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The Homogeneous Helmholtz Equation

Recall as you read this that WIYF and MWIYFin addition to the treatment of this available in Jackson, chapters 2, 3, 6, and 8 of Wyld, and doubtless Arfkin, Morse and Feshback, and probably six other sources if you look. Very important stuff, can't know it too well.

Recall from above the Homogeneous Helmholtz Equation (HHE):

\begin{displaymath}
(\nabla^2 + k^2) \chi(\mbox{\boldmath$x$}) = 0.
\end{displaymath} (11.79)

We assume that11.6:
\begin{displaymath}
\chi(\mbox{\boldmath$x$}) = \sum_L f_\ell (r) Y_L(\theta,\phi).
\end{displaymath} (11.80)

We reduce the HHE with this assumption to the radial differential equation
\begin{displaymath}
\left[ \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} + k^2 - \frac{\ell(\ell
+ 1)}{r^2} \right] f_\ell (r) = 0.
\end{displaymath} (11.81)

If we substitute
\begin{displaymath}
f_\ell (r) = \frac{1}{r^{1/2}} u_\ell (r)
\end{displaymath} (11.82)

we transform this into an equation for $u_\ell (r)$,
\begin{displaymath}
\left[ \frac{d^2}{dr^2} + \frac{1}{r} \frac{d}{dr} + k^2 - \frac{(\ell +
\frac{1}{2})^2}{r^2} \right] u_\ell (r) = 0.
\end{displaymath} (11.83)

The is Bessel's differential equation. See Wyld, (2-6) or Jackson in various places (see key on back inside cover) for more detail. Or your own favorite Math Physics book.

Two linearly independent solutions on $\rm I \hspace{-.180em} R^3$ minus the origin to this radial DE are:

$\displaystyle f_\ell (r)$ $\textstyle =$ $\displaystyle j_\ell(kr) \quad \hbox{\rm and}$ (11.84)
$\displaystyle f_\ell (r)$ $\textstyle =$ $\displaystyle n_\ell(kr),$ (11.85)

the spherical bessel function and spherical neumann functions respectively. They are both real, and hence are stationary in time (why?). The $j_\ell(kr)$ are regular (finite) at the origin while the $n_\ell(kr)$ are irregular (infinite) at the origin. This is in exact analogy with the situation for the homogeneous Laplace equation (which is a special case of this solution).

The following is a MINIMAL table of their important properties. A better table can be found in Wyld between chps. 6 and 7 and in Morse and Feshbach (I can't remember which volume).



Subsections
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Next: Properties of Spherical Bessel Up: Radiation Previous: The Far Zone   Contents
Robert G. Brown 2013-01-04