 Solution

Away from the line source, insert a leading wavefront form $u\sim a(\mathbf{x})W(t-T(\mathbf{x}))$ into the shear-horizontal wave equation and match the two most singular orders. The first gives the eikonal equation $|\nabla T|=1/\beta$. The next givesEquivalently, SH ray-tube amplitude transport conserves $\rho \beta a^{2}J$, where $J$ is the transverse width of a two-dimensional ray tube. A line source has straight radial elastic rays and $J=rd\theta$ for fixed angular width, soThe same geometrical spreading applies to the first nonzero derivative discontinuity or leading wavefront singularity; the source history determines which waveform $W$ and normalization $C$ occur. In three dimensions a point source instead has $J\propto r^2$ and amplitude proportional to $r^{-1}$.

This does not contradict the SH line-source near-field singularity. Ray transport describes an arrival-front limit with $t-r/\beta$ small, or the high-frequency field outside the immediate source neighborhood. The logarithm describes $r\downarrow 0$ at fixed elapsed time and includes the tail behind the two-dimensional wavefront. For example, a causal step force $F_{0}H(t)$ has the exact radial displacementFor $r\ll \beta t$, this is $F_0\log (2\beta t/r)/(2\pi \mu )$, the near-source logarithm. For $t=r/\beta +\tau$ with $0<\tau \ll r/\beta$, it isThe coefficient of the leading arrival singularity therefore has exactly the ray-theory dependence. A force impulse gives the corresponding $r^{-1/2}H(\tau )/\sqrt{\tau }$ leading field. The limits describe different regions of one causal solution; neither is a uniform approximation to the other.