Radial acoustic spectrum between rigid concentric spheres ($(1+k^{2}ab)\tan k(b-a)=k(b-a)$)

A velocity potential obeying the wave equation with constant speed of sound $c_0$ has spherical normal modes $\varphi =e^{-i\omega t}[A\cos kr+B\sin kr]/r$, $k=\omega /c_0$. Zero radial velocity at radii $a,b$ gives Neumann conditions on the potential, not on its numerator. The inner condition is built into $F(r)=C[\cos k(r-a)+(ka{)}^{-1}\sin k(r-a)]/r$. The outer condition gives $k(b-a)\cos k(b-a)-(1+k^{2}ab)\sin k(b-a)=0$, equivalent to the displayed equation for nonzero oscillatory modes. The formal zero root is a constant potential gauge, rather than a nonzero-frequency acoustic wave. Applying a one-dimensional sine condition to the numerator would miss the spherical geometric factors.