In What Sense Do Slow Waves Carry Negative Energy?

Abstract

It has been found in the theory of electron tubes that, according to the ``small‐amplitude power theorem,'' the fast and slow space‐charge waves carry positive and negative energy, respectively. Similar analysis of different systems leads to similar results, leading one to conjecture that there is some sense in which one might assert that, for a wide class of dynamical systems, slow waves carry negative energy. In a one‐dimensional model, ``slow'' and ``fast'' waves in a moving propagating medium refer to waves of which the phase velocity does or does not change sign, respectively, on transforming from the moving frame to the stationary frame. Small‐amplitude disturbances of any dynamical system may be described by a quadratic Lagrangian function, from which one may form the canonical stress‐tensor, elements of which are quadratic functions of the variables which appear in the linearized equations of motion. For any pure wave in this system, the energy density E and the momentum density P, as they appear in the canonical stress tensor, are related to the frequency ω and wave number κ by E=Jω, P=Jκ, where 2πJ is the action density. The rules for Galilean transformation now show that the energy densities, as measured in the stationary frame, of fast and slow waves have positive and negative sign, respectively, if (as is usually the case) the energy densities of both waves are positive in the moving frame. Similar arguments explain the signs of the energy density of the two ``synchronous'' waves which arise in the analysis of transverse disturbances of an electron beam in a magnetic field.