Dispersion of Elastic Waves in Solid Circular Cylinders

Abstract

The theory of elastic vibrations in solid circular cylindrical rods of homogeneous isotropic materials is redeveloped from the general equations of elasticity with the following general results: 1. For any mode of vibration the ratio of the velocity of any elastic wave traveling along the rod to the velocity $c_0$ of shear waves is the same for any two rods whose Poisson ratios are equal and whose ratios of circumference to shear wave-length are equal; 2. If the velocity of propagation for any particular mode remains less than $c_0$ as the frequency or radius is increased indefinitely, this velocity approaches that of Rayleigh surface-waves; 3. If the velocity of propagation for any particular mode remains greater than $c_0$ as the frequency or radius is increased indefinitely, this velocity approaches $c_0$ in the limit; and 4. A considerable simplification is introduced into the method of computing dispersion curves for any mode. This investigation not only generalizes and extends the work of D. Bancroft on elongational waves but further includes the computation of an exact table of dispersion curves for the flexural mode of vibration. The dispersion curves for magnesium are compared with the experimental results of Shear and Focke. Excellent agreement between theory and experiment is obtained for the first elongational and flexural branches.