Abstract:

Realistic description of sound interaction with the ocean bottom requires environmental models having piecewise-continuous dependence of their parameters on position. For a boundary-value problem to be well posed for a parabolic equation (PE), unlike the wave equation, type, and even the number of boundary conditions (BCs) to be imposed at an interface, depends on the interface geometry and the PE itself. In this paper, the boundary conditions at staircase and sloping interfaces are considered for a class of PEs for which the operator square root is approximated by a rational function. Boundary conditions are derived from the PEs themselves thereby ensuring consistency between the equation and the BCs. In particular, energy-conserving PEs produce energy-conserving BCs. Physically, this procedure can be viewed as the substitution of an interface by an infinitesimally thin transition layer. The existence and uniqueness of solutions for the boundary-value problem for a wide-angle PE in the presence of interfaces are both illustrated by considering plane-wave reflection and transmission. Moreover, this fundamental problem provides insight into asymptotic accuracy of the parabolic approximation in media with piecewise-continuous parameters. [Work supported by NSERC.]