Spherically Symmetric Solutions

Spherically symmetric solutions are those whose form is characterized by $Y(\theta,\phi)\equiv$ const. so that $\lambda =0$ . Consequently, the radial part $j(r)$ of the solution obeys

$$\left[\frac{1}{r}~\frac{d^2}{dr^2} r+k^2\right] j=0$$

so that

$$j(r)=\frac{e^{ikr}}{r},\frac{e^{-ikr}}{r}, \frac{\sin kr}{r},\frac{\cos kr} {r}\,,$$

or any of their linear combinations. Which one it is, and what the allowed values of $k$ are, depends entirely on the given boundary conditions. The concomitant spherically symmetric normal modes have the form

$$\Psi = e^{-i\omega t} \frac{e^{ikr}}{r}, e^{-i\omega t} \frac{e^{-ikr}}{r},\textrm{etc.}$$

For example, the amplitudes of the spherically symmetric normal modes confined to the interior of a hard sphere of radius $a$ are

$$\Psi=e^{-i\omega t} \frac{\sin n\pi r/a}{r} ~~~\omega=\frac{n\pi}{ca},~~n=1,2,\cdots$$

A pure sound note in a spherical resonator is an example of such a normal mode. It vibrates with frequency $\frac{\omega}{2\pi}=\frac{n}{2ca}$ .