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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.

At a welded interface, elastic-interface boundary conditions require continuous SH displacement and shear traction. Incident and reflected vertical slownesses have signs $q_1,-q_1$, givingThus the local displacement coefficient isIt multiplies the leading waveform discontinuity as well as a harmonic displacement. At the interface the incident amplitude from SH ray-tube amplitude transport is $C/\sqrt{r_1}$, where $r_1=h/({\beta }_1{q}_1)$ is the incident path length and $C$ is the source normalization from the previous part.

The transmitted geometrical spreading is not simply its total path length. Differentiate the ray position with respect to $p$ at fixed travel time and project onto the unit direction transverse to the transmitted ray, ${\mathbf{m}}_2=({\beta }_2{q}_2,-{\beta }_{2}p)$. Since $d{x}_I/dp=h/({\beta }_1^2{q}_1^3)$, the transverse width per unit $p$ isThe derivative of $t_1$ contributes only in the ray direction and drops out of this projection. At entry,Within uniform medium 2 the leading amplitude varies as $J_2^{-1/2}$, so the required refraction of a cylindrical SH wavefront amplitude isThe local transmission coefficient and ray-tube spreading together incorporate both impedance contrast and refraction. Indeed the normal energy transmission fraction is ${\mu }_2{q}_2|T_{\mathrm{SH}}{|}^2/({\mu }_1{q}_1)$, while the change of transverse width supplies the refraction factor in the conserved tube flux. If the media coincide, $q_1=q_2$ and $T_{\mathrm{SH}}=1$, reducing this expression to $C/\sqrt{{\beta }_{1}t}$ as required. The formula holds for the stated propagating slownesses, without a critical or head-wave contribution.

Place a line source a distance $h$ above a plane welded interface and put $q_j=({\beta }_j^{-2}-p^2{)}^{1/2}$, $t_1=h/({\beta }_1^2{q}_1)$ and $r_1=h/({\beta }_1{q}_1)$. The transmitted elastic ray is $x=hp/q_1+{\beta }_2^{2}p(t-t_1)$, $z={\beta }_2^2{q}_2(t-t_1)$. Elastic-interface boundary conditions give $T_{\mathrm{SH}}=2{\mu }_1{q}_1/({\mu }_1{q}_1+{\mu }_2{q}_2)$. Differentiating the ray positions across neighboring values of $p$ and applying SH ray-tube amplitude transport gives the displayed leading amplitude. Both $q_j$ must be positive; critical and evanescent incidence require different asymptotics.

Integrating the shear-horizontal wave equation with force $F(t){\delta }^{(2)}(\mathbf{x})$ over a shrinking circle gives $2\pi \mu r{\partial }_{r}u\to -F(t)$. Thus the local displacement field has the displayed logarithmic singularity. This is a fixed-time near-source limit, distinct from the wavefront limit underlying SH ray-tube amplitude transport.