The outgoing monopole field of a sound-soft sphere

Substituting [F2] into the Hankel ratio cancels the factor $-i\mathrm{/}k$ and gives $u(r)=(a\mathrm{/}r)e^{ik(r-a)}$, so $u(a)=1$. Direct differentiation yields $u^{\mathrm{\prime }}=(ik-r^{-1})u$ and $u^{\mathrm{\prime }\mathrm{\prime }}=(-k^2-2ik\mathrm{/}r+2\mathrm{/}{r}^2)u$. In three dimensions a radial function satisfies $\mathrm{\Delta }u=u^{\mathrm{\prime }\mathrm{\prime }}+2u^{\mathrm{\prime }}\mathrm{/}r$; hence $u^{\mathrm{\prime }\mathrm{\prime }}+2u^{\mathrm{\prime }}\mathrm{/}r+k^{2}u=0$ by cancellation of each coefficient, in agreement with [F3] after setting $z=kr$.

The same derivative identity gives $r(u^{\mathrm{\prime }}-iku)=-u(r)$ and $\mathrm{\mid }u(r)\mathrm{\mid }=a\mathrm{/}r\to 0$. Thus the outgoing condition holds, and the displayed mode is an outgoing monopole solution.

If $w$ is any other classical outgoing solution with boundary value $1$, then $w-u$ is a classical outgoing solution with zero Dirichlet trace on $r=a$. By [F4], $w-u=0$. Hence the displayed field is the unique outgoing monopole solution.