= Group-velocity minimum of a high-contrast Love wave
{title2=$\omega_{\min,n}\sim\beta'(n+\tfrac12)\pi/h$}
With $\epsilon=\beta'/\beta\ll1$, $\eta=\mu'/\mu=O(\epsilon^2)$ and a fixed mode index, put $y=qh$, $W=\omega h/\beta'$ and $K=kh$. The exact <Love wave> branch has $W=y\sqrt{1+\eta^2\tan^2y}/\sqrt{1-\epsilon^2}$ and $K=y\sqrt{\epsilon^2+\eta^2\tan^2y}/\sqrt{1-\epsilon^2}$. Its minimum <group velocity> occurs near $y=(n+1/2)\pi$, with transition width $W-(n+1/2)\pi$ of order $(\eta^2(n+1/2)\pi)^{1/3}$. Well above that width, the <rigid-base approximation to Love waves> captures both velocities. Density ratios and the mode index must be controlled in this estimate.
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