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Green's Functions and Free Spherical Waves

We expect, for physical reasons11.7 that the wave emitted by a time dependent source should behave like an outgoing wave far from the source. Note that inside the bounding sphere of the source that need not be true. Earlier in this chapter, we used an ``outgoing wave Green's function'' to construct the solution to the IHE with this asymptotic behavior. Well, lo and behold:

\begin{displaymath}
G_\pm(\mbox{\boldmath$x$},\mbox{\boldmath$x$}') = \mp \frac...
...\pm_0 (k\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}'\vert)
\end{displaymath} (11.107)

For stationary waves (useful in quantum theory)
\begin{displaymath}
G_0(\mbox{\boldmath$x$},\mbox{\boldmath$x$}') = \frac{k}{4\pi} n_0(k\vert\mbox{\boldmath$x$}- \mbox{\boldmath$x$}'\vert).
\end{displaymath} (11.108)

This extremely important relation forms the connection between free spherical waves (reviewed above) and the integral equation solutions we are interested in constructing.

This connection follows from the addition theorems or multipolar expansions of the free spherical waves defined above. For the special case of $L = (0,0)$ these are:

\begin{displaymath}
N_0({\bf r} - {\bf r'}) = n_0(k \vert {\bf r} - {\bf r'}) \...
...} = \sqrt{4 \pi} \sum_{L} N_L({\bf r}_>) J_L({\bf r}_<)^\ast
\end{displaymath} (11.109)

and
\begin{displaymath}
H^\pm_0({\bf r} - {\bf r'}) = h_0^\pm (k \vert {\bf r} - {\...
...qrt{4 \pi} \sum_{L} H^\pm_L({\bf r}_>) J_L({\bf
r}_<)^\ast .
\end{displaymath} (11.110)

From this and the above, the expansion of the Green's functions in free spherical multipolar waves immediately follows:
\begin{displaymath}
G_0({\bf r} - {\bf r'}) = k \sum_{L} N_L({\bf r}_>) J_L({\bf
r}_<)^\ast
\end{displaymath} (11.111)

and
\begin{displaymath}
G_\pm({\bf r} - {\bf r'}) = \mp ik \sum_{L} H^\pm_L({\bf r}_>)
J_L({\bf r}_<)^\ast .
\end{displaymath} (11.112)

Note Well: The complex conjugation operation under the sum is applied to the spherical harmonic (only), not the Hankel function(s). This is because the only function of the product $Y_L(\hat{\mbox{\boldmath$r$}})Y_L(\hat{\mbox{\boldmath$r$}}')^\ast$ is to reconstruct the $P_\ell(\Theta)$ via the addition theorem for spherical harmonics. Study this point in Wyld carefully on your own.

These relations will allow us to expand the Helmholtz Green's functions exactly like we expanded the Green's function for the Laplace/Poisson equation. This, in turn, will allow us to precisely and beautifully reconstruct the multipolar expansion of the vector potential, and hence the EM fields in the various zones exactly11.8.

This ends our brief mathematical review of free spherical waves and we return to the description of Radiation.


next up previous contents
Next: Electric Dipole Radiation Up: The Homogeneous Helmholtz Equation Previous: General Solutions to the   Contents
Robert G. Brown 2013-01-04