Exact closed-form solutions for Lamb's problem (III): The case for buried source and receiver

TL;DR

This paper derives an exact closed-form solution for elastic wave displacement in a half-space caused by a buried point force, extending Lamb's problem to include buried sources and receivers, with validated numerical results.

Contribution

It provides the first explicit closed-form solutions for Lamb's problem with buried sources and receivers, generalizing previous surface-only solutions.

Findings

Closed-form solutions match numerical results perfectly.

The solutions involve elementary functions and elliptic integrals.

Validation confirms the accuracy of the derived formulas.

Abstract

In this article, we derive the exact closed-form solution for the displacement in the interior of an elastic half-space due to a buried point force with Heaviside step function time history. It is referred to as the tensor Green's function for the elastic wave equation in a uniform half-space, also a natural generalization of the classical 3-D Lamb's problem, for which previous solutions have been restricted to the cases of either the source or the receiver or both are located on the free surface. Starting from the complex integral solutions of Johnson, we follow the similar procedures presented by Feng and Zhang to obtain the closed-form expressions in terms of elementary functions as well as elliptic integrals. Numerical results obtained from our closed-form expressions agree perfectly with those of Johnson, which validates our explicit formulae conclusively.