Elastic wave polarization ($a=a_L+a_T$)

The polarization of an elastic wave in an isotropic solid specifies the direction of its displacement field amplitude relative to its wavevector. The longitudinal wave component is parallel to the wavevector; the transverse wave component is perpendicular. For a real unit propagation vector $n$, $a_L=(a\cdot n)n$ and $a_T=a-a_L$. An evanescent constituent has a complex spatial derivative vector, so the corresponding amplitude ratio can be imaginary. The longitudinal and transverse constituents of a Rayleigh wave combine into ellipse particle motion at the free surface.