On degeneracy of dispersive waves at the bulk wave velocities

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.


Introduction
The paper is devoted to analyzing and correcting solutions of the Pochhammer -Chree wave equation at phase velocities coinciding with the longitudinal ( 1 c ) and shear ( 2 c ) bulk

Principle equations Equation Section 2
Equation of motion for an isotropic medium at absence of body forces can be represented in a form where u is the displacement field, 12 , cc are velocities of bulk longitudinal and shear waves respectively: (2.2) ,  are Lame's constants, and  is a material density.The Helmholtz representation for the displacement field u yields rot    u  , where  and  are scalar and vector potentials respectively.In cylindrical coordinates representation (2.3) for the physical components of the displacement field, becomes   In (2.4) it is assumed that coordinate z directs along central axis of the rod.It is also assumed that the displacement field is axially symmetric, that yields 0 u   .(2.6)For a plane harmonic wave propagating along axis z , potentials (2.6) can be represented in a form ( ) ( ) 00 ( ) , ( ) where, as before,  is the wave number related to the phase speed c and circular frequency  by equation c   . (2.8) x is the (vector) coordinate in the cross section of a rod ( () where n is the wave vector; and z  nx.
(2.10)  potentials.The corresponding solution for the scalar potential has the form   is satisfied by the following equations [4]   0

.24)
And accounting that for a solid cylinder the corresponding displacement field should be finite at where the multiplier  is defined by the following expression   , see [11].

, cc phase velocities Equation Section (Next)
The traction-free boundary conditions on a lateral cylindrical surface at where  is the (outward) surface normal.
rr rr rR rz rz rR q J q r C tI q C q rJ q r J q r r i C q rJ q r J q r i q C J q r i C J q r t i q C J q r q C J q r Equations ( 3.2) can be rewritten in a matrix form where A is a square and non-symmetric 22  matrix with complex coefficients: At deriving (3.4) from (3.2) the following identity was used Suppose initially that 0  , then direct analysis of Eq. (3.8) 2 reveals that either 12 ( ) 0 J q R  or 22 2 0 q    .The first option is inconsistent with Eq. (3.8) 1 , as Bessel functions 0 J and 1 J have no common roots.The second option in view of (2.13) means q  .Note, that the case 0  at this stage is not considered.But, condition (3.9)   means a very specific material with vanishing Poisson's ratio, that flows out from analyzing expressions (2.2).Now, assuming that (3.9) takes place, Eq. (3.8) 1 yields the following equation which satisfies at some specific values of the radius R independently of 0  .Thus, the case (3.9) at 0  does not lead to any meaningful dispersion relation.Suppose now that 0  , then direct analysis of Eqs. (3.8) reveals that in such a case both equations are satisfied identically, ensuring that a pair satisfies the dispersion relation and that is the unique solution irrelevant of physical properties.Thus, at 1 cc  all the dispersion curves vanish.

Dispersion equation at degenerate case
( ) ( ) 0 rr rr rR rz rz rR q J q r C i C tI q q J q r r J q r C i C t i q C J q r i C r i q C J q r Presumably, the most interesting is disappearing all the terms not containing Bessel functions in the resulting Eq. (3.13).

Conclusions
The exact solutions of the linear Pochhammer -Chree equation for propagating harmonic axisymmetric longitudinal waves (0, ) Lm in a cylindrical body, were analyzed, revealing that at the phase velocities coinciding with the bulk wave velocities

(2. 5 )
Substituting representation (2.3) into equation of motion (2.1 Two-dimensional (right) eigenvectors related to vanishing eigenvalues (kernel eigenvectors) of matrix A define polarization of the corresponding Pochhammer -Chree waves.Substituting components (3.4) into Eq.(3.6) yields the dispersion equation in the form[7] In view of Eqs.(2.25), the surface traction components at rR  become

1 c and 2 c
, the Pochhammer -Chree equation becomes degenerate leading to solutions involving functions other than Bessel.