An Approximate Model for Wave Propagation in Rectangular Rods and Their Geometric Limits

Abstract

An approximate model for wave propagation in rods of rectangular cross section was developed that is based on neglecting the dependence of the shear stresses and the longitudinal displacement on one, or both, of the in-cross-sectional coordinates. The resulting approximate system of equations, together with the appropriate boundary conditions, are then solved exactly, leading to a simple, but general, characteristic dispersion equation. The degenerate geometric limiting cases of infinite media and flat plates are obtained as special cases. The model also predicts all of the general features exhibited in Morse's experimental data (1948). These include the correct low- and high-frequency limits of the wave speeds, the cut-off frequencies, and the common point of crossing of the higher group of modes.