The calculation of surface-wave dispersion is difficult when the waves propagate in media whose physical properties change with depth, and only a few solutions are available for fairly simple cases.
These computations may now be performed with the aid of high-speed computers, even for media whose material constants change arbitrarily with depth. The dispersion of both Love waves and Rayleigh waves has been obtained for such cases by the numerical specification of surface displacement followed by numerical solution of the equations of motion.
For example, with respect to the problem of Love waves, besides the ordinary boundary condition that the stress vanishes at the free surface, an extra condition is stated which requires that the displacement amplitude be unity at the surface. The equation of motion is then solved numerically for tentative values of frequency and wave number, and this solution produces the distribution of displacement amplitude in the half space. For all combinations of frequency and wave number which are not solutions, the values of computed displacement do not converge and tend to become positively or negatively infinite for increasing depth below the free surface. To obtain a solution, one of the parameters—for instance, wave number—is fixed, and frequency is varied in small steps until the computed displacement converges to zero at great depths. This combination of parameters fulfills all the standard boundary conditions and is the required solution. The problem of sound waves in an elastic liquid can also be solved with only a minor change in the physical properties.
The dispersion of Rayleigh waves propagating in a heterogeneous substance can also be obtained by a similar method. In this case, another parameter is needed, namely, the ratio of the amplitude of horizontal and vertical components of displacement at the free surface. Denote this quantity by a, and the phase velocity by c. Wave number is fixed, and a two-dimensional search in the a-c plane is used to locate the point that produces a convergent solution. For limited media a solution is required which satisfies the boundary conditions at the other surface.
Love waves in a medium with constant density and linearly increasing rigidity.
Sound waves in a medium whose density and velocity are given by experimental curves.
Rayleigh waves in a medium having constant density and equal rates of increase for λ and μ. Using an IBM “650,” it takes a few minutes to get a point for cases 1 and 2, and from thirty minutes to two hours per point for Rayleigh-wave dispersion.