= Slowness curvature and wave-surface cusps
{title2=$\mathbf g\prime=(v+v\prime\prime)\mathbf t$}
In a symmetry-plane cross-section, write $\mathbf p=r(\theta)\mathbf n(\theta)$, $v=1/r$, and $\mathbf t=\mathbf n\prime$. The <elastic wave surface> is $\mathbf g=v\mathbf n+v\prime\mathbf t$. Its derivative vanishes when $v+v\prime\prime=(r^2+2r\prime^2-r r\prime\prime)/r^3=0$, exactly an <inflection point> of the regular <slowness surface>. At a simple zero the second and third derivatives of $\mathbf g$ are linearly independent, giving an ordinary <cusp>. Smooth nonconvex <slowness surface> sheets can therefore generate cusped <wavefronts>.
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