= Solution
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, and
$$
v_{tt}=c_s^2(v_{xx}+v_{zz}),\qquad c_s=\sqrt{\mu/\rho}.
$$
Rigid fixed boundaries require $v=0$ at $z=0,h$. The modes are $v=A\sin(q_nz)\cos(kx-\omega_nt)$ with $q_n=n\pi/h$, $n=1,2,\ldots$, and
$$
\boxed{\omega_n(k)=c_s\sqrt{k^2+q_n^2},\qquad\omega_{n,\mathrm{cut}}=c_sq_n.}
$$
There is no $n=0$ displacement mode with these clamped boundary conditions. For $k>0$, the <phase velocity> and <group velocity> are
$$
\boxed{c(k)=c_s\frac{\sqrt{k^2+q_n^2}}k,\qquad c_g(k)=c_s\frac{k}{\sqrt{k^2+q_n^2}}.}
$$
The <phase velocity> decreases from infinity to $c_s$, while the <group velocity> increases from zero to $c_s$; their product is $c_s^2$.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2015/ii/paper-2-sh-waveguide.png]
{title=Dispersion, phase velocity and group velocity of a clamped shear-horizontal waveguide mode}
For a mode of frequency $\omega$, the appropriate time and cross-sectional average is $\langle A\rangle=\omega/(2\pi h)\int_0^{2\pi/\omega}\int_0^h A\,dz\,dt$, 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 \b[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> with
$$
|k_*|=\frac{q_n|V|}{\sqrt{c_s^2-V^2}},\qquad\omega_n''(k_*)>0.
$$
The <stationary phase method> gives a \b[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 \b[exactly zero after a sufficiently long time]. These are the requested subsonic and super-shear-speed regimes.
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