Analysis of long compressional elastic waves in rods of arbitrary cross section and elastic wave fronts in plates and circular rods

Abstract

Long waves in elastic rods of arbitrary cross section are studied by writing a general expansion of the exact solution for three dimensional linear elasticity. The solution holds for transient excitation of the end of a semi-infinite cylinder and is in terms of the harmonic modes of wave propagation for the infinite elastic cylinder. The major contribution to the solution for large distances from the end of the rod is found by making approximations to the infinitely long wave length part of the solution. This is aided by using a perturbation method for long wave length to study the modes of propagation. An approximate theory for rods of arbitrary cross section is developed and compared to the exact theory for harmonic waves of infinitely long wave lengths. The amplitudes and locations of all wave fronts caused by certain suddenly applied loads on elastic plates and circular rods are presented. Both end loads on the rod and plate as well as normal line and point forces on the plate are considered. The problems are solved by expanding double transforms into a series of terms, each term representing the disturbance following a single wave front. Evaluation of the terms for the wave front behavior is accomplished by Cagniard's method and the saddle point method. Ray theory aids in the interpretation of the results and also serves to verify most of the formulas. The solution by Cagniard's method is exact for the plane strain problems studied and is plotted and compared to experiments.