Time-harmonic solution of the elastic head wave problem incorporating the influence of Rayleigh poles

Abstract

The classical Lamb problem of a time-harmonic line load applied to the surface of an elastic solid is considered in this paper. In the past, ordinary asymptotic methods have been applied to the branch-cut evaluation of the time-harmonic compressional and shear head waves. It is shown here that the application of these methods generally leads to erroneous results. The problem of the shear branch cut is associated with the Rayleigh pole, and the problem of the compressional branch cut is associated with the other two roots of the Rayleigh equation, which lie in the neighborhood of the compressional branch point. These two roots affect the compressional branch-cut integration and can contribute a residue to the total solution. To evaluate the time-harmonic fields, we utilize the following methods of integration—the modified saddle-point method, the method of numerical integration along the branch cut, and the method of numerical integration along the Sommerfeld contour. With the inclusion of the effects of all three roots of the Rayleigh equation, the results from the three methods are in excellent agreement.