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We consider the third-order wide-angle `parabolic' equation of underwateracoustics in a cylindrically symmetric fluid medium over a bottom ofrange-dependent bathymetry. It is known that the initial-boundary-value problemfor this equation may not be well posed in the case of (smooth) bottom profilesof arbitrary shape if it is just posed e.g. with a homogeneous Dirichlet bottomboundary condition. In this paper we concentrate on downsloping bottom profilesand propose an additional boundary condition that yields a well posed problem,in fact making it $L^2$-conservative in the case of appropriate realparameters. We solve the problem numerically by a Crank-Nicolson-type finitedifference scheme, which is proved to be unconditionally stable andsecond-order accurate, and simulates accurately realistic underwater acousticproblems.

A Finite Difference method for the Wide-Angle `Parabolic' equation in a waveguide with downsloping bottom