The velocity of energy through a dissipative medium

The velocity of seismic and electromagnetic signals depends on properties such as elastic moduli, density, porosity, viscosity, dielectric permittivity, and conductivity. Hence, the identification of the correct velocity of energy transport is essential to obtain the characteristics of the medium. The energy and group velocities, defined for a monochromatic plane wave, are compared to the centrovelocity, related to the centroid of the pulse in the time and spatial domains. The comparison is performed for a 1D medium and a band-limited pulse with a given dominant frequency and taking into account that the centroid of the spectrum decreases with increasing distance. For a lossless medium, the three velocities coincide. In absorbing media, the centrovelocity is closer to the group velocity at short travel distances, in which the wave packet retains its shape. At a given distance, the centrovelocity equals the energy velocity, and beyond that distance this velocity becomes a better approximation. This is generally the case for the propagation of acoustic and electromagnetic waves in earth materials. In other cases, such as electromagnetic propagation at the atomic scale (Lorentz model), the meaning of the energy velocity needs to be revisited, and concepts such as the signal velocity are required.