Past exam of the mathematics course of the University of Cambridge / 2003 / iii / Paper 78 / 2 / i

Let $u(x,z,t)$ be displacement along the source line and let $F(t)$ be its force per unit length. The shear-horizontal wave equation isRotational symmetry about the line makes $u=u(r,t)$. Integrate over a disk of radius $r$ and use the divergence theorem:At a fixed regular time, a logarithmic singularity and its time derivatives are locally integrable; the inertial integral is $O(r^2|\log r|)$ and vanishes as $r\downarrow 0$. The leading balance is consequently $r{u}_r=-F(t)/(2\pi \mu )$, giving the SH line-source near-field singularityThe reference length $r_{\ast }$ only changes the bounded term; boundary conditions and source history determine that term. This is a local fixed-time statement, interpreted distributionally in time if the source force has jumps. It establishes logarithmic spatial dependence directly, without extrapolating far-field ray spreading into the source.