Energy Propagation in Dissipative Systems Part I: Centrovelocity for Linear Systems

In this paper we deal with the almost classical problem of the energy propagation associated with a given dispersive wave motion. Usually, in the literature the energy-flux velocity is assumed to specify this propagation, and it is known simply as energy velocity. For homogeneous monochromatic waves propagating in a linear isotropic system with dispersion but without dissipation, this velocity coincides with the group velocity as is well known; the statement can be proved in different ways (see, e.g., [1-8]). In order to be able to treat energy propagation for nonperiodic waves with finite energy and to incorporate dissipative effects as well, the concept of centrovelocity of energy (or, simply centrovelocity) has proved to be useful (see, e.g., [9-12]). The centrovelocity, defined as the velocity of the center of gravity of some density, here the energy density, has a clear physical interpretation. Van Groesen [ 12] has shown that, for linear dispersive waves governed by a scalar and conservative wave equation, this velocity can be related to an appropriate average of the classical group velocity. Furthermore, it was shown there that this concept can also be applied to nonlinear wave equations such as Korteweg-de Vries (KdV) and BenjaminBona-Mahony (BBM) equations. In this paper we will restrict the analysis to uniaxial waves. Starting from a given balance equation, we recall in Section 2 the concepts of energy-flux velocity (denoted by Vr) and of centrovelocity (denoted by Ve). In Section 3 a quite general expression for the centrovelocity is derived. It is shown that this velocity differs from the energy-flux velocity by some term that is due to the presence of dissipation. For a class of solutions, which we refer to as uniformly-damped solutions, we prove the coincidence of Ve with Vf. Only in such a case will we use the common term "energy velocity" to denote the two velocities. After this general part we restrict our attention to linear wave equations. In Section 4 we derive, for a linear equation of first-order in time,