Conservative initial‐boundary value problems for the Wide‐Angle PE in waveguides with variable bottoms

Abstract

We consider the third‐order, Claerbout‐type Wide‐Angle Parabolic Equation (PE) in the context of Underwater Acoustics in a cylindrically symmetric medium consisting of water over a soft bottom B of range‐dependent topography. There are strong indications, that the initial‐boundary value problem for this equation with just a homogeneous Dirichlet boundary condition on B, may not be well‐posed, for example when B is downsloping. In previous work we proposed an additional bottom boundary condition that, together with the zero field condition on B, yields a well‐posed problem. In the present paper we continue our investigation of additional bottom boundary conditions that yield well‐posed, physically correct problems. Motivated by the fact that the solution of the wide‐angle PE in a domain with horizontal layers conserves its L2 norm in the absence of attenuation, we seek additional boundary conditions on a variable‐topography bottom, that yield L2‐ conservative solutions of the problem. We identify a family of such boundary conditions after a range‐dependent change of the depth variable that makes B horizontal. We discretize the continuous problems by second‐order accurate Crank‐Nicolson type finite difference schemes, and show, by means of numerical experiments, that the new models yield accurate simulations of the acoustic field in standard, wedge‐type domains with upsloping and downsloping bottoms.