The PDE relevant to 1-D wave propagation is given by
where “u” is particle displacement, “t” is time, “x” is the coordinate in the direction of wave propagation, is the stiffness , and is the damping . Equation (1) reduces to the elastic wave equation when the damping value, . In that case, the phase velocity is constant for all frequencies, and the wave does not experience any decay (for a 1-D plane wave). In the elastic case, the phase velocity will be .In the more general case, , and there will be both velocity dispersion and exponential, inelastic amplitude decay. A solution of equation (1) is
Michaels (10) shows that the inelastic decay of a plane wave will be given by
The values for and can be expressed in terms of the following : and
Determination of and is by nonlinear joint inversion of the phase velocity, c, and inelastic decay, , over a range of frequencies. The inversion is currently performed in the Octave procedure, cainv3.m. Initial estimates of stiffness and damping are obtained at the frequency corresponding to the largest measured by bamp. First, is found by evaluation of equation (4). In that computation, . Then, is estimated from equation (5).