Elastic Wave Propagation in Rods of Arbitrary Cross Section

A study is made of the problem of transient elastic wave propagation when a load is applied to one end of an infinitely long rod of arbitrary cross section. Fourier and Laplace transforms aid in developing a general solution in the form of a sum over all modes of harmonic wave propagation and integrals over all wavelengths. An expansion at long wavelength reveals two types of modes of propagation which are called longitudinal and radial shear modes. For a suddenly applied pressure, the long-distance, long-time response is found to be an integral of the Airy integral.