SH ray-tube amplitude transport ($\rho \beta a^{2}J=\mathrm{constant}$)

For a leading wavefront singularity or discontinuity in a smooth isotropic shear-horizontal wave field, the ray transport equation conserves the displayed quantity along a tube of elastic rays. Here $J$ is transverse tube width in two dimensions and area in three. Cylindrical spreading has $J\propto r$, so the leading wave amplitude is proportional to $r^{-1/2}$. This rule excludes the immediate source neighborhood and caustics.

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.