Solution to Lamb's Problem (lamb)

The BSU distribution includes some synthetic seismogram capabilities. The solution of Lamb's problem (Lamb (6) ) is computed by lamb. The program, lamb, is a recreation of a program originally described by Mooney (15) . The algorithm closely follows that described in Mooney's paper, with the exception that an autoregressive operator is used to generate a minimum phase wavelet to filter the data (Mooney convolved with a zero phase wavelet). Mooney's work cites an earlier solution as the basis for his program (Pekeris (16) ), and that paper is well worth examining for clarification on any points not entirely clear in Mooney's publication. The current BSU version only implements the case for a Poisson's ratio of $\frac{1}{4}$ . Modification to other cases is relatively straight forward, and will probably be done at some point in the future.

Briefly, Lamb's problem is the solution for the inhomogeneous waves that travel on the surface of an elastic half-space due to a vertical, point impact source. Lamb referred to these waves as major and minor tremors. Today, we would call these the Rayleigh (major), P- and S-waves (minor). A test of the lamb program was to recreate previously published solutions plotted in unit less time. The unit less time coordinate is given by

$\displaystyle \tau=\frac{V_{s}\cdot t}{R},$ (44)

where R is the range (m) from the source, t is time (s), and $V_{s}$ is the shear wave velocity (m/s). This solution is shown in Figure 52, and compares favorably with previously published solutions (Mooney (15) figure2; Pekeris (16) figures 3 and 4; and Richards (18) figures 2 and 3, integrals $I_{3}$ and $I_{4}$ respectively). The GSL library provides the necessary elliptical integrals.

Figure 52: Solution to Lamb's Problem (after Mooney, 1974 (15)). Step function source.
\includegraphics[scale=.6]{Figure18}



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