On evaluation of Lamb's integrals for waves in a two-dimension elastic half-space

In this paper, a modified version of the method of steepest descent is prop osed for the evaluation of Lamb's integrals which can be considered as basi s functions dealing with the development of the transition matrix method wh ich can be used to study the wave scattering in a two-dimensional elastic h alf-space. The formal solutions of the generalized Lamb's problem are studi ed and evaluated on the basis of the proposed method. After defining a phas e function which presents in wavenumber integral, an exact mapping and an i nverse mapping can be obtained according to the phase function. Thus, the o riginal 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 inte gral of Hermite type. This integral can be efficiently evaluated numericall y in spite of either near- to far-field or low to high frequency. At the sa me time, the asymptotic value can easily be obtained by applying the propos ed method. The numerical results for generalized Lamb's solutions are calcu lated and compared with analytic, asymptotic or other existing data, the ex cellent agreements are found. The properties of generalized Lamb's solution s are studied and discussed in details. Their possible applications for wav e scattering in elastic half-space are also pointed out.