 Elastic slowness surface

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 $c_{ijkl}{a}_i{a}_k{p}_{l}d{p}_j=0$.