 SH-wave travel times in a crustal waveguide

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.