The governing differential equation for this problem is a 3rd order PDE that is formulated as a 1-D plane wave problem (hence the need for the BAMP program to correct for beam divergence.
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 (4) 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 (4) is
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 (7). In that computation, . Then, is estimated from equation (8).
RUNNING CAINV3: Start an octave session, type cainv3 GUI, use mouse to pick min and max frequencies, click OK and then use the mouse. Horizontal position is all that is read. Focus one of the panels. You can exclude some frequencies, and that will create an fbx vector. If you include all frequencies, you may get an error statement (since it can't write out something that does not exist). Typically not a problem when you run caplot3.m later. Don't worry about it. GUI, C1=stiffness, C2=damping initial estimate for the 3rd order wave equation. GUI, Choose weighting GUI, Choose balance between damping and velocity, .5 good idea Plots, update as inversion progresses GUI, continue LSQE plot GUI, continue Chi squared plot GUI, save results to disk, yes if you want to run caplot3.m
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After running cainv3, you may wish to make nice plots. For this, there is program caplot3 (8.2.8).