The propagation characteristics of longitudinal modes are described by the Pochhammer frequency equation in the case of elastic cylinders and the Rayleigh-Lamb frequency equation in the case of elastic plates. The longitudinal modes in both geometries are closely analogous. In both cases, the second branch of the mode spectrum determined by the roots of the applicable frequency equation shows a region in which phase and group velocities have opposite signs. Wave motions having phase and group velocities of opposite sign occur in certain guided-electromagnetic-wave motions and are often referred to as backward-wave motions. This paper reports experimental observations of stress pulses traveling in backward elastic-wave motions in cylinders and plates. Also included in the paper is a detailed description of how the distribution of the energy flux density changes as the wave motion corresponding to the second branch of roots goes from the region of forward-wave motion to the region of backward-wave motion.
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A.H. Meitzler (1965) studied this question.