)}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}$ (11.43)
$\displaystyle G_+(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0)$ $\textstyle =$ $\displaystyle \frac{-e^{+ik\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}$ (11.44)
$\displaystyle G_-(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0)$ $\textstyle =$ $\displaystyle \frac{-e^{-ik\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}}{4\pi\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}$ (11.45)

As before, one can add arbitrary bilinear solutions to the HHE, $(\nabla^2
+ k^2)F(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0) = (\nabla^2_0 + k^2)F(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0) = 0$ to any of these and the result is still a Green's function. In fact, these forms are related by this sort of transformation and superposition:

\begin{displaymath}
G_0(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0) = \frac{1}{2...
...math$x$}_0) + G_-(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0))
\end{displaymath} (11.46)

or
$\displaystyle G_+(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0)$ $\textstyle =$ $\displaystyle F(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0) + G_0(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0)$ (11.47)
  $\textstyle =$ $\displaystyle \frac{-i\sin(k\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\ver...
...}-
\mbox{\boldmath$x$}_0\vert} + G_0(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0)$ (11.48)

etc.

In terms of any of these:

$\displaystyle \phi(\mbox{\boldmath$x$})$ $\textstyle =$ $\displaystyle \chi_0(\mbox{\boldmath$x$}) - \frac{1}{\epsilon_0}\int_V \rho(\mbox{\boldmath$x$}_0)
G(\mbox{\boldmath$x$},\mbox{\boldmath$x$}_0)d^3x_0$ (11.49)
  $\textstyle =$ $\displaystyle \chi_0(\mbox{\boldmath$x$}) + \frac{1}{4 \pi \epsilon_0}\int_V
...
...ldmath$x$}_0\vert}}{\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}_0\vert}d^3x_0$ (11.50)

where $(\nabla^2+ k^2)\chi_0(\mbox{\boldmath$x$}) = 0$ as usual.

We name these three basic Green's functions according to their asymptotic time dependence far away from the volume $V$. In this region we expect to see a time dependence emerge from the integral of e.g.

\begin{displaymath}
\phi(\mbox{\boldmath$x$},t) \sim e^{ik r - i\omega t}
\end{displaymath} (11.51)

where $r = \vert\mbox{\boldmath$x$}\vert$. This is an outgoing spherical wave. Consequently the Green's functions above are usually called the stationary wave, outgoing wave and incoming wave Green's functions.

It is essential to note, however, that any solution to the IHE can be constructed from any of these Green's functions! This is because the form of the solutions always differ by a homogeneous solution (as do the Green's functions) themselves. The main reason to use one or the other is to keep the form of the solution simple and intuitive! For example, if we are looking for a $\phi(\mbox{\boldmath$x$},t)$ that is supposed to describe the radiation of an electromagnetic field from a source, we are likely to use an outgoing wave Green's function where if we are trying to describe the absorption of an electromagnetic field by a source, we are likely to use the incoming wave Green's function, while if we are looking for stationary (standing) waves in some sort of large spherical cavity coupled to a source near the middle then (you guessed it) the stationary wave Green's function is just perfect.

[As a parenthetical aside, you will often see people get carried away in the literature and connect the outgoing wave Green's function for the IHE to the retarded Green's function for the Wave Equation (fairly done - they are related by a contour integral as we shall see momentarily) and argue for a causal interpretation of the related integral equation solutions. However, as you can clearly see above, not only is there no breaking of time symmetry, the resulting descriptions are all just different ways of viewing the same solution! This isn't completely a surprise - the process of taking the Fourier transform symmetrically samples all of the past and all of the future when doing the time integral.

As we will see when discussing radiation reaction and causality at the very end of the semester, if anything one gets into trouble when one assumes that it is always correct to use an outgoing wave or retarded Green's function, as the actual field at any point in space at any point in time is time reversal invariant in classical electrodynamics - absorption and emission are mirror processes and both are simultaneously occurring when a charged particle is being accelerated by an electromagnetic field.]


next up previous contents
Next: Green's Function for the Up: Green's Functions for the Previous: Poisson Equation   Contents
Robert G. Brown 2013-01-04