Solid-fluid P-wave transmission coefficient ($T_P=\frac{2\rho {\alpha }_f\cos 2{\theta }_S\cos {\theta }_P}{{\rho }_f{\alpha }_f\cos {\theta }_P+\rho \alpha E\cos {\theta }_f}$)

For incidence from a solid with compressional/shear speeds $\alpha ,\beta$ into a fluid of sound speed ${\alpha }_f$, let $E={\cos }^{2}2{\theta }_S+({\beta }^2/{\alpha }^2)\sin 2{\theta }_P\sin 2{\theta }_S$. The displayed coefficient multiplies the transmitted displacement field potential relative to the incident P wave potential, using the same retarded-time waveform. Snell law for elastic and acoustic waves matches the tangential slowness. Continuity of normal displacement field, zero tangential traction, and continuity of normal traction give the coefficient. It is not a displacement-amplitude coefficient: longitudinal displacement field acquires an additional ratio $\alpha /{\alpha }_f$.