Discrete Transparent Boundary Conditions for Wide Angle Parabolic Equations in Underwater Acoustics

Abstract

This paper is concerned with transparent boundary conditions (TBCs) for wide angle “parabolic” equations (WAPEs) in the application to underwater acoustics (assuming cylindrical symmetry). Existing discretizations of these TBCs introduce slight numerical reflections at this artificial boundary and also render the overall Crank–Nicolson finite difference method only conditionally stable. Here, a novel discrete TBC is derived from the fully discretized whole-space problem that is reflection-free and yields an unconditionally stable scheme. While we shall assume a uniform discretization in range, the interior depth discretization (i.e. in the water column) may be nonuniform, and we shall discuss strategies for the “best exterior discretization” (i.e. in the sea bottom). The superiority of the new discrete TBC over existing discretizations is illustrated on several benchmark problems. In the literature different WAPEs (or WAPE and the standard “parabolic” equation) have been coupled in the water and the sea bottom. We analyze under which conditions this yields a hybrid model that is conservative for the acoustic field.