Parabolic approximation of nonlocal boundary conditions for elastic seabeds

Abstract

A parabolic approximation of the nonlocal boundary condition is constructed to model the elastic seabed of an ocean acoustic waveguide. The construction departs from the Neumann to Dirichlet (NtD) map that transfers the seabed dynamics at the interface of the coupled acousto-elastic system. The main tool is an appropriate asymptotic decomposition of the kernel of the NtD map, which reveals the outgoing and incoming compo nents of the trace of the wavefleld on the interface far from the sources. This decomposition, in combination with the narrow-angle parabolic approximation of the field in the interior of the waveguide, leads to a Volterra-type nonlocal boundary condition for the parabolic equation (impedance bound ary condition). This boundary condition is radically different from the con dition derived by assuming from the beginning the narrow-angle parabolic approximation of the wave field in the elastic seabed.

Keywords:

• Parabolic equations
• nonlocal boundary conditions
• Neumann to Dirichlet map
• ocean acoustics

AMS:

• 73D10
• 35S99