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# 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): (13.12)

We assume that13.6: (13.13)

We reduce the HHE with this assumption to the radial differential equation (13.14)

If we substitute (13.15)

we transform this into an equation for , (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 minus the origin to this radial DE are:   (13.17)   (13.18)

the spherical bessel function and spherical neumann functions respectively. They are both real, and hence are stationary in time (why?). The are regular (finite) at the origin while the 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).

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Robert G. Brown 2017-07-11