Transmitted jump at a collimating solid-fluid interface ($[{\partial }_t^k{u}_z]=-\frac{T_P(R)}{{\alpha }_{f}r(R)}[f^{(k+1)}]$)

The spherical source potential $f(t-r/\alpha )/r$ gives leading longitudinal displacement field $-f^{\prime }(t-r/\alpha )/(\alpha r)$. 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 $f^{\prime }$ 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.