Endpoint approximation for a rigid-backed shear mode ($A-\theta \sim {ϵ}^{2}A/(2{\tan }^{2}A)$)

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