A higher-order energy-conserving parabolic equqation for range-dependent ocean depth, sound speed, and density

Outgoing solutions of the wave equation, including parabolic equation (PE) and normal-mode solutions, are usually formulated so that pressure is continuous with range for range-dependent problems. The accuracy of normal-mode solutions has been improved by conserving energy rather than maintaining continuity of pressure [Porter et al., “The problem of energy conservation in one-way equations,” J. Acoust. Soc. Am. 89, 1058–1067 (1991)]. This approach is applied to derive a higher-order energy-conserving PE that provides improved accuracy for problems involving large ocean bottom slopes and large range and depth variations in sound speed and density. A special numerical approach and complex Padé coefficients are applied to suppress Gibbs’ oscillations. The back-propagated half-space field, an improved PE starter, is applied to handle wide propagation angles. Reference solutions generated with a complex ray model and with the rotated PE are used for comparison.