Strict cutoff of the second Love-wave mode ($kh>\pi /\sqrt{c_s^2/{\bar{c}}_s^2-1}$)

For a slow elastic layer above a faster half-space, put $q=\sqrt{c^2/{\bar{c}}_s^2-1}$ and $q_{\max }=\sqrt{c_s^2/{\bar{c}}_s^2-1}$. The positive branches of $\tan (khq)$ intersect the positive strictly decreasing traction ratio once whenever their left endpoint $m\pi$ is strictly below $kh{q}_{\max }$. The fundamental exists for every positive $kh$; the second requires the displayed strict inequality. At equality $c=c_s$ and the half-space decay exponent vanishes, so that endpoint is a cutoff, not a trapped mode satisfying decay at minus infinity.