On reciprocity and energy conservation for one-way acoustic wave equations

Abstract

Acoustic reciprocity in stationary media at rest and energy conservation are widely recognized as fundamental properties of the sound field to be inherited by any successful paraxial approximation. Rigorous validity of the reciprocity principle for a parabolic equation (PE) solution in motionless media is of particular importance for the PE to be capable of modeling subtle nonreciprocal acoustic effects due to ocean currents. In this paper, reciprocity and energy conservation are considered for a class of two-dimensional PEs for fluids with piecewise continuous range dependence. In the case of lossless media, equivalence of the reciprocity and the energy conservation is demonstrated. Uniqueness of reciprocity and energy conservation corollaries of a PE and their relation to respective properties of the true acoustic field are discussed. A wide-angle PE for moving fluid and appropriate boundary conditions are derived which lead to the exact fulfillment of a reciprocity principle and a conservation law. In a motionless layered fluid of constant density the PE reduces to the Claerbout PE. Some features of the new PE are analyzed. Its ability to model the nonreciprocal component of the acoustic field in the ocean is compared to that of other existing paraxial approximations. [Work supported by NSERC.]