It is shown that under very general conditions there is a rigorous identity between the group velocity and the velocity of energy transport in nonhomogeneous media with or without anomalous dispersion. The medium is assumed nondissipative and the parameters such as density, rigidity, dielectric constant, etc., vary from point to point but are independent of the Cartesian coordinate lying in the direction of the mode propagation. This covers propagation in any type of wave guide, surface waves, propagation in stratified media, etc. The identity is established for fluids, for isotropic and anisotropic solids with or without prestress, and for electromagnetic waves. Anomalous dispersion is assumed to result from hidden coordinates such as electron oscillators. A new variational formulation of field theory is introduced. An interesting application is to wave propagation in an electron gas and it is shown that such wave propagation obeys the relativistic Schrödinger equation for a mass particle.