An analytical solution for the transient response of a semi-infinite elastic medium with a buried arbitrary cylindrical line source

This study develops an analytical solution for the transient response of a semi-infinite medium subjected to an arbitrary cylindrical line source buried at a certain depth by using the Laplace transform and Cagniard's method. The analytical solution is presented in a simple closed form and each term represents a transient physical wave. During the solution procedure, the arbitrary source pulse is first decomposed into a compressional wave (P) and a tangential wave (S), and then analytical solutions in the Laplace domain for the P and S pulse sources are derived. By applying Cagniard's inverse Laplace transform method and the convolution theorem, analytical solutions in the time domain are obtained. Finally, the analytical solution for arbitrary source pulse can be expressed as the superposition of the solutions for the P and S pulse sources. Numerical examples are provided to illustrate some interesting features of the cylindrical P and S wave propagation in a semi-infinite media with a free surface. A head wave can be observed in a certain area of the semi-infinite media due to the S pulse, which is not found in the P pulse problem.