Exact closed-form solutions for Lamb's problem. Issue 1 (14th April 2018)

• Record Type: Journal Article
• Authors: Feng, Xi; Zhang, Haiming
• Is Part Of: Geophysical journal international. Volume 214:Issue 1(2018:Jul.)
• Page Start: 444
• Page End: 459
• Publication Date: 2018-04-14
• Subjects: Theoretical seismology -- Wave propagation -- Body waves; Geophysics -- Periodicals; 550
• DOI: 10.1093/gji/ggy131
• Languages: English
• ISSNs: 0956-540X

Abstract

SUMMARY: In this paper, we report on an exact closed-form solution for the displacement at the surface of an elastic half-space elicited by a buried point source that acts at some point underneath that surface. This is commonly referred to as the 3-D Lamb's problem for which previous solutions were restricted to sources and receivers placed at the free surface. By means of the reciprocity theorem, our solution should also be valid as a means to obtain the displacements at interior points when the source is placed at the free surface. We manage to obtain explicit results by expressing the solution in terms of elementary algebraic expression as well as elliptic integrals. We anchor our developments on Poisson's ratio 0.25 starting from Johnson's integral solutions which must be computed numerically. In the end, our closed-form results agree perfectly with the numerical results of Johnson, which strongly confirms the correctness of our explicit formulae. It is hoped that in due time, these formulae may constitute a valuable canonical solution that will serve as a yardstick against which other numerical solutions can be compared and measured.