SH-wave

An SH-wave is polarized horizontally and perpendicular to the vertical propagation plane.

For a flat layer of thickness $h$ and shear speed ${\beta }_1$ above a faster half-space of speed ${\beta }_2$, unfolding $m$ down-and-up traversals gives the displayed reflected-ray travel time. Its slope is the horizontal ray slowness. At $x=2mh\tan i_c$, with $\sin i_c={\beta }_1/{\beta }_2$, the corresponding head-wave branch $t=x/{\beta }_2+2mh\sqrt{{\beta }_1^{-2}-{\beta }_2^{-2}}$ is tangent to this curve.

For subcritical incidence, a reflected SH-wave loses energy into the faster substrate. Postcritical incidence has an evanescent substrate tail and no outgoing normal energy flux in the ideal lossless model. Repeated reflections can then interfere to form Love waves. Individual reflected rays are contributions to the field; normal-mode frequencies require the reflection-phase condition and are not supplied by a geometrical travel-time curve alone.

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.

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.

For a layer $0<z<h$ with rigid lower boundary and traction-free upper boundary, an SH mode
$u_y=Y(z)e^{i(\kappa x-\omega t)}$ hasand dispersionIts cutoff is ${\omega }_n=c_s{q}_n$, its phase speed is $\omega /\kappa$, and its group speed is $c_s^2\kappa /\omega$.