Past exam of the elasticity and plasticity course of the University of Oxford / 2011 / 2 / b

Resolve each amplitude by the orthogonal projectionsThe acoustic tensor acts by $\lambda +2\mu$ on the longitudinal line and by $\mu$ on its perpendicular plane. ThusThe longitudinal component is a longitudinal wave, while the two independent transverse polarizations are transverse waves. Each projection obeys its own wave equation, and superposition adds the corresponding solutions.

A general planar disturbance is therefore a sum of the two types. For a single sinusoid with a common speed, both projections cannot be nonzero unless the two speeds coincide: projecting the eigenvalue equation gives $(\rho c^2-\lambda -2\mu )a_L=0$ and $(\rho c^2-\mu )a_T=0$. This distinguishes superposition of waves with different speeds from a mixed polarization traveling rigidly at one speed.