A Note on Exact Closed-Form Solutions for Lamb’s Problem with an Arbitrary Poisson’s Ratio

Abstract

Abstract Lamb’s Problem, a cornerstone in theoretical seismology, investigates the displacement of an elastic half-space due to surficial or buried point sources. Although the exact closed-form solution for surface displacements from surficial sources is well established, the solutions for buried sources in the 3D case have remained limited, either confined to integral forms or applicable only for certain Poisson’s ratios. This study addresses this gap by generalizing the existing closed-form solutions to accommodate Poisson’s ratio for buried sources. Our solutions are expressed in terms of elementary algebraic expressions, elliptic integrals, and Gauss hypergeometric functions. The derived solution is validated by numerical tests. With the newly derived solution, we also explore the properties of the leaking mode, which stems from the complex conjugate roots of the Rayleigh equation.