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,
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
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).