Three-dimensional, energy-conserving, and reciprocal one-way acoustic wave equations

Abstract

Within the parabolic approximation, preserving energy conservation and reciprocity intrinsic to acoustic fields in stationary media is known to be vital for accurate amplitude predictions in shallow-water environments and for modeling subtle nonreciprocal acoustic effects due to ocean currents. Recently, a wide-angle parabolic equation (PE) has been derived that ensures energy conservation and reciprocity of its solutions in fluids with range-dependent, piecewise-continuous sound speed and density [O. A. Godin, J. Acoust. Soc. Am. 100, 2835(A) (1996)]. In this paper, the theory is extended to 3-D problems having considerable azimuth coupling. Moreover, a technique is proposed to derive energy-conserving and reciprocal PEs with arbitrary high wide-angle capability. Formulations in cylindrical and Cartesian coordinates are compared. The ability of the 3-D energy-conserving PE to account for cross-range inhomogeneities and horizontal refraction is discussed. The asymptotic accuracy of the energy-conserving parabolic approximation is addressed and compared to that of other paraxial approximations. It is concluded that energy conservation, reciprocity, and an improved description of mode coupling can be achieved simultaneously within the 3-D parabolic approximation without adversely affecting phase accuracy. [Work supported by NSERC.]