The Homogeneous Helmholtz Equation

Recall as you read this that and in 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):

$$(\nabla^2 + k^2) \chi(\vx) = 0.$$ (13.12)

We assume that^13.6:

$$\chi(\vx) = \sum_L f_\ell (r) Y_L(\theta,\phi).$$ (13.13)

We reduce the HHE with this assumption to the radial differential equation

$$\left[ \frac{d^2}{dr^2} + \frac{2}{r} \frac{d}{dr} + k^2 - \frac{\ell(\ell + 1)}{r^2} \right] f_\ell (r) = 0.$$ (13.14)

If we substitute

$$f_\ell (r) = \frac{1}{r^{1/2}} u_\ell (r)$$ (13.15)

we transform this into an equation for $u_\ell (r)$ ,

$$\left[ \frac{d^2}{dr^2} + \frac{1}{r} \frac{d}{dr} + k^2 - \frac{(\ell + \frac{1}{2})^2}{r^2} \right] u_\ell (r) = 0.$$ (13.16)

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 $\RE^3$ minus the origin to this radial DE are:

$$f_\ell (r) = j_\ell(kr) \quad \hbox{\rm and}$$ (13.17)

$$f_\ell (r) = n_\ell(kr),$$ (13.18)

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).