Elastic mode classification by vertical energy flux ($J_3=\frac{{\omega }^2}{2}\mathrm{Re}[a^{\ast }(Tq+R)a]$)

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.