Polarization of Pochhammer–Chree Waves: Axisymmetric Longitudinal Modes

Abstract

Exact solutions to the Pochhammer–Chree equation describing harmonic wave propagation in an elastic cylindrical bar are analyzed. For the first time, spectral analysis of the matrix of dispersion equation is performed for axisymmetric longitudinal modes. Analytic expressions for wave polarization are derived. For the fundamental axisymmetric longitudinal mode, polarization coefficients of the corresponding waves at the bar surface are determined and frequency dependent variations of the coefficients are analyzed. It is found that, when the phase velocity of the fundamental axisymmetric longitudinal mode coincides with the velocity of transverse waves, all the displacement components determined at the lateral bar surface simultaneously vanish, which is significant for acoustic waveguide design.