3.2.1 Acoustic Wave in a Wave Guide with an Obstacle

In this section we consider the acoustic imaging problem in a homogeneous, shallow-water wave guide. The shallow-water wave guide is denoted by , where h is the depth of the ocean. Let ? be a bounded obstacle imbedded in the wave guide. The total acoustic field u from a point source x _s = ( x ^s ₁, x ^s ₂) satisfies

u also satisfies the outgoing radiation condition

where x ^s ₁ is the location of the acoustic source, k > 0 is the wavenumber, and u _n is the nth normal propagation mode; i.e., if

then u has representation

On the boundary of ?, denoted by ??, u satisfies some unknown boundary condition of the Dirichlet, Neumann, and impedence types. We denote the boundary conditions as

We consider the total wave u as the combination of a prime field u ⁱ and a scattered field u ^s, u = u ⁱ + u ^s where u ⁱ is the Green's function for the parallel wave guide without any obstacle:

where

From (3.44) (3.47), we have

On ??, u ^s satisfies some unspecified boundary condition

where (3.55) may be data of the following types, namely Dirichlet:

Neumann:

Robin (impedence):

where ? > 0. We assume that ??