Elastic wave

An elastic wave propagates a disturbance through an elastic solid. Small displacements in linear elasticity satisfy a vector wave equation governed by mass density and the two Lamé parameters.

In a homogeneous elastic layer $-H<z<H$ with rigid boundaries, in-plane propagating modes mix P- and SV-waves. For phase speed $c>c_P>c_S$, one even-horizontal, odd-vertical family obeys

A shear-horizontal wave mode has ${\omega }_n(k)=c_s\sqrt{k^2+(n\pi /h{)}^2}$. At $x=Vt$ with $0<|V|<c_s$, a nondegenerate stationary Fourier phase gives generic $t^{-1/2}$ decay. Faster rays have no stationary point; compactly supported data give eventual zero by finite propagation speed.

Divergence of the elastic equation isolates dilatational motion, while curl isolates rotational motion, producing P and S wave equations with different speeds.

A P-wave is longitudinal, carries nonzero dilatation, and has speed $c_P=\sqrt{(\lambda +2\mu )/\rho }$ in an isotropic solid.

In a longitudinal plane wave, displacement is parallel to the wavevector and the curl vanishes.

An S-wave is transverse, carries rotation without dilatation, and has speed $c_S=\sqrt{\mu /\rho }$.

In a transverse plane wave, displacement is perpendicular to the wavevector and its divergence vanishes.

An SV-wave is polarized in the vertical plane containing its wavevector and perpendicular to that wavevector.

An SH-wave is polarized horizontally and perpendicular to the vertical propagation plane.

For a layer $0<z<h$ with rigid lower boundary and traction-free upper boundary, an SH mode
$u_y=Y(z)e^{i(\kappa x-\omega t)}$ hasand dispersionIts cutoff is ${\omega }_n=c_s{q}_n$, its phase speed is $\omega /\kappa$, and its group speed is $c_s^2\kappa /\omega$.

At a perfectly bonded interface, all displacement and traction components are continuous.

An obliquely incident in-plane P-wave generally produces reflected and transmitted P and SV waves; SH polarization decouples.

Phase matching fixes a common frequency and tangential wavenumber for every incident, reflected, and transmitted elastic mode.

For two inviscid media, continuity of normal displacement and normal stress determines the reflected and transmitted longitudinal amplitudes.

Equal normal velocities and the membrane tension-mass balance determine complex pressure amplitudes. With identical fluids on both sides the reflected and transmitted squared amplitudes sum to one, expressing conserved mean energy flux.

Equating the absolute reflected and transmitted displacement amplitudes, then using acoustic Snell's law, gives an algebraic condition on incidence angle and the two moduli-over-speed impedances.