Dispersion of Compressional Waves in Rods of Rectangular Cross Section

Abstract

An earlier paper [J. Acous. Soc. Am. 20, 833 (1948)] reported measurements of the phase velocity of compressional waves as a function of frequency in rectangular rods where the cross-sectional dimensions are of comparable magnitude. Two modes of propagation were observed each being determined by one of the lateral dimensions, there being a discontinuity in the dispersion curve determined by the larger side. The present paper is an attempt to explain the behavior of these curves using the general methods of the theory of elasticity applied to the symmetrical vibrations of an infinitely long isotropic rod. Harmonic wave solutions of the displacement equations of motion are constructed which satisfy the free-surface boundary conditions for certain wave-length regions in the general case and for all wave-lengths in certain limiting configurations. Dispersion curves are calculated and compared with the observed ones. The frequencies at which the experimental discontinuities were found to occur are closely predicted and the theoretical curves are in good general agreement with the measured values.