ABSTRACTIn this paper, a modified version of the method of steepest descent is proposed for the evaluation of Lamb's integrals which can be considered as basis functions dealing with the development of the transition matrix method which can be used to study the wave scattering in a two-dimensional elastic half-space. The formal solutions of the generalized Lamb's problem are studied and evaluated on the basis of the proposed method. After defining a phase function which presents in wavenumber integral, an exact mapping and an inverse mapping can be obtained according to the phase function. Thus, the original integration path can be deformed into an equivalent admissible path, namely, steepest descent path which passed through the saddle point, and then mapped onto a real axis of mapping plane, finally, resulted in an integral of Hermite type. This integral can be efficiently evaluated numerically in spite of either near- to far-field or low to high frequency. At the same time, the asymptotic value can easily be obtained by applying the proposed method. The numerical results for generalized Lamb's solutions are calculated and compared with analytic, asymptotic or other existing data, the excellent agreements are found. The properties of generalized Lamb's solutions are studied and discussed in details. Their possible applications for wave scattering in elastic half-space are also pointed out.
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