Elastic slowness surface ($\det (c_{ijkl}{p}_j{p}_l-\rho {\delta }_{ik})=0$)

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.