Transparent boundary conditions for parabolic equation solutions of radiowave propagation problems

Perfectly transparent boundary conditions are derived for truncating the integration domain when solving radiowave propagation problems with a parabolic equation (PE) method, The boundary conditions are nonlocal: they are expressed as a convolution integral involving the field at all previous ranges. The convolution kernel is matched to the refractive index vertical gradient at the boundary. The boundary conditions include an incoming energy term which can model an arbitrary incident field, In particular, they may be used with plane-wave incidence, or with a point-source located below or above the domain boundary, If required, the solution can be extended to heights above the boundary with a generalized horizontal PE method. Closed-form solutions for the incoming energy term are given for plane-wave incidence and for Gaussian sources when the refractive index above the boundary is constant or linear, The resulting finite-difference algorithms provide efficient solutions to problems involving airborne sources, Numerical examples are given, showing excellent agreement with a pure split-step/Fourier PE algorithm.