Elastic-wave energy flux

For displacement velocity $\dot{u}$ and stress $\sigma$, the instantaneous elastic-energy flux isFor complex harmonic amplitudes, its average over one temporal period is

An incident SV-wave generally reflects as both an SV-wave and a P-wave. A rigid boundary determines their amplitudes by requiring both displacement components to vanish, while phase matching preserves frequency and tangential wavenumber.

For incident SV angle $\theta$, tangential phase matching gives $\sin \varphi =(c_P/c_S)\sin \theta$ for the reflected P-wave. It is evanescent when $\sin \theta >c_S/c_P$.

When the converted P-wave is evanescent, its amplitude decays away from the boundary and the reflected SV amplitude has modulus one. The P field stores reactive energy near the boundary but carries zero mean normal power.

At a perfectly rigid plane, every component of the displacement field vanishes.

A wavevector points in the direction of phase propagation; its magnitude is the wavenumber and its scalar product with position gives the spatial phase.

For a slowly varying local dispersion relation $\omega =\Omega (\mathbf{k};\mathbf{x},t)$, wave-packet rays obey

The WKB method seeks a rapidly oscillating field in the formand determines the phase and slowly varying amplitude order by order in $\epsilon$.

The leading WKB phase obeys an eikonal equation. For a local dispersion relation $\omega =\Omega (k;x,t)$ and $k=\nabla \theta$, it is

Along a ray $\mathbf{x}(t)$, the derivative of a field $f(\mathbf{x},t)$ is

An evanescent wave has an imaginary component of its wavevector and therefore decays exponentially in that direction instead of transporting energy away as a propagating wave.