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Substitute the harmonic plane waveinto the elastic wave equation. The vector identitygivesIf $\mathbf{A}$ is parallel to $\mathbf{K}$, this becomesand gives a longitudinal P wave. If $\mathbf{A}\cdot \mathbf{K}=0$, it becomesand gives either polarization of an S wave.

For $\mathbf{K}=(k,0,m)$, convenient polarization vectors areThe SV-wave lies in the $xz$ propagation plane, whereas the SH-wave is perpendicular to that plane.

Let $k_S=\omega /c_S$ and $k_P=\omega /c_P$. Snell law for elastic waves fixes the reflected waves' common tangential wavenumber:Choose the reflected SV-wave and converted reflected P wave asThe polarizations are respectively perpendicular and parallel to their wavevectors. Phase matching makes all three exponential factors equal at $z=0$. The rigid boundary condition for an elastic wave ${\mathbf{u}}_I+{\mathbf{u}}_{RS}+{\mathbf{u}}_{RP}=0$ there givesSolving this two-by-two linear system yieldsThese fields give the Reflection of an SV-wave from a rigid plane.

An SV-wave is transverse but polarized in the vertical plane containing the wavevector:It is perpendicular to $\hat{k}$ and has $\omega =c_s\kappa$.

Take horizontal interfaces, water depth $h_w=50\mathrm{m}$, and sediment thickness $h_s=20\mathrm{m}$. Both source and receiver are just below the sea surface. Every angle below is measured from the vertical interface normal. Snell's law preserves the horizontal seismic ray parameterA basement seismic head wave of speed $V_h$ has horizontal propagation at the basement interface, so $p=1/V_h$. The two possibilities are:

The water supports only P waves. At oblique incidence the seabed can convert a P wave into an SV-wave in the sediment, and the return SV-wave can convert back to a P wave in the water. Further seismic mode conversion can occur at the basement. Thus the down-going and up-going sediment legs can independently be P or S. The S entries refer to SV-waves; a fluid P wave source in this two-dimensional isotropic geometry does not excite the decoupled SH-wave sector.

Let $L_h\ge 0$ be the distance travelled along the basement interface. For sediment legs $a,b\in \{P,S\}$,The first-appearance distance is the limiting value at $L_h=0$:

For example, the unconverted P branch gives $2(50+20)\tan 36.870^{\circ }=105\mathrm{m}$. The earliest geometrically permitted basement P seismic head wave is the doubly converted path with sediment S legs, at approximately $90\mathrm{m}$. The corresponding basement S family first appears at approximately $297\mathrm{m}$. These limiting distances describe existence of ray branches; a finite detectable wave amplitude also depends on conversion efficiency and acquisition noise.

There is no separate finite-distance P head wave at the water–sediment interface: $V_{p,w}=V_{p,s}=1500\mathrm{m}{\mathrm{s}}^{-1}$, so the nominal critical angle is $90^{\circ }$, and a ray from a source 50 m above the interface cannot reach it at finite horizontal distance. Nor can the slower sediment S speed, $900\mathrm{m}{\mathrm{s}}^{-1}$, support a seismic head wave launched from water: it would require $\sin {\theta }_w=1500/900>1$. A seismic reflection still exists at the seabed because the densities, hence the acoustic impedances, differ.

First appearance and first arrival differ. For each permitted branch the travel time isThe all-P basement P branch has $t_h=x/2500+0.0746667\mathrm{s}$. It overtakes the direct water P wave, $t_d=x/1500$, only at $x=280\mathrm{m}$. The converted P-head branches have larger seismic intercept times, so even where they exist at shorter offsets they are later arrivals. The S-head branches are likewise later than the all-P P-head branch wherever they exist in this model. Reflections and layer reverberations can supply additional later phases.

For real frequency $\omega >0$ and both vertical wavenumbers $p=k_{\alpha }>0$, $s=k_{\beta }>0$, let $D=k^2+ps$, $A=\rho \omega p/D$, $B=\rho \omega s/D$, and $C=k(2\mu D-\rho {\omega }^2)/(\omega D)$. With traction $\mathbf{t}=({\sigma }_{xz},{\sigma }_{zz})$ and particle velocity $\mathbf{v}=(v_x,v_z)$, each directional P wave/SV-wave field obeys $\mathbf{t}=-Z_{\pm }\mathbf{v}$. Direct displacement-potential substitution yields $Z_{-}=-Z_{+}^{†}$. The Hermitian part of $Z_{+}$ is positive diagonal; thus forward/backward fluxes have opposite signs and mixed directional flux cross terms cancel.

In a uniform isotropic material, these scalar P-SV displacement potentials separate longitudinal P waves from vertically polarized SV-waves in an $x,z$ plane. They satisfy ${\varphi }_{tt}={\alpha }^2\Delta \varphi$ and ${\psi }_{tt}={\beta }^2\Delta \psi$, with compressional and shear speeds $\alpha ,\beta$. Reversing the sign convention for $\psi$ changes its polarization column but not the physical P-SV directional impedance matrix or the separate modal energy fluxes.