= Hyperbolic interface collimation of P-waves
{title2=$r/\alpha-Z(R)/\alpha_f=z_0/\alpha-z_0/\alpha_f$}
A solid-to-fluid interface can turn spherical source <P waves> into parallel vertical rays if its height $Z(R)$ satisfies the displayed travel-time identity, with $r=\sqrt{R^2+Z^2}$. The positive branch through $Z(0)=z_0$ is a hyperboloid when $\alpha>\alpha_f$. Differentiating the travel-time identity enforces <Snell law for elastic and acoustic waves>. Travel time above the interface is then $z_0/\alpha+(z-z_0)/\alpha_f$, independent of cylindrical radius. The leading <wavefront> is planar although its <wave amplitude> need not be uniform across that plane.
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