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 wave in an isotropic elastic material has displacement perpendicular to its wavevector, with zero divergence for a plane wave. The Navier-Cauchy equation with inertia then reduces to $\rho {\mathbf{u}}_{tt}=\mu \Delta \mathbf{u}$, giving the displayed wave speed. At a boundary it can be generated by mode conversion from a compressional wave, even when the incident displacement is purely longitudinal.

The polarization of an elastic wave in an isotropic solid specifies the direction of its displacement field amplitude relative to its wavevector. The longitudinal wave component is parallel to the wavevector; the transverse wave component is perpendicular. For a real unit propagation vector $n$, $a_L=(a\cdot n)n$ and $a_T=a-a_L$. An evanescent constituent has a complex spatial derivative vector, so the corresponding amplitude ratio can be imaginary. The longitudinal and transverse constituents of a Rayleigh wave combine into ellipse particle motion at the free surface.

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 $w_{tt}=c_S^2(w_{xx}+w_{yy})$.

The SH input impedance is the negative ratio of vertical traction to particle velocity for a shear-horizontal wave field, with positive depth and tension-positive traction. A superposition of upward and downward waves generally has a depth-dependent complex ratio. A downward progressive wave in a uniform medium has $Z=\mu p_{\beta }$, while the upward wave has $Z=-\mu p_{\beta }$. The input ratio includes reflection from deeper structure and can have poles at velocity nodes.

With $e^{ikx-i\omega t}$ phasors, positive depth $z$, shear modulus $\mu =\rho {\beta }^2$, vertical traction $T$ and particle velocity $V$, the shear-wave equations give $V^{\prime }=-i\omega T/\mu$ and $T^{\prime }=-i\omega \mu p_{\beta }^{2}V$, where $p_{\beta }^2={\beta }^{-2}-k^2/{\omega }^2$. Define $Z=-T/V$ wherever $V\ne 0$. Differentiation gives the displayed Riccati equation and $V^{\prime }=i\omega ZV/\mu$. A pole of $Z$ can be an ordinary zero of $V$, not a singular physical wavefield.

In a uniform layer, put $q=\mu p_{\beta }$, $\kappa =\omega p_{\beta }$ and $d=z-z_r$. Initial impedance $Z_r$ gives the displayed fractional-linear solution and $V(z)=V_r[\cos \kappa d+i(Z_r/q)\sin \kappa d]$. Expanding into exponentials yields forward and backward shear-horizontal waves. Negative $d$ propagates a bottom load upward. If the denominator vanishes, the nonsingular pair $(V,T)$ must be propagated through the impedance pole.

Set the lower half-space load to $q_b={\mu }_b{p}_{\beta ,b}$ using the causal vertical wavenumber branch, which excludes a wave arriving from depth infinity. Apply the uniform-layer SH impedance update upwards through every layer, keeping particle velocity and traction continuous. For upper half-space impedance $q_a$, incident velocity $V_0$ and reflected velocity $R{V}_0$, the top values are $V=V_0(1+R)$ and $T=-q_a{V}_0(1-R)$. Their ratio gives the displayed reflection coefficient, incorporating every underlying interface and multiple reflection.

For positive real mass density and shear modulus, real $k$ and $\mathrm{Im}\omega >0$, the displayed derivative of $S_z=-\frac{1}{2}\mathrm{Re}(V^{\ast }T)$ follows from the two first-order shear equations. Every nontrivial solution has strictly decreasing $S_z$. Since $S_z=\frac{1}{2}\mathrm{Re}Z|V{|}^2$, positive-real impedance at a deeper point remains positive at all shallower points; negative-real impedance at a shallower point remains negative deeper down. A zero field is the necessary exception to strictness. At complex frequency this is a spectral flux diagnostic, reducing to a physical cycle average on the real axis.

For real horizontal wavenumber $k$, positive real wave speed $\beta$ and $\omega =u+iv$ with $v>0$, write $k_{\beta }=a+ib$ with $k_{\beta }^2={\omega }^2/{\beta }^2-k^2$. Its imaginary part gives $ab=uv/{\beta }^2$. The number $b$ cannot vanish, and $\mathrm{Re}(k_{\beta }/\omega )=v[b+u^2/({\beta }^{2}b)]/|\omega {|}^2$ has the sign of $b$. The branch $b>0$ makes $e^{i{k}_{\beta }z}$ decay towards increasing depth and analytically continues the outgoing propagating or evanescent branch to the upper frequency half-plane.

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 $c_S{q}_n$.

A layer with shear modulus ${\mu }_1$ and wave speed $c_1$ lies between a rigid support and a half-space with shear modulus ${\mu }_2$ and wave speed $c_2>c_1$. A trapped antiplane shear normal mode has layer amplitude $A\sin (my)$ and exterior amplitude $A\sin (mh)e^{-\ell (y-h)}$, where $m=k\sqrt{c^2/c_1^2-1}$ and $\ell =k\sqrt{1-c^2/c_2^2}$. Continuity of displacement field and traction gives the displayed dispersion relation. It requires $c_1<c<c_2$ and differs from a Love wave with a free boundary.

For $A,ϵ>0$, consider $\tan \theta =-ϵ\theta /\sqrt{A^2-{\theta }^2}$ on $0<\theta <A$. On each negative branch of the tangent, the left side increases strictly, while the right side decreases strictly. A unique intersection exists precisely when the branch starts at $p_n=(n-1/2)\pi <A$, since the difference increases from negative infinity to a positive value or positive infinity. Consequently the number of trapped normal modes is $\lceil A/\pi -1/2\rceil$. The floor function expression $\lfloor A/\pi +1/2\rfloor$ agrees except at $A=p_n$, where it counts an inadmissible cutoff state with zero exterior exponential decay rate.

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.

Conversion of an incident S wave into a P wave at a material contrast. It provides a complementary receiver function constraint because its travel-time ordering and interference with reverberations differ from P-to-S conversion.

Conversion of an incident P wave into an S wave at a material contrast. The converted arrival's relative time and amplitude can constrain subsurface structure through a receiver function, subject to the velocity model and interface geometry.

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 $\alpha /{\alpha }_f$.

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, $Z\cos \theta$.

Equality of the relevant wave impedances across an interface. In the elementary normal-incidence model this makes the reflection coefficient vanish, even when other material properties differ.

The product $Z_s=\rho V_s$ of bulk density and S wave speed. At a normal-incidence boundary it controls the corresponding stress-amplitude reflection coefficient.

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.

In a coherent specular model, let $r_b$ be the effective lower pressure reflection coefficient referenced to $z=0$, and let the planar upper interface lie at $z=d$. With layer and upper normal impedances $Z_1,Z_2$, $r_{12}=(Z_2-Z_1)/(Z_2+Z_1)$ and $t_{21}=2Z_1/(Z_1+Z_2)$. Summing repeated round trips gives the displayed downward amplitude at the upper interface. The mean layer field is $D{e}^{i\alpha x}[e^{-i\beta (z-d)}+r_b{e}^{2i\beta d}{e}^{i\beta (z-d)}]$. Feedback through diffuse roughness channels must be neglected in this scalar model or included in a context-dependent effective reflection coefficient.

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 ${\rho }^{\prime }{c}^{\prime }\cos \theta =\rho c\cos {\theta }^{\prime }$.