Governing Differential Equation

The PDE relevant to 1-D wave propagation is given by

$\displaystyle \frac{\partial^{2}u}{\partial t^{2}}=C_{1}\frac{\partial^{2}u}{\partial x^{2}}+C_{2}
\frac{\partial^{3}u}{\partial t\partial x^{2}}$ (1)

where “u” is particle displacement, “t” is time, “x” is the coordinate in the direction of wave propagation, $C_{1}$is the stiffness $\left(\frac{m^{2}}{s^{2}}\right)$, and $C_{2}$ is the damping $\left(\frac{m^{2}}{s}\right)$. Equation (1) reduces to the elastic wave equation when the damping value, $C_{2}=0$. 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 $\sqrt{C_{1}}$ .

In the more general case, $C_{2}\neq0$, and there will be both velocity dispersion and exponential, inelastic amplitude decay. A solution of equation (1) is

$\displaystyle u(x,t)=\exp\left(-\alpha x\right)\cdot\cos\left(\beta x-\omega t\right),$

where the wavenumber is complex and given by $\beta+i\alpha$ .

Michaels (10) shows that the inelastic decay of a plane wave will be given by

$\displaystyle \alpha=\frac{4\sqrt{D}\omega^{2}C_{2}}{\left(2\omega C_{2}\right)^{2}+D^{2}}$

where $\omega$ is angular frequency (rad/s) and the quantity, D, is given by

$\displaystyle D=2\left(C_{1}+\sqrt{C_{1}^{2}+\omega^{2}C_{2}^{2}}\right).$ (2)

The phase velocity, c, varies with frequency according to the following relationship

$\displaystyle c=\frac{2\omega^{2}C_{2}}{D\alpha}.$ (3)

The values for $C_{1}$ and $C_{2}$ can be expressed in terms of the following :

$\displaystyle C_{1}=\frac{\left(\beta^{2}-\alpha^{2}\right)\omega^{2}}{\left(\beta^{2}+\alpha^{2}\right)^{2}},$ (4)

and

$\displaystyle C_{2}=\frac{2\alpha\beta\omega}{\left(\beta^{2}+\alpha^{2}\right)^{2}}.$ (5)

Determination of $C_{1}$ and $C_{2}$ is by nonlinear joint inversion of the phase velocity, c, and inelastic decay, $\alpha$ , 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 $\alpha$ measured by bamp. First, $C_{1}$ is found by evaluation of equation (4). In that computation, $\beta=\frac{\omega}{c}$ . Then, $C_{2}$ is estimated from equation (5).