The analytical method such as the boundary integral (element) method or the series expansion method is usually used to solve the wave scattering problem. In these methods, the singular Green’s function should be determined firstly; the main difficulty of the use of the Lamb’s singular solutions in integral form to represent the diffracted fields is the numerical implementation for the evaluation of those improper integrals. The integrands of these integrals are highly irregular and oscillatory. In this paper, a technique is proposed to calculate the integral in wave-number domain based on the method of steepest descent. After replacing the original integration path by steepest decent path, the wave-number integral results in a Gauss-Hermite type quadrature, so the oscillating characteristics of the original integrand can be removed and results a non-oscillating integrand, it is very helpful in computing efficiency.
On Evaluation of Lamb’s Integrals for Seismic Waves in a Three-Dimension Elastic Half-Space
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Liao, W, & Teng, T. "On Evaluation of Lamb’s Integrals for Seismic Waves in a Three-Dimension Elastic Half-Space." Proceedings of the ASME 2005 Pressure Vessels and Piping Conference. Volume 8: Seismic Engineering. Denver, Colorado, USA. July 17–21, 2005. pp. 55-61. ASME. https://doi.org/10.1115/PVP2005-71448
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