Abstract
Degeneracy of the linear dispersion wave equation at the phase velocities coinciding with the bulk wave velocities is observed and analysed. Spectral analysis of Pochhammer – Chree equation is performed. The corrected analytical solutions for components of the displacement fields are constructed, accounting degeneracy of the secular equations and the corresponding solutions.
Highlights
The paper is devoted to analyzing and correcting solutions of the Pochhammer – Chree wave equation at phase velocities coinciding with the longitudinal ( c1 ) and shear ( c2 ) bulk wave velocities, at which degeneracy of the Bessel equation occurs
Since the first derivation of the Pochhammer – Chree equation for harmonic waves propagating in a cylindrical rod [1 – 3] and numerous subsequent works [4 – 14] it was assumed that the solution of the dynamic equations for harmonic waves in a circular rod reduces to the Bessel equations regardless of the phase velocity. As it will be shown later on, at the phase velocities coinciding with c1 and c2 bulk wave velocities, the corresponding dynamic equations do not lead to Bessel equations, and the solutions for the dynamic equations and the dispersion equations should be reworked
For a plane harmonic wave propagating along axis z, potentials (2.6) can be represented in a form
Summary
The paper is devoted to analyzing and correcting solutions of the Pochhammer – Chree wave equation at phase velocities coinciding with the longitudinal ( c1 ) and shear ( c2 ) bulk wave velocities, at which degeneracy of the Bessel equation occurs. Since the first derivation of the Pochhammer – Chree equation for harmonic waves propagating in a cylindrical rod [1 – 3] and numerous subsequent works [4 – 14] it was assumed that the solution of the dynamic equations for harmonic waves in a circular rod reduces to the Bessel equations regardless of the phase velocity. As it will be shown later on, at the phase velocities coinciding with c1 and c2 bulk wave velocities, the corresponding dynamic equations do not lead to Bessel equations, and the solutions for the dynamic equations and the dispersion equations should be reworked.
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