The exact solutions of the linear Pochhammer-Chree equation for propagating harmonicwaves in a cylindrical rod, are analyzed. Spectral analysis of the matrix dispersionequation for longitudinal axially symmetric modes is performed. Analytical expressionsfor displacement fields are obtained. Variation of wave polarization on the free surfacedue to variation of Poisson's ratio and circular frequency is analyzed. It is observed thatat the phase speed coinciding with the bulk shear wave speed all the components of thedisplacement eld vanish, meaning that no longitudinal axisymmetric Pochhammer-Chree wave can propagate at this phase speed.