Conservation and balance equations for waves in dissipative media

It is shown that conservation equations for various wave equations may be constructed by means of a simple technique. The averaged conservation equations are obtained automatically by using complex conjugate solutions. Formerly such conservation equations were obtained either by averaging the exact conservation equations for a single solution, or by averaging the Lagrangian density appropriate for the problem at hand and using the variational principle as postulated by Whitham. The present method allows to construct nonconservation equations, describing the balance of the quantity at hand. For conservative systems we are invariably led to the correct averaged conservation equations. These involve the group velocity. In dissipative media averaged conservation equations are obtained, which define a new complex group velocity. With respect to the conventional complex group velocity, balance equations are obtained. Consequently the present method facilitates the description of the ray intensity in inhomogeneous, dissipative, time-dependent linear systems.