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