Pochhammer-Cree Longitudinal Waves: Anomalous Polarization

Abstract

The exact solutions of the Pochhammer-Cree wave equation, which describes the propagation of harmonic waves in an elastic cylindrical rod, are analyzed. For longitudinal axially symmetric modes, a spectral analysis of the matrix of the dispersion equation is carried out for the first time. Analytical expressions for wave polarization are obtained. On the surface of the rod for the fundamental longitudinal axially symmetric mode, the polarization coefficients of the corresponding waves are determined and the variation of these coefficients depending on the frequency is analyzed. It was found that at the phase velocity of the fundamental axially symmetric longitudinal mode, which coincides with the shear wave velocity, all components of displacements on the side surface of the rod simultaneously vanish, which seems to be extremely important for the design of acoustic waveguides.