2.1.1 Point Source Response in a Constant Depth, Homogeneous Ocean

In a homogeneous ocean of constant depth, the response to the point source time-harmonic acoustic wave (Green's function) satisfies the nonhomogeneous equation

Here, the source is located at ( x ₀, 0) in a cylindrical coordinate system, where ? ₃ is the 3D Laplace operator.

Assuming the ocean to be of constant depth h, the surface conditions are

z = 0 is referred to as a pressure-release boundary, and z = h is a totally reflecting boundary. Using the method of separation of variables with the boundary conditions (2.2), we may represent the Green's function as

where H ⁽¹⁾ ₀ and H ⁽²⁾ ₀ are Hankel functions of order zero of the first and second kind, respectively. Since we restrict our attention to outgoing waves, the appropriate form of the radiation condition is

Here the coefficients a _n are the eigenvalues of the separated modal solutions ? _n, i.e.,

Throughout this book, we shall refer to this condition (and some of its variations) as the outgoing radiation condition, and the corresponding Green's function shall be referred to as the outgoing Green's function. The outgoing Green's function has several equivalent representations [6], including the normal mode representation

and the...