Solutions for compressional waves in an isotropic rod of rectangular cross section are formulated from the general elastic equations of motion. Two general modes of propagation are constructed which satisfy the boundary conditions for the free vibrations of a long rod for which the width 2d is considerably greater than the thickness 2a. The thickness mode is shown to have an infinite phase velocity (cut-off frequency) at that frequency for which the half wave-length in an infinitely wide rod would be equal to 2d. The phase velocity of both of these propagation modes is shown to approach that of Rayleigh surface waves when the wave-lengths are much smaller than the lateral dimensions. The calculated thickness mode cut-off frequencies are compared with measurements reported earlier and the agreement is shown to be good. A comparison between the predicted and measured phase velocity dispersion is made for various values of a/d. The agreement is found to be satisfactory as long as d is several times greater than a.

Morse, R. W. (1950). The Velocity of Compressional Waves in Rods of Rectangular Cross Section. The Journal of the Acoustical Society of America, 22(2), 219–223.