A COMMUTATION PROBLEM

So long as R is isotropic, then it will commute with $\nabla ^{T} C \nabla$ and there will be a Christoffel-like representation. If R is not isotropic then the following trick will get around the problem. First, premultiply the equation by R^-1/2, which I define as the symmetric (but possibly complex) matrix whose eigenvalues have a positive real part, and are the inverse of the square-root of the eigenvalues of the matrix R, which, for physical reasons (causality, no energy gain), are also constrained. So equation (4), dropping the $(i \omega)$ notation, can be written as,

$$R^{-1/2} \nabla ^{T} C \nabla R^{-1/2}R^{1/2} u + \omega ^{2} R^{-1/2} u = 0,$$ (5)

and this can be further simplified by replacing the displacement field variable, u, by an energy-like field variable, w, defined equal to R^1/2u. Our final form is now,

$$R^{-1/2} \nabla ^{T} C \nabla R^{-1/2} w + \omega ^{2} \ddot{w} = 0.$$ (6)

This new field variable is quite similar to the energy flux variable introduced by Clint Frasier and reported in Aki & Richards (1980), although our motivations are somewhat different.