Nonlocal boundary conditions for finite-difference parabolic equation solvers

Theoretical/numerical models of underwater sound propagation usually incorporate a downgoing radiation condition on the transverse component of the acoustic field. Most one-way or parabolic equation (PE) solvers approximate this radiation condition by appending an absorbing layer to the computational mesh and setting the field to zero at the base of this layer. Papadakis et al. [J. Acoust. Soc. Am. 92, 2030–2038 (1992)] replaced such approximate treatments with a nonlocal boundary condition (NLBC) that exactly transforms the semi-infinite PE problem to an equivalent one in a bounded domain. Papadakis’ approach requires the evaluation of a spectral (wave number) integral whose integrand is inversely proportional to the impedance of the subbottom medium. In this paper, an alternate procedure is analyzed for obtaining NLBCs directly from the z-space Crank–Nicolson solvers for both the Tappert and Claerbout PEs. Formulas for the field ψ at range r+Δr are derived in terms of the known field at the previously calculated range values from 0 to r by expanding the appropriate “vertical wave number” operator for the downgoing field in powers of the translation operator R=exp(−Δr∂r). The effectiveness of these NLBCs is examined for several numerical examples relevant to one-way underwater sound propagation.