Dispersion of Compressional Waves in Isotropic Rods of Rectangular Cross Section

Abstract

The compressional wave velocity in rectangular brass rods is measured a function of frequency by means of standing wave patterns made visible by the use of lycopodium powder. An x-cut quartz crystal is used as a driver between frequencies of 200 and 650 kilocycles per second. Results are obtained on rods of different width-to-thickness ratios, the thickness being the same in each case. The dispersion curves are made up of two branches with a discontinuity occurring one of them when the half-wave-length is close to the larger dimension. Each of these branches is shown to be determined by one of the lateral dimensions and a qualitative interpretation is offered in terms of a lateral resonant effect. In the case of a square rod the two branches coincide, and the dispersion is the same as that encountered in a circular rod of the same cross-sectional area. Theoretical calculations are made for the case in which the width is much larger than the thickness. Numerical solutions are found for the dispersion curves of the first symmetrical mode for different values of Poisson's ratio. Calculated values are in good agreement with measured values when the cross-sectional dimensions are in the ratio of 8:1 and the wave-length is smaller than the larger dimension.