A derivation of the Christoffel equation with damping

This paper provides a derivation of the Christoffel eigenvalue equation for acoustic wave propagation in an acoustically damped piezoelectric medium. The damping tensor is shown to couple into both the stress and displacement constitutive equations. Application of the quasi-static approximation leads to an additional term in the Christoffel equation that generates a complex k-vector, due both to introduction of a complex term and to breaking of symmetry in the left-hand side of the eigenvalue equation, subsequently resulting in damping and a phase shift for a plane wave solution. Shown are the effects of damping on the eigenvalues of the piezoelectrically stiffened Christoffel equation for plane wave propagation in unconstrained quartz over a 1 MHz to 1 GHz frequency range.