= Strict cutoff of the second Love-wave mode
{title2=$kh>\pi/\sqrt{c_s^2/\bar c_s^2-1}$}
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 $khq_{\max}$. 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.
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