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Away from the line source, insert a leading wavefront form $u\sim a(\mathbf{x})W(t-T(\mathbf{x}))$ into the shear-horizontal wave equation and match the two most singular orders. The first gives the eikonal equation $|\nabla T|=1/\beta$. The next givesEquivalently, SH ray-tube amplitude transport conserves $\rho \beta a^{2}J$, where $J$ is the transverse width of a two-dimensional ray tube. A line source has straight radial elastic rays and $J=rd\theta$ for fixed angular width, soThe same geometrical spreading applies to the first nonzero derivative discontinuity or leading wavefront singularity; the source history determines which waveform $W$ and normalization $C$ occur. In three dimensions a point source instead has $J\propto r^2$ and amplitude proportional to $r^{-1}$.

This does not contradict the SH line-source near-field singularity. Ray transport describes an arrival-front limit with $t-r/\beta$ small, or the high-frequency field outside the immediate source neighborhood. The logarithm describes $r\downarrow 0$ at fixed elapsed time and includes the tail behind the two-dimensional wavefront. For example, a causal step force $F_{0}H(t)$ has the exact radial displacementFor $r\ll \beta t$, this is $F_0\log (2\beta t/r)/(2\pi \mu )$, the near-source logarithm. For $t=r/\beta +\tau$ with $0<\tau \ll r/\beta$, it isThe coefficient of the leading arrival singularity therefore has exactly the ray-theory dependence. A force impulse gives the corresponding $r^{-1/2}H(\tau )/\sqrt{\tau }$ leading field. The limits describe different regions of one causal solution; neither is a uniform approximation to the other.

Let $u(x,z,t)$ be displacement along the source line and let $F(t)$ be its force per unit length. The shear-horizontal wave equation isRotational symmetry about the line makes $u=u(r,t)$. Integrate over a disk of radius $r$ and use the divergence theorem:At a fixed regular time, a logarithmic singularity and its time derivatives are locally integrable; the inertial integral is $O(r^2|\log r|)$ and vanishes as $r\downarrow 0$. The leading balance is consequently $r{u}_r=-F(t)/(2\pi \mu )$, giving the SH line-source near-field singularityThe reference length $r_{\ast }$ only changes the bounded term; boundary conditions and source history determine that term. This is a local fixed-time statement, interpreted distributionally in time if the source force has jumps. It establishes logarithmic spatial dependence directly, without extrapolating far-field ray spreading into the source.

With vertical shear modulus ${\rho }^{\prime }{\beta }_z^{\prime 2}$ and horizontal shear modulus ${\rho }^{\prime }{\beta }_x^{\prime 2}$, force balance therefore gives the anisotropic shear-horizontal wave equationFor $u_y=Y(z)e^{i(kx-\omega t)}$, the layer vertical wavenumber and substrate decay exponent areThe free top still gives $Y=A\cos (qz)$; the substrate still has $Y=B{e}^{-\kappa (z-h)}$. Welded displacement continuity and the now anisotropic traction continuity giveHence the anisotropic Love-wave dispersion relation isEquivalently, writing $c=\omega /k$,This replaces the isotropic speed in the layer equation by separate horizontal and vertical speeds and replaces the layer traction modulus by ${\rho }^{\prime }{\beta }_z^{\prime 2}$. No additional SH/P-SV coupling arises for the aligned symmetry axes and constitutive law here.

A rigid base still imposes $q_n={\chi }_n/h$, but nowThusThe vertical speed sets the resonance frequencies, while the horizontal speed sets the high-frequency limits and $c_n{U}_n={\beta }_x^{\prime 2}$.

For the elastic substrate, put ${ϵ}_x={\beta }_x^{\prime }/\beta$, $\zeta ={\rho }^{\prime }{\beta }_x^{\prime }{\beta }_z^{\prime }/(\rho {\beta }^2)$, $W=\omega h/{\beta }_z^{\prime }$ and $K=kh{\beta }_x^{\prime }/{\beta }_z^{\prime }$. The exact parameterization isIt reproduces the isotropic analysis with ${\beta }^{\prime }$ replaced by ${\beta }_z^{\prime }$ in the frequency scale, by ${\beta }_x^{\prime }$ in the velocities, and by the two effective contrast parameters above. In particular the true cutoffs are ${\omega }_{c,n}=n\pi {\beta }_z^{\prime }/[h\sqrt{1-{ϵ}_x^2}]$, the group minima occur near ${\omega }_{n,z}$, and a sufficient rigid-limit validity range isFor comparable horizontal and vertical layer speeds, a bounded density ratio and fixed mode index, the minimum estimate becomes ${\delta }_{\min }\sim ({\zeta }^2{\chi }_n/2{)}^{1/3}$ and $U_{\min }/{\beta }_x^{\prime }\sim 3(\zeta /{\chi }_n{)}^{1/3}/2^{2/3}$. If their ratio is extreme, that more specific minimum-depth estimate additionally needs $\zeta /{\delta }_{\min }\gg {ϵ}_x$; the exact parameterization and load criterion remain valid without it. As in the isotropic case, neither a slow horizontal nor a slow vertical speed alone justifies the rigid boundary at cutoff or within the strong-dispersion transition.

Let the shear-horizontal wave displacement field be $U(z)e^{ikx-i\omega t}$. Its velocity is $V=-i\omega U$, and its traction is $T=\mu U^{\prime }$. The scalar shear-wave equation givesWith $p=k/\omega$ and $p_{\beta }^2={\beta }^{-2}-p^2$, the first-order velocity–traction system isThese equations remain valid for depth-dependent $\mu ,\rho$; differentiating $\mu U^{\prime }$ is essential and one must not incorrectly use $\mu U^{\prime \prime }$ when $\mu$ varies.

Where $V\ne 0$, define the SH input impedance $Z=-T/V$. Differentiating $T=-ZV$ and substituting the two equations gives the SH impedance Riccati equation:The second relation is a one-way scalar equation once the complete SH input impedance is known. It does not discard reflections: $Z$ describes the actual superposition of both wave directions. At a zero of $V$, $Z$ may have a pole while the velocity–traction system stays regular.

Take shear-horizontal waves with displacement $\mathbf{u}=(0,0,w(x,y,t))$. This is divergence-free, so the elastic equation reduces to $\rho w_{tt}=\mu (w_{xx}+w_{yy})$. Rigid boundaries require $w=0$ at $y=0,h$, giving modes $w=A\sin (n\pi y/h)\cos (kx-\omega t)$, $n\ge 1$. With shear speed $c_s=\sqrt{\mu /\rho }$, the dispersion relation isFor $k>0$, the phase velocity is $v_p=\omega /k>c_s$ and the group velocity is $v_g=d\omega /dk=c_s^{2}k/\omega <c_s$, with $v_p{v}_g=c_s^2$.

The kinetic energy density is $T=\rho w_t^2/2$. The only strains are $e_{xz}=w_x/2$ and $e_{yz}=w_y/2$, together with their symmetric counterparts; thus the isotropic linear-elastic energy density is $W=\mu (w_x^2+w_y^2)/2$. Average over one temporal period and integrate across the layer. The sine- and cosine-squared transverse integrals are both $h/2$, givingThe guided shear-horizontal mode has average equipartition after both operations; a time average at a fixed transverse point alone need not give equality.

The linear elastic wave equation is $\rho {\mathbf{u}}_{tt}=(\lambda +\mu )\nabla (\nabla \cdot \mathbf{u})+\mu \Delta \mathbf{u}$. For a shear-horizontal wave, take $\mathbf{u}=(0,v(x,z,t),0)$ with no $y$ dependence. Its divergence vanishes, andRigid fixed boundaries require $v=0$ at $z=0,h$. The modes are $v=A\sin (q_{n}z)\cos (kx-{\omega }_{n}t)$ with $q_n=n\pi /h$, $n=1,2,\dots$, andThere is no $n=0$ displacement mode with these clamped boundary conditions. For $k>0$, the phase velocity and group velocity areThe phase velocity decreases from infinity to $c_s$, while the group velocity increases from zero to $c_s$; their product is $c_s^2$.

For a mode of frequency $\omega$, the appropriate time and cross-sectional average is $⟨A⟩=\omega /(2\pi h){\int }_0^{2\pi /\omega }{\int }_0^{h}Adzdt$, per unit transverse width. Its average kinetic energy density is $\rho A^2{\omega }^2/8$. The only nonzero strains are $e_{xy}=e_{yx}=v_x/2$ and $e_{yz}=e_{zy}=v_z/2$, while $e_{kk}=0$. Thus the elastic energy density is $\mu (v_x^2+v_z^2)/2$, whose average is $\mu A^2(k^2+q_n^2)/8$. The dispersion relation proves equality of average kinetic and elastic energies.

For the ray asymptotics of a clamped elastic waveguide mode, expand the localized initial displacement in transverse sine modes and Fourier transform in $x$. For a displacement released from rest, each mode evolves with $\cos ({\omega }_n(k)t)$. At $x=Vt$ the oscillatory phases are $t(kV\pm {\omega }_n(k))$. If $0<|V|<c_s$, one phase has a stationary point withThe stationary phase method gives a generic oscillatory amplitude proportional to $t^{-1/2}$ for each contributing mode. If the initial Fourier coefficient vanishes at the stationary point, that mode can decay faster; smooth localized data make the modal sum well behaved. If $|V|>c_s$, there is no stationary point. Repeated integration by parts gives rapid decay for smooth rapidly decaying data; for initially compactly supported displacement, finite propagation speed makes the displacement on that ray exactly zero after a sufficiently long time. These are the requested subsonic and super-shear-speed regimes.

Take a shear-horizontal wave polarized in the $z$ direction,Then $\nabla \cdot \mathbf{u}=0$, and the elastic-wave equation reduces toFor $w=Y(y)e^{i(kx-\omega t)}$, the rigid boundary gives $Y(0)=0$, while vanishing shear traction at the free surface gives $Y^{\prime }(h)=0$. ThusThe dispersion relation of each guided shear-horizontal mode is

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.

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.

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 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.

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.