Past exam of the mathematics course of the University of Cambridge / 2003 / ib / Paper 4 / 2D (Methods)

Put $v(r,t)=ru(r,t)$. Multiplication of the radial wave equation by $r$ gives $v_{tt}=c^2{v}_{rr}$ for $r>0$. In the characteristic coordinates $\xi =r+ct$, $\eta =r-ct$, this becomes $v_{\xi \eta }=0$. Integrating first in $\xi$ and then in $\eta$ gives $v=f(\xi )+g(\eta )$. ConsequentlyThe two terms are incoming and outgoing radial disturbances. If the field is finite at the origin, their numerators must cancel there; arbitrary independent choices generally produce a $1/r$ singularity.

For the single-frequency response, use separation of variables with $u=R(r)\sin \omega t$. Set $k=\omega /c$ and $V=rR$; then $V^{\prime \prime }+k^{2}V=0$, so $V=C\sin kr+D\cos kr$. A finite origin value forces $D=0$, since $\cos kr/r$ is singular. Since $\sin kr/r\to k$, the prescribed origin amplitude gives $C=A/k$. The regular time-harmonic spherical wave is thereforewith the quotient defined by its limit at zero.

This answers the literal finite-origin condition in the class of regular single-frequency responses. It is a standing combination of inward and outward waves: the numerator is proportional to $\cos (kr-\omega t)-\cos (kr+\omega t)$. A purely outgoing wave from an ideal point source would instead be proportional to $r^{-1}\sin [\omega (t-r/c)]$, which has no finite prescribed value $u(0,t)$. Such a source must be specified by a singularity strength, for example ${\lim }_{r\downarrow 0}ru$. The source's wording supplies neither an initial field nor an outgoing singular-source normalization; without restriction to the stated harmonic response, a general initial-value problem is not uniquely determined by the one origin condition.