Boundary conditions for the parabolic equation in a range-dependent duct

Abstract

The parabolic wave equation in a range-dependent duct with a rigid bottom is considered. The range dependence will introduce amplitude errors in addition to the usual phase error of the local modes. It is shown that the amplitude error increases by local mode angle and slope of the bottom, and is inversely proportional to the depth. Imposing the vanishing of the normal derivative of the pressure at the bottom, which is the physically correct boundary condition, might also lead to ill conditioning of the parabolic model. Various ways to replace the boundary condition ∂u/∂n=0 are discussed, and their accuracy is assessed both numerically and theoretically.