On Group Velocity of Elastic Waves, in an Anisotropic Plate

The equivalence of the energy velocity and the group velocity is often taught in university physics. An early proof of the equivalence of the two velocities for one-dimensional water waves was given by Rayleigh [1]. In the 1950s and 1960s many studies [2}4] were done to show the equivalence of the two velocities for more general systems. Later, in the 1990s, Awati and Howes raised a question asking for a general proof of the relationship between the group velocity and the energy velocity, and a number of people [5}8] responded to their question. However, several cases need particular considerations. There are some researchers who tried to show that the energy velocity and the group velocity are not equal for acoustic waves in piezoelectrics, but their result was proved to be incorrect later [9]. Another ambiguous case is that of the backward waves where the group velocity is negative while the wave number is positive [10}12]; in other words, the group velocity and the wave number have opposite signs. Some people believe that when the group velocity is negative, it is not equal to the energy velocity [13]. Tolstoy and Usdin [10] studying the Rayleigh}Lamb dispersion equation in 1957, numerically found that the group velocity may be negative for some modes of wave propagation in a plate. Meitzler [11] in 1965 reported an experimental result showing the existence of the mode for which the group velocity could be negative in a plate. In a recent study, photo-elastic pictures of wave modes where there may exist negative group velocity were presented [12]. People who believe that the energy and group velocities are equivalent interpret this phenomenon in such a way that the group velocity should be positive and the wave number should be negative. By the 1970s, a proof was o!ered by Achenbach [14] for the equivalence of the two velocities for wave in a plate under a simple assumption that the amplitude of the wave is independent of frequency and wave number k along a branch of the frequency spectrum. However, his proof is not valid for the backward waves. In this paper, a general proof is presented to show the equivalence of the energy velocity and the group velocity for waves in an anisotropic homogenous plate and it is valid for the backward waves. The expressions for both the energy velocity and the group velocity are obtained in terms of the Lagrangian density.