= Transmitted jump at a collimating solid-fluid interface
{title2=$[\partial_t^k u_z]=-\frac{T_P(R)}{\alpha_fr(R)}[f^{(k+1)}]$}
The spherical source potential $f(t-r/\alpha)/r$ gives leading longitudinal <displacement field> $-f'(t-r/\alpha)/(\alpha r)$. Apply the local <solid-fluid P-wave transmission coefficient> to its potential at a smooth interface. In <hyperbolic interface collimation of P-waves>, transmitted rays are parallel, so <ray-tube conservation for a P-wave jump> makes the leading coefficient constant along each vertical ray. This yields the displayed jump for the first nonzero <derivative> discontinuity of the waveform. Smooth transverse <derivatives> multiply $f$ rather than $f'$ and are one order more regular. The construction presupposes the indicated one-sided <derivative> jump exists; a source smooth to every order may have no jump.
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