Energy dispersion of localized disturbance in continuum and discrete media

We present analytical and numerical investigations of the size and shape evolution of localized disturbances during their propagation in elastic media (furthermore, we refer to this process as energy dispersion). We consider two models of media: the continuum, described by the multidimensional wave equation, and the simple scalar harmonic lattice. The presented research is based on the energy dynamics approach. We describe the evolution of the disturbance by the concept of the energy ellipsoid, which characterizes the eigenvectors and eigenvalues of the second moment of the energy distribution. The wave packet is considered as an example of a localized disturbance. For this case, the time evolution of the energy ellipsoid is obtained. It is shown that the velocities and directions of the energy dispersion are determined by the second derivative of the dispersion relation with respect to the wave vector. The analytical results are numerically confirmed for the case of a square lattice. The obtained results demonstrate that both the propagation and the dispersion of the wave packet energy can be described analytically in the framework of the energy dynamics.