= Solution
For a homogeneous lossless elastic solid, insert the <plane wave> $u_i=\operatorname{Re}\{U a_i e^{i(\mathbf k\cdot\mathbf x-\omega t)}\}$ into the equations of <linear elasticity>. Put $\mathbf k=\kappa\mathbf n$, $|\mathbf n|=1$, and $v=\omega/\kappa$. The resulting <eigenvalue> problem is
$$
\Gamma_{ik}(\mathbf n)a_k=\rho v^2 a_i,\qquad
\Gamma_{ik}=c_{ijkl}n_jn_l.
$$
The minor and major symmetries make this <acoustic tensor> a real <symmetric matrix>. For a stable material it is <positive-definite>, so its three positive <eigenvalues> give three real positive <phase velocities> and therefore \b[three <slownesses> $s_a=1/v_a$]. Coincident <eigenvalues> are possible, so there need not be three distinct speeds. For distinct <eigenvalues> the displacement <polarization (waves)> are orthogonal; in an <anisotropic> solid they are generally quasi-longitudinal and quasi-transverse rather than exactly longitudinal and transverse. Stability, or at least <strong ellipticity in elasticity>, is required for the assertion about positive real speeds.
The <elastic slowness surface> is the locus of the three vectors $\mathbf p=\mathbf n/v_a(\mathbf n)=\mathbf k/\omega$. Its equation is
$$
\det[c_{ijkl}p_jp_l-\rho\delta_{ik}]=0.
$$
For each <wavefront> normal there are three radial sheets, which may meet at degeneracies. To obtain the <energy flux> directly, multiply the elastic equation by $\dot u_i$. For
$$
W=\tfrac12\rho\dot u_i\dot u_i+\tfrac12 c_{ijkl}e_{ij}e_{kl},
$$
the symmetry of <stress> gives $\partial_t W=\partial_j(\sigma_{ij}\dot u_i)$. Thus the instantaneous <elastic-wave energy flux> is $F_j=-\sigma_{ij}\dot u_i$. With a real branch <polarization (waves)> $\mathbf a$, harmonic averaging gives
$$
\langle F_j\rangle=\frac12\omega|U|^2c_{ijkl}k_l a_i a_k
=\frac12\omega^2|U|^2c_{ijkl}p_l a_i a_k.
$$
Now differentiate $c_{ijkl}p_jp_l a_k=\rho a_i$ along a regular <slowness surface> sheet and contract with $a_i$. The terms containing $d\mathbf a$ cancel by symmetry and the original <eigenvalue> equation, leaving
$$
2c_{ijkl}a_i a_k p_l\,dp_j=0.
$$
Every tangent $d\mathbf p$ is consequently perpendicular to $\langle\mathbf F\rangle$: \b[the <energy flux> is normal to the <slowness surface>]. At a degenerate point this statement is made on the limiting branches, rather than assuming a unique normal.
For completeness, equality of the mean kinetic and <strain> energies gives $\langle W\rangle=\rho\omega^2|U|^2|\mathbf a|^2/2$. Differentiating the <acoustic tensor> equation in the <wavevector> variables similarly gives
$$
g_j=\frac{\partial\omega}{\partial k_j}
=\frac{c_{ijkl}k_l a_i a_k}{\rho\omega|\mathbf a|^2},\qquad
\boxed{\langle\mathbf F\rangle=\langle W\rangle\mathbf g.}
$$
Thus the <group velocity> is the <elastic energy velocity>. The homogeneous elastic <dispersion relation> is degree one in $\mathbf k$, so differentiating $\omega(\lambda\mathbf k)=\lambda\omega(\mathbf k)$ at $\lambda=1$ gives $\mathbf p\cdot\mathbf g=1$.
The <elastic wave surface> is the unit-time locus of these <group velocities>. From an impulsive point source, the propagating <wavefronts> at time $t$ are its sheets scaled to $\mathbf x=t\mathbf g$. A particular source may fail to excite a mode in some directions, and its amplitudes require the source <polarization (waves)> and geometrical spreading; the <wave surface> determines the arrival geometry. It does not specify all near-field motion between the radiative fronts. Geometrically it is the envelope of the planes $\mathbf p\cdot\mathbf x=1$: tangent variation gives $d\mathbf p\cdot\mathbf x=0$, so $\mathbf x$ is the appropriately normalized normal to the <slowness surface>.
In a symmetry-plane cross-section write $\mathbf p=r(\theta)\mathbf n$, $\mathbf n=(\cos\theta,\sin\theta)$ and $\mathbf t=(-\sin\theta,\cos\theta)$. The two envelope conditions give
$$
\mathbf g=\frac{\mathbf n}{r}-\frac{r'}{r^2}\mathbf t
=v\mathbf n+v'\mathbf t,\qquad
\mathbf g'=(v+v'')\mathbf t
=\frac{r^2+2r'^2-r r''}{r^3}\mathbf t.
$$
The numerator is the signed curvature numerator of the <slowness surface> cross-section. At a simple <inflection point>, $h=v+v''=0$ and $h'\ne0$. Then $\mathbf g''=h'\mathbf t$ and $\mathbf g'''=h''\mathbf t+2h'\mathbf t'$ are linearly independent. These are precisely the local conditions for an ordinary <cusp>. The <slowness curvature and wave-surface cusps> relation explains why a smooth nonconvex <slowness surface> can generate a cusped <wavefront> and multiple ray arrivals.
The example below uses a positive orthotropic stiffness in normalized units: $\rho=1$, $C_{11}=C_{22}=C_{33}=4$, $C_{44}=C_{55}=1.2$, $C_{66}=1$, and all normal off-diagonal stiffnesses zero. In the symmetry plane $n_3=0$, its two in-plane <acoustic tensor> <eigenvalues> are $v_\pm^2=(5\pm\sqrt{5+4\cos4\theta})/2$, and its out-of-plane shear speed is $\sqrt{1.2}$. The in-plane shear sheet is nonconvex and produces the marked <cusps>; the construction uses an actual stable elastic tensor.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2002/iii/paper-54-slowness-wave.png]
{title=Slowness and group-velocity wave surfaces of a stable anisotropic elastic tensor, with shear-wave cusps}
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