Reciprocity and energy conservation within the parabolic approximation

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 crucial for the PE to be capable of modeling subtle nonreciprocal acoustic effects due to ocean currents. The energy-conserving property of a PE is of particular importance for correct modeling of the field amplitude in shallow water with significant bathymetric variations and/or considerable range-dependence of geoacoustic parameters. In this paper, reciprocity and energy conservation are considered for two-dimensional (2D) PEs for fluids with piecewise-continuous range-dependence. In the case of lossless media, equivalence of the reciprocity and the energy conservation is demonstrated. A new class of wide-angle PEs is derived such that its members ensure energy conservation and reciprocity, and possess higher asymptotic accuracy in range-dependent environments than previously known PEs. Energy-conserving boundary conditions consistent with the parabolic approximation are obtained for stair-case and sloping interfaces.