Formulas for Phase Velocity and Damping of Longitudinal Waves in Thick-Walled Viscoelastic Tubes

Abstract

The frequency equation for longitudinal waves in a viscoelastic tube of infinite length filled with a viscous fluid is solved for the complex wavenumber, k(=k1+ik2). Thus for two modes, Young's mode and Lamb's mode, an explicit expression for k is obtained which yields the phase velocity c(=ω/k1, ω=radial frequency) and the damping constant k2. The formulas for k, c, and k2 are valid for wall thicknesses up to the size of the inner radius of the tube. The approximation as compared with results obtained numerically with an IBM 7040 digital computer from the original frequency equation involves an error which in almost all cases is considerably less than 1.5%. For Lamb's mode, as it was found earlier for the torsional mode, an increase of the fluid viscosity increases the damping at the higher frequencies but decreases the damping at the lower frequencies. The theoretical results are compared with those of experiments in which the longitudinal waves are generated in the fluid and the fluid in its turn moves the wall. The experimental curves thus obtained show the pattern of Young waves, and the agreement with the Young curves obtained from the formulas mentioned above is satisfactory. For torsional waves, formulas for k, c, and k2 analogous to these for longitudinal waves are given.