Exact transient solution of the buried line source problem

The disturbance due to a line source buried within an elastic, homogeneous and isotropic half-space represents a most interesting problem of theoretical seismology. Exact formal solutions have been published by a number of investigators, but the integrals which appeared in their solutions could be evaluated only approximately at large distances from the source. As a result, the behaviour at intermediate distances or close to the epicentre could not be investigated analytically. This paper discusses a different approach to the problem. It is shown that the use of Laplace transform techniques and of a suitable contour in the complex plane reduce the complexity of the problem to a point where no integration is required. The surface displacements as a function of time and position are found to be expressible in terms of algebraic functions in closed form. The case of a source displacement consisting of a sudden rise followed by a gradual recovery is considered in detail. Graphs are presented which show the vertical and horizontal surface displacement as a function of time, at different distances from the epicentre and for different values of Poisson’s ratio.