= Endpoint approximation for a rigid-backed shear mode
{title2=$A-\theta\sim\epsilon^2 A/(2\tan^2 A)$}
In the <rigid-backed shear-wave dispersion> equation $\tan\theta=-\epsilon\theta/\sqrt{A^2-\theta^2}$, take fixed $(n-1/2)\pi<A<n\pi$ and let $\epsilon\to0^+$. The highest intersection approaches the endpoint $A$. Setting $\delta=A-\theta$ and using $A^2-\theta^2\sim2A\delta$ gives the displayed <asymptotic expansion>. At $A=n\pi$ the appropriate balance is instead $\delta\sim(\epsilon^2n\pi/2)^{1/3}$. Near that transition, with $\Delta=A-n\pi$, the leading equation is $\delta^2(\Delta+\delta)=\epsilon^2n\pi/2$, where now $\theta=n\pi-\delta$ and $\Delta+\delta>0$.
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