Derivation of an exact energy balance for Galbrun equation in linear acoustics

In the frame of mixed Eulerian–Lagrangian description, the propagation of linear waves in inhomogeneous moving fluids can be described by Galbrun equation, governing the particle displacement caused by the wave. A Lagrangian density for Galbrun equation is built from the perturbations of the potential and kinetic energy. Using Hamiltonian formalism, an exact acoustic energy balance can be derived from this Lagrangian density. Some important physical properties of this acoustic pseudo-energy are illustrated on a few simple cases and compared with other expressions given by different authors.