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 an isotropic half-space $y<0$ with zero displacement at $y=0$, an incident longitudinal wave directed along $(\cos \alpha ,\sin \alpha )$ reflects into both longitudinal and shear waves. Let $c_p^2=(\lambda +2\mu )/\rho$ and $c_s^2=\mu /\rho$. Horizontal phase matching gives $\sin \beta =(c_s/c_p)\cos \alpha$ for a reflected shear wavevector proportional to $(\sin \beta ,-\cos \beta )$ and polarization $(\cos \beta ,\sin \beta )$. The two zero-displacement equations at the boundary give the displayed coefficients. Since $c_s<c_p$ in a stable isotropic solid, both reflected branches are propagating for every genuine incident angle $0<\alpha <\pi$, with no critical-angle obstruction for this incident-wave type. Changing to a stress-free boundary changes the amplitude equations and coefficients.

For an isotropic elastic layer $|y|<h$ between two identical elastic half-spaces, an antiplane displacement $w=f(y)e^{i(kx-\omega t)}$ has $m^2={\omega }^2/c_{s1}^2-k^2>0$ in the layer and exterior decay rate ${\ell }^2=k^2-{\omega }^2/c_{s2}^2>0$. Hence a trapped mode has phase speed between the layer and cladding shear speeds. Continuity of displacement and shear traction gives the first displayed equation for even modes $f\propto \cos (my)$ and the second for odd modes $f\propto \sin (my)$.

In a rigid-cladding limit with nonzero decay rate, the interfaces become displacement nodes. Even transverse wavenumbers are $(n+1/2)\pi /h$, $n\ge 0$, and odd ones are $n\pi /h$, $n\ge 1$. The minimum even-mode dispersion branch is ${\omega }^2=c_{s1}^2[k^2+{\pi }^2/(4h^2)]$. When the cladding speed is finite, exterior decay remains a separate requirement; satisfying this lower cutoff alone is insufficient.

At a material interface, incident P or SV energy can scatter into both P and SV modes, subject to conserved horizontal slowness and traction/displacement conditions. SH is decoupled in a planar isotropic model; some conversions vanish at symmetry directions such as normal incidence.

A critically refracted wave associated with a faster lower layer that travels along an interface and radiates back into the slower upper layer. Its travel-time branch has the lower-layer horizontal slowness and a positive intercept time.

For a P ray incident from a layer of speed ${\alpha }_1$ onto a faster layer ${\alpha }_2$, the transmitted ray is tangential when $\sin i_c={\alpha }_1/{\alpha }_2$. Angles are measured from the interface normal.

In a two-layer P wave model, the head-wave branch overtakes the direct upper-layer arrival at $X_c=2H\sqrt{({\alpha }_2+{\alpha }_1)/({\alpha }_2-{\alpha }_1)}$. It differs from the smaller minimum offset needed for the critical-ray construction.

For small torsional displacement $\varphi (x,t)$, the transmitted torque is $C=G{J}_p{\varphi }_x$. A slice of length $dx$ receives net torque $G{J}_p{\varphi }_{xx}dx$ and has axial moment of inertia $\rho J_{p}dx$. Its angular equation therefore gives the displayed wave equation, with wave speed $\sqrt{G/\rho }$. The same geometric factor cancels from inertia and stiffness, making this ideal torsional speed independent of rod radius. This long-wave circular-rod model neglects additional elastic modes and dissipation.

The local elastic strain energy and rotational kinetic energy per length are $u_s=G{J}_p{\theta }_x^2/2$ and $u_k=\rho J_p{\theta }_t^2/2$. Multiplying the torsional wave equation in a circular shaft by $\rho J_p{\theta }_t$ and integrating the product derivative gives ${\partial }_t(u_s+u_k)+{\partial }_x\mathcal{P}=0$. Thus the energy flux is torque times angular speed, with the sign set by the positive direction. The inequality $|\mathcal{P}|=2\sqrt{G/\rho }\sqrt{u_s{u}_k}\le \sqrt{G/\rho }(u_s+u_k)$ follows from a square. A one-way travelling wave has $u_s=u_k$; a steadily rotating twisted shaft need not. Stored energy divided by transit time is not a separate bound for each energy component in a general superposition.

For a slender homogeneous circular shaft in small-amplitude linear elasticity, torque is $G{J}_p{\theta }_x$ and axial angular inertia per length is $\rho J_p$. Their common geometric factor cancels in the angular equation of motion, leaving the displayed wave equation. Here $J_p=\int r^{2}dA$ is the polar second moment of area, rather than the mass moment of inertia itself.

In a right-going simple wave, the state emitted at boundary time $s$ travels along $X=c_b(s)(t-s)$. The mapping from emission time to position first becomes singular when ${\dot{c}}_b(t-s)-c_b=0$, giving the displayed shock time and position $X=c_b^2/{\dot{c}}_b$. A smooth increase of traction produces steepening only if it increases the propagation speed. This describes the first gradient singularity; continuation as a shock wave needs admissible jump conditions.

For homogeneous stable linear elasticity, the three positive eigenvalues of the acoustic tensor give three phase velocities for each wavefront normal. Plotting the corresponding three vectors $\mathbf{p}=\mathbf{n}/v$ gives the three sheets of the elastic slowness surface. Sheets can meet at degeneracies. The elastic-wave energy flux is normal to each regular sheet: differentiating its eigenvalue equation along a sheet tangent $d\mathbf{p}$ gives $c_{ijkl}{a}_i{a}_k{p}_{l}d{p}_j=0$.

Fix horizontal slowness $p_{\alpha }$ and define $T_{ip}=c_{i3p3}$, $R_{ip}=c_{i3p\alpha }{p}_{\alpha }$ and $Q_{ip}=c_{i\alpha p\beta }{p}_{\alpha }{p}_{\beta }$. Under strong ellipticity in elasticity, $T$ is positive definite, so the vertical-slowness determinant has degree six with real coefficients. Real roots are intersections of a vertical line with the elastic slowness surface; complex-conjugate pairs give evanescent waves.

The unit-time elastic wave surface is the locus of group velocities on the three branches of a homogeneous elastic wave. At time $t$, its scaled sheets describe the propagating wavefronts from an impulsive point source when that source excites the corresponding modes. Homogeneity of the dispersion relation gives $\mathbf{p}\cdot \mathbf{g}=1$, so each regular sheet is the envelope of planes $\mathbf{p}\cdot \mathbf{x}=1$. It is generally different from the radial reciprocal of the elastic slowness surface.

In a symmetry-plane cross-section, write $\mathbf{p}=r(\theta )\mathbf{n}(\theta )$, $v=1/r$, and $\mathbf{t}=\mathbf{n}\prime$. The elastic wave surface is $\mathbf{g}=v\mathbf{n}+v\prime \mathbf{t}$. Its derivative vanishes when $v+v\prime \prime =(r^2+2r{\prime }^2-rr\prime \prime )/r^3=0$, exactly an inflection point of the regular slowness surface. At a simple zero the second and third derivatives of $\mathbf{g}$ are linearly independent, giving an ordinary cusp. Smooth nonconvex slowness surface sheets can therefore generate cusped wavefronts.

For a real branch polarization $\mathbf{a}$ and complex displacement amplitude $U$, the cycle averages are $⟨F_j⟩=\omega |U{|}^2{c}_{ijkl}{a}_i{a}_k{k}_l/2$ and $⟨W⟩=\rho {\omega }^2|U{|}^2|\mathbf{a}{|}^2/2$. Differentiating the acoustic tensor eigenvalue equation with respect to $k_j$ gives their ratio as the group velocity. This identifies the energy flux direction with the normal to the elastic slowness surface in a lossless nondispersive elastic medium.

For a mode $u=a{e}^{i\omega (p_{\alpha }{x}_{\alpha }+q{x}_3-t)}$ with $x_3$ positive downwards, the reduced traction is $(Tq+R)a$. Its mean vertical acoustic energy flux is the displayed expression. Propagating modes are downgoing for positive flux and upgoing for negative flux; phase-slowness sign need not agree in anisotropy. Depth-decaying evanescent modes have $\mathrm{Im}q>0$. At grazing or repeated roots use limiting radiation modes.

Select the three downgoing or depth-decaying elastic modes for a solid below a plane free surface. For each mode use its reduced traction column $b=(Tq+R)a$. Zero total boundary traction gives $BA=-b_0$ for an incident mode of reduced traction $b_0$. Invertibility gives the displayed amplitudes. A singular matrix instead permits an outgoing homogeneous surface mode, so a radiation-limit or solvability analysis is necessary.

The causal Green function for isotropic linear elasticity is tensor-valued: column $k$ is the displacement caused by a point impulsive body force in direction $k$. Its radiative terms are longitudinal P waves at $t=R/\alpha$ and transverse S waves at $t=R/\beta$, proportional to $R^{-1}$. A term supported between the arrivals carries the intermediate/near-field response. Convolution in time and space constructs the displacement for a general source.

In a homogeneous isotropic solid, the far-field P wave is the radial projection of the force and the S wave is its transverse projection. For a unit force axis $d$, the magnitudes are $|\cos \theta |$ and $\sin \theta$. The approximation requires distance large compared with both reduced wavelengths; angular nodes can still be controlled by a smaller near-field contribution.

Choose a surface normal $\mathbf{n}$ from its minus to its plus side and jumps $[u]=u^{+}-u^{-}$, $[t]=({\sigma }^{+}-{\sigma }^{-})\mathbf{n}$. The equivalent force density is $-[t]$ and the dipole density is $M_{jk}=c_{jk\ell m}[u_{\ell }]n_m$. This follows by applying the elastic wave operator to a piecewise smooth displacement field: its distributional source is $-[t_i]{\delta }_{\Sigma }-{\partial }_j(c_{ijkl}[u_k]n_l{\delta }_{\Sigma })$. Integration by parts against the elastodynamic Green tensor gives the displayed signs. At a material interface the dipole term is interpreted through the continuous reciprocal Green traction, rather than an undefined product of a discontinuous stiffness with a surface delta.

Integrating the equivalent dipole body force against the elastodynamic Green tensor by parts gives the displayed time convolution. For a fixed seismic moment tensor shape $M_{kj}=K_{kj}V(t)$, its $R^{-1}$ terms are $[\mathbf{e}(\mathbf{e}\cdot K\mathbf{e})\dot{V}(t-R/\alpha )/{\alpha }^3+(K\mathbf{e}-\mathbf{e}(\mathbf{e}\cdot K\mathbf{e}))\dot{V}(t-R/\beta )/{\beta }^3]/(4\pi \rho R)$. Derivatives of the angular projectors and the retarded near-field integral additionally produce $R^{-2}$ and $R^{-4}$ coefficients, yielding the final static field after the source stops.

A localized zero-resultant seismic moment tensor represents the leading force-dipole part of an elastic wave source through the displayed equivalent body force. For a prescribed displacement jump $b_i$ across a surface with normal $n_j$, its moment density is $c_{ijkl}{b}_k{n}_l$, where $c$ is the elastic stiffness tensor. Spatial integration gives the point-source moment when the source is small compared with the wavelengths.

For uniform slip on a shear fault, scalar seismic moment is $M_0=\mu Ad$, with shear modulus $\mu$, fault area $A$ and slip magnitude $d$. More generally it integrates rigidity-weighted slip. It measures source strength rather than radiated energy alone.

A double-couple seismic source has a moment tensor proportional to $\mathbf{b}\otimes \mathbf{n}+\mathbf{n}\otimes \mathbf{b}$, with the slip direction $\mathbf{b}$ perpendicular to the fault plane's normal $\mathbf{n}$. Its eigenvalues are proportional to $(1,0,-1)$. A general symmetric seismic moment tensor need not have this restricted form.

For mass-normalized normal modes, an instantaneous step in a point seismic moment tensor excites mode $\alpha$ with $C_{\alpha }=\mathbf{M}:{\mathbf{e}}_{\alpha }^{\ast }({\mathbf{x}}_s)$ and $\tau =t-t_s$. Convolving the causal oscillator Green's function with the step gives $a_{\alpha }=C_{\alpha }[1-\cos ({\omega }_{\alpha }\tau )]H(\tau )/{\omega }_{\alpha }^2$. The expression contains a static response and undamped oscillations; attenuation requires an extension of the conservative model.

A stress glut tensor describes a seismic source through an equivalent body-force density $\mathbf{f}=-\nabla \cdot \mathbf{\Gamma }$. For a symmetric stress glut tensor supported away from the boundary, integration by parts makes its coupling to a normal mode equal to $\int \mathbf{\Gamma }:{\mathbf{e}}^{\ast }dV$, where $\mathbf{e}$ is that mode's strain.

An ideal compact volume source has $M_{ij}(t)=M(t){\delta }_{ij}$. In an unbounded homogeneous isotropic solid its far-field body radiation is P only. Boundaries and heterogeneity can create converted SV and surface-wave energy; a real explosion can have nonisotropic components.

A description of earthquake source geometry, often strike, dip and rake for a double-couple shear fault. First-motion polarities constrain the radiation sign pattern, with a fundamental interchange ambiguity between fault and auxiliary planes.

The dependence of radiated P wave or S wave amplitude and polarity on direction from an earthquake source. For an ideal double couple it has nodal directions, so amplitude differences between stations do not measure distance or directivity alone.

The negative and positive principal directions of a double-couple moment tensor. They bisect the nodal-plane quadrants, but are not automatically the principal axes of regional stress. A stress inversion needs additional faulting assumptions and multiple mechanisms.

Interchanging the slip direction and fault normal leaves a double-couple moment tensor unchanged. First-motion polarities therefore cannot identify the physical fault plane without information such as mapped rupture, aftershock geometry or geodetic displacement.

An imagined sphere around a seismic point source on which source-to-station ray takeoff directions are plotted. A lower-hemisphere equal-area projection commonly displays first-motion polarities and nodal planes.

The sign of the initial direct P wave motion after instrument, component and ray-direction conventions are calibrated. It constrains the sign of ${\hat{r}}^{\mathsf{T}}M\hat{r}$ at the source takeoff direction. Near-nodal arrivals and uncertain takeoff angles need special treatment.

An isotropic point moment is a dilatational gradient source. Its radial projection gives direction-independent P wave radiation, whereas its transverse projection vanishes. Thus it produces no far-field S wave in a homogeneous isotropic solid; conversion at an interface can change that conclusion.

Tangential fault slip $\mathbf{b}$ in an isotropic elastic material has the displayed moment density. It is symmetric and trace-free, with eigenvalues $\mu |b|,-\mu |b|,0$. The two off-diagonal force dipoles form equal couples with opposite torques, producing zero resultant force and torque. For a uniform patch of area $A$, the scalar seismic moment is $M_0=\mu A|b|$. Traction continuity removes the single-force layer in the elastodynamic surface-jump representation.

For the seismic moment tensor $M=e_1{e}_2^{\mathsf{T}}+e_2{e}_1^{\mathsf{T}}$, source differentiation of the outgoing elastodynamic Green tensor gives $A_P=n^{\mathsf{T}}Mn$ and ${\mathbf{A}}_S=(I-n{n}^{\mathsf{T}})Mn$. In the $x_1{x}_2$ plane the signed radial P wave and tangential S wave factors are $\sin 2\varphi$ and $\cos 2\varphi$. The full S wave magnitude squared is $n_1^2+n_2^2-4n_1^2{n}_2^2$.

For orthogonal slip and fault-normal unit vectors $s,n$, the P radiation amplitude is $2M_0(\hat{r}\cdot s)(\hat{r}\cdot n)$. Its zeros are the two planes normal to $s$ and $n$, producing four alternating polarity quadrants.

A small planar crack of area $S$ opening normally by $d(t)$ has volume change $V=Sd$ and the displayed seismic moment tensor. Its principal dipoles are $\lambda V,\lambda V,(\lambda +2\mu )V$: two act in the crack plane and the largest acts normally for positive opening and ordinary positive Lamé parameters. The definition follows by contracting the isotropic elastic stiffness tensor with two copies of the crack normal.

For a planar tensile-crack seismic source, $\theta$ is measured from its normal. The P wave displacement is radial with the first displayed amplitude. The S wave displacement is tangential in the plane containing the observation direction and crack normal, with four alternating signed lobes and nodes at the poles and equator. Both radiative fields depend on the opening rate and vanish once opening stops; the static displacement field depends on the final opening.

The difference of two solutions with the same body force, boundary traction and initial data satisfies a homogeneous elastic equation with zero boundary work. Multiplication by its velocity and integration by parts conserve the displayed positive energy. Zero initial energy forces zero velocity for all times, and zero initial displacement field then forces zero displacement field. Rigid zero-strain motions do not cause nonuniqueness after initial data are fixed. This assumes positive mass density, stable real elastic moduli and sufficient regularity for the energy identity.

For a time-independent symmetric negative semidefinite boundary matrix $k$, the difference of two linear elasticity solutions with the same data has traction $kw$. Integration by parts conserves the displayed nonnegative energy. Zero initial data force zero velocity and then zero displacement. The matrix $-k$ is the stiffness of boundary springs to a prescribed support; null directions are traction-free.

For a one-dimensional shear elastic wave, the seismic impedance is the magnitude of the ratio of stress to particle velocity in a progressive wave. With the positive coordinate aligned with elastic wave propagation and tension-positive stress, $\sigma =-Zv$ for the forward wave and $\sigma =Zv$ for the backward wave. Its instantaneous elastic-wave energy flux is $-v\sigma$.

At normal incidence, a uniform lossless elastic wave layer of thickness $h_j$, speed $c_j$ and seismic impedance $q_j={\rho }_j{c}_j$ has wave phase thickness ${\delta }_j=\omega h_j/c_j$. With phasors proportional to $e^{-i\omega t}$, particle velocity $V$ and compressive traction $P=-T$, the displayed matrix maps $(V,P)$ from the top to the bottom. It has determinant one. Multiplying these matrices in physical order accounts for all coherent multiple reflections. Continuity of $V,P$ connects adjacent layers without another propagation factor.

The two-layer cell seismic layer transfer matrix has reciprocal eigenvalues because its determinant is one. Writing them as $e^{\pm iKd}$ gives the displayed half-trace relation. At a real frequency, an absolute half-trace exceeding one forces an exponentially growing/decaying pair instead of propagating cell modes. A long finite stack then suppresses transmission in this stop band, without dissipating energy: the missing transmitted flux is reflected. Quarter-wave layers of unequal seismic impedance give half-trace $-\frac{1}{2}(q_1/q_2+q_2/q_1)<-1$.

These characteristic wave amplitudes split a one-dimensional elastic wave into forward and backward components. Their instantaneous net elastic-wave energy flux is ${\varphi }_{+}^2-{\varphi }_{-}^2$. For real-part harmonic phasors the mean flux is $(|{\varphi }_{+}{|}^2-|{\varphi }_{-}{|}^2)/2$. In a static graded medium with real positive seismic impedance, put $g=(\log Z{)}^{\prime }/2$; then ${\partial }_x{\varphi }_{\pm }\pm {\beta }^{-1}{\partial }_t{\varphi }_{\pm }=g{\varphi }_{\mp }$. Seismic impedance gradients couple the two directions while preserving net flux.

At zero frequency the graded-layer transfer equation integrates to $\exp [L\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right)]$, where $L=\frac{1}{2}\log (Z_b/Z_a)$. Eliminating the outgoing wave amplitude at the left endpoint gives the displayed incoming-to-outgoing scattering matrix. It is real orthogonal, so it preserves total incoming and outgoing flux. It coincides with continuity of velocity and traction at a sharp seismic impedance interface; the profile between the endpoints enters at nonzero frequency.

A Love wave is a horizontally polarized shear wave trapped in a slower elastic surface layer above a faster elastic half-space. Zero shear traction at the free surface, displacement continuity and traction continuity at the interface produce its dispersion relation. A trapped mode has phase speed between the two shear speeds and decays exponentially into the half-space.

For an aligned anisotropic surface layer, $q^2=({\omega }^2-{\beta }_x^{\prime 2}{k}^2)/{\beta }_z^{\prime 2}$, while the isotropic substrate has ${\kappa }^2=k^2-{\omega }^2/{\beta }^2$. Zero top traction and welded displacement/traction continuity give the displayed dispersion relation. The horizontal speed sets the limiting phase velocity; the vertical speed sets rigid-base cutoff frequencies. The effective contrast for the high-contrast transition is ${\rho }^{\prime }{\beta }_x^{\prime }{\beta }_z^{\prime }/(\rho {\beta }^2)$.

A free top and rigid base impose transverse wavenumbers $(n+1/2)\pi /h$ on guided SH modes between rigid and free planes. Their phase velocity and group velocity are $c={\beta }^{\prime }/\sqrt{1-{\omega }_n^2/{\omega }^2}$ and $U={\beta }^{\prime }\sqrt{1-{\omega }_n^2/{\omega }^2}$ for $\omega >{\omega }_n$. A real Love wave replaces the rigid base by an evanescent elastic load; the approximation becomes reliable sufficiently above the corresponding strong-dispersion transition, not at the actual mode cutoff.

With $ϵ={\beta }^{\prime }/\beta \ll 1$, $\eta ={\mu }^{\prime }/\mu =O({ϵ}^2)$ and a fixed mode index, put $y=qh$, $W=\omega h/{\beta }^{\prime }$ and $K=kh$. The exact Love wave branch has $W=y\sqrt{1+{\eta }^2{\tan }^{2}y}/\sqrt{1-{ϵ}^2}$ and $K=y\sqrt{{ϵ}^2+{\eta }^2{\tan }^{2}y}/\sqrt{1-{ϵ}^2}$. Its minimum group velocity occurs near $y=(n+1/2)\pi$, with transition width $W-(n+1/2)\pi$ of order $({\eta }^2(n+1/2)\pi {)}^{1/3}$. Well above that width, the rigid-base approximation to Love waves captures both velocities. Density ratios and the mode index must be controlled in this estimate.

For a slow uniform layer over a faster half-space, the nth Love wave branch reaches the half-space shear wave speed at the displayed threshold. Its half-space decay exponent then vanishes, so the threshold is a limiting nonlocalized state, not a trapped finite-energy mode. The fundamental has zero threshold and exists at every positive frequency; overtones require frequency strictly above their positive threshold. At high frequency each fixed branch approaches the layer shear wave speed from above.

For a slower surface layer supporting propagating SH-waves with vertical wavenumber $q>0$ above an evanescent half-space with decay exponent $p>0$, continuity of displacement field and traction gives the displayed reflection coefficient for a downward incident displacement field wave. Its modulus is one and its wave phase is $-2\arctan (\mu p/({\mu }_{0}q))$. A traction-free top reflects displacement field with coefficient $+1$. The round-trip condition $R_h{e}^{2iqh}=1$ then gives $\tan (qh)=\mu p/({\mu }_{0}q)$, equivalent to direct Love wave mode matching.

For a slow elastic layer above a faster half-space, put $q=\sqrt{c^2/{\bar{c}}_s^2-1}$ and $q_{\max }=\sqrt{c_s^2/{\bar{c}}_s^2-1}$. The positive branches of $\tan (khq)$ intersect the positive strictly decreasing traction ratio once whenever their left endpoint $m\pi$ is strictly below $kh{q}_{\max }$. The fundamental exists for every positive $kh$; the second requires the displayed strict inequality. At equality $c=c_s$ and the half-space decay exponent vanishes, so that endpoint is a cutoff, not a trapped mode satisfying decay at minus infinity.

A Rayleigh wave propagates along a stress-free boundary of an isotropic elastic half-space and decays into its interior. It combines an evanescent longitudinal wave with an evanescent vertically polarized shear wave. If its speed is $0<c<c_S<c_P$, the decay factors are $\alpha =\sqrt{1-c^2/c_P^2}$ and $\beta =\sqrt{1-c^2/c_S^2}$. Vanishing shear and normal traction gives the displayed determinant condition.

For a Rayleigh wave at a free surface of a homogeneous isotropic elastic half-space, define $s=c^2/c_T^2$ and $\eta =c_T^2/c_L^2$. Cancelling the two surface tractions between the constituents with Rayleigh wave decay exponents gives the displayed equation with $0<s<1$. Equivalently $(b_T^2+k^2{)}^2=4k^2{b}_L{b}_T$. Squaring gives a factor $s$ times $P(s)=s^3-8s^2+(24-16\eta )s-16(1-\eta )$. For stable Lamé parameters, $0<\eta <3/4$, so $P(0)<0$, $P(1)=1$, and $P^{\prime \prime }(s)=6s-16<0$ on $(0,1)$. Its concavity ensures exactly one crossing in this interval. Both sides of the unsquared equation are positive, so this root gives the physical speed. The zero root is a degenerate static cancellation, not a propagating Rayleigh wave.

In a homogeneous isotropic elastic half-space $Y>0$, a Rayleigh wave with factor $e^{ik(X-ct)}$ has two depth factors $e^{-b_{L}Y}$ and $e^{-b_{T}Y}$. Substitution into the Navier-Cauchy equation factors the amplitude determinant into longitudinal and transverse factors, giving the displayed exponents. Strict exponential decay requires $0<c<\min (c_L,c_T)$. The longitudinal constituent has $(A,B)\propto (ik,-b_L)$; the transverse constituent has $ikA-b_{T}B=0$. The free-surface traction conditions select a combination of both constituents.

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.

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.

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

Let $t=T(\mathbf{x})$ be a smooth wavefront and suppose the first jump in time derivatives of displacement field is $[{\partial }_t^k\mathbf{u}]=a^{[k]}\mathbf{n}$, with $k\ge 2$. Substitution of $\mathbf{u}=a^{[k]}\mathbf{n}(t-T{)}_{+}^k/k!+\cdots$ into the elastic equation first gives the eikonal equation $|\nabla T|=1/\alpha$ and longitudinal polarization $\mathbf{n}=\alpha \nabla T$. Projecting the next-order compatibility condition onto $\mathbf{n}$ gives $2\rho \alpha \mathbf{n}\cdot \nabla a^{[k]}+a^{[k]}\nabla \cdot (\rho \alpha \mathbf{n})=0$, hence the displayed conservation law. Coefficients and wavefront must be sufficiently differentiable for this classical transport equation; its local form fails at a caustic.

The spherical source potential $f(t-r/\alpha )/r$ gives leading longitudinal displacement field $-f^{\prime }(t-r/\alpha )/(\alpha r)$. Apply the local solid-fluid P-wave transmission coefficient to its potential at a smooth interface. In hyperbolic interface collimation of P-waves, transmitted rays are parallel, so ray-tube conservation for a P-wave jump makes the leading coefficient constant along each vertical ray. This yields the displayed jump for the first nonzero derivative discontinuity of the waveform. Smooth transverse derivatives multiply $f$ rather than $f^{\prime }$ and are one order more regular. The construction presupposes the indicated one-sided derivative jump exists; a source smooth to every order may have no jump.

A tube of neighboring elastic rays follows the wavefront normal. If $J$ is its infinitesimal transverse area, its expansion is $d\log J/ds=\nabla \cdot \mathbf{n}$ along arclength. The P-wavefront discontinuity transport law therefore gives the displayed conserved flux. Geometrical spreading, mass density and wave speed determine the leading jump wave amplitude together; the rule is valid before the ray tube reaches a caustic.

For a stable isotropic solid in $y<0$, use $u_x={\varphi }_x+{\psi }_y$, $u_y={\varphi }_y-{\psi }_x$. An incident P wave with frequency $\omega$, tangential wavenumber $\kappa$ and upward normal wavenumber $a$ reflects into a P potential $A{e}^{i(\kappa x-ay-\omega t)}$ and an SV potential $B{e}^{i(\kappa x-by-\omega t)}$, where $b^2={\omega }^2/c_s^2-{\kappa }^2$. Vanishing of both displacements at the rigid plane gives $\kappa (1+A)-bB=0$, $a(1-A)-\kappa B=0$. Hence $A=(ab-{\kappa }^2)/(ab+{\kappa }^2)$ and $B=2a\kappa /(ab+{\kappa }^2)$. Because $c_p>c_s$ and ${\kappa }^2\le {\omega }^2/c_p^2$, $b^2>0$ and the converted S wave propagates rather than being evanescent.

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 $c_S=\sqrt{\mu /\rho }$.

For an isotropic layered medium, SH displacement is perpendicular to the vertical propagation plane and SV displacement lies within it. P-SV coupling occurs at interfaces, while SH remains a separate polarization sector.

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 flat layer of thickness $h$ and shear speed ${\beta }_1$ above a faster half-space of speed ${\beta }_2$, unfolding $m$ down-and-up traversals gives the displayed reflected-ray travel time. Its slope is the horizontal ray slowness. At $x=2mh\tan i_c$, with $\sin i_c={\beta }_1/{\beta }_2$, the corresponding head-wave branch $t=x/{\beta }_2+2mh\sqrt{{\beta }_1^{-2}-{\beta }_2^{-2}}$ is tangent to this curve.

For subcritical incidence, a reflected SH-wave loses energy into the faster substrate. Postcritical incidence has an evanescent substrate tail and no outgoing normal energy flux in the ideal lossless model. Repeated reflections can then interfere to form Love waves. Individual reflected rays are contributions to the field; normal-mode frequencies require the reflection-phase condition and are not supplied by a geometrical travel-time curve alone.

For a leading wavefront singularity or discontinuity in a smooth isotropic shear-horizontal wave field, the ray transport equation conserves the displayed quantity along a tube of elastic rays. Here $J$ is transverse tube width in two dimensions and area in three. Cylindrical spreading has $J\propto r$, so the leading wave amplitude is proportional to $r^{-1/2}$. This rule excludes the immediate source neighborhood and caustics.

Place a line source a distance $h$ above a plane welded interface and put $q_j=({\beta }_j^{-2}-p^2{)}^{1/2}$, $t_1=h/({\beta }_1^2{q}_1)$ and $r_1=h/({\beta }_1{q}_1)$. The transmitted elastic ray is $x=hp/q_1+{\beta }_2^{2}p(t-t_1)$, $z={\beta }_2^2{q}_2(t-t_1)$. Elastic-interface boundary conditions give $T_{\mathrm{SH}}=2{\mu }_1{q}_1/({\mu }_1{q}_1+{\mu }_2{q}_2)$. Differentiating the ray positions across neighboring values of $p$ and applying SH ray-tube amplitude transport gives the displayed leading amplitude. Both $q_j$ must be positive; critical and evanescent incidence require different asymptotics.

Integrating the shear-horizontal wave equation with force $F(t){\delta }^{(2)}(\mathbf{x})$ over a shrinking circle gives $2\pi \mu r{\partial }_{r}u\to -F(t)$. Thus the local displacement field has the displayed logarithmic singularity. This is a fixed-time near-source limit, distinct from the wavefront limit underlying SH ray-tube amplitude transport.

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 $c_s^2\kappa /\omega$.