Elastic wave in an isotropic solid

For density $\rho$ and Lamé parameters $\lambda ,\mu$, the displacement satisfiesIts longitudinal and transverse wave speeds are

A shear-horizontal wave propagating in the $x$ direction has displacement perpendicular to the sagittal $xy$ plane, for example $\mathbf{u}=w(x,y,t){\mathbf{e}}_z$. It is divergence-free and obeys the scalar shear-wave equation $w_{tt}=c_S^2(w_{xx}+w_{yy})$.

Between parallel boundaries, a guided shear-horizontal mode has a discrete transverse wavenumber $q_n$ and dispersion relation ${\omega }_n^2=c_S^2(k^2+q_n^2)$. Its cutoff frequency is $c_S{q}_n$.

An obliquely incident in-plane P- or SV-wave generally produces reflected and transmitted P- and SV-waves. Continuity of displacement and traction supplies four scalar conditions for their four amplitudes, while frequency and tangential wavenumber are shared.

Phase matching at a planar interface preserves frequency and tangential wavenumber. Thus waves of speeds $c_j$ and angles ${\theta }_j$ from the normal satisfy

The acoustic impedance of a fluid is $Z=\rho c$. For a progressive plane acoustic wave, pressure amplitude $P$ and velocity amplitude $U$ in its propagation direction satisfy $P=ZU$. At an oblique planar interface, zero reflection requires a matching condition involving the normal impedances $Z/\cos \theta$ or, equivalently for displacement amplitudes, $Z\cos \theta$.

The normal acoustic impedance of a progressive acoustic wave at a planar boundary is the pressure amplitude divided by the normal-velocity amplitude. For propagation angle $\theta$ from the normal,The distinction from the acoustic impedance $\rho c$ matters at oblique incidence: interface reflection depends on the ratio of normal impedances. For an evanescent outgoing wave $k_n$ is imaginary and the normal impedance is reactive, giving no mean normal acoustic energy flux.

At an acoustic boundary, the surface acoustic impedance is the pressure amplitude divided by the velocity amplitude directed into that boundary. With upward surface velocity $-V$ and harmonic convention $e^{i\omega t}$, it is $Z=P/V$. Reflection compares this impedance with the incident fluid's normal acoustic impedance. The velocity orientation and harmonic convention must be fixed before interpreting signs of inertial, elastic, and radiation terms.

With pressure amplitude $P=ZV$ and velocity $V$ directed into a boundary, the mean absorbed acoustic energy flux is $\frac{1}{2}\mathrm{Re}(P{V}^{\ast })=\frac{1}{2}\mathrm{Re}Z|V{|}^2$. A passive boundary cannot supply net energy, so $\mathrm{Re}Z\ge 0$ at a real frequency. Its reflection from a propagating acoustic plane wave has magnitude at most one. A pole continued to a complex incidence angle or frequency must be interpreted separately from real-frequency passive scattering.

A layer of impedance $Z_1$, wavenumber $k_1$, and thickness $L$ between identical media of impedance $Z_0$ has pressure-amplitude transmission coefficientIt transmits perfectly at the layer resonances $k_{1}L=n\pi$, irrespective of the impedance mismatch.

For a P-wave incident at angle $\theta$ from a liquid $(\rho ,c,\lambda )$ onto $({\rho }^{\prime },c^{\prime },{\lambda }^{\prime })$, with transmitted angle ${\theta }^{\prime }$ obeying Snell law for elastic and acoustic waves,No reflection is equivalent to ${\rho }^{\prime }{c}^{\prime }\cos \theta =\rho c\cos {\theta }^{\prime }$.