In theoretical/numerical models of underwater sound propagation, a downgoing radiation condition is usually imposed on the acoustic field as z->(infinity). For most implementations of parabolic equation (PE) solvers, this condition is approximated by appending an absorbing layer to the computational mesh. Such a layer acts to attenuate any energy that reaches the grid boundary before the waves undergo reflection and return to the ocean region. As shown by Papadakis [J. Acoust. Soc. Am. 92, 2030--2038 (1992)], this approximate treatment can be replaced with a nonlocal boundary condition (NLBC) that exactly transforms the semi-infinite PE problem to an equivalent one in a bounded domain. Papadakis' method involves evaluating a spectral (wave number) integral of a singular kernel that is inversely proportional to the impedance of the subbottom medium. In this paper, an alternate procedure is described for obtaining NLBCs directly from the z-space Crank--Nicolson formulations of both theTappert and Claerbout PEs. Formulas for the boundary field at range r+(Delta)r are derived in terms of the known field along 0->r by expanding the appropriate ``vertical wave number'' operator in powers of R=exp(-(Delta)r(cursive beta)[inf r]) and applying R [sup j](psi)(r,z)=(psi)(r-j(Delta)r,z). The effectiveness of these NLBCs is compared to Berenger's matched-layer technique [J. Comp. Phys. 114, 185--200 (1994)] for several numerical examples relevant to one-way underwater sound propagation.