An integral equation approach to range‐dependent boundary value problems in ocean acoustics

Abstract

In ocean acoustics, the introduction of range discontinuities, for example, the water‐to‐ice canopy surface, creates a mixed boundary value problem. In this paper, an exact solution of certain mixed boundary value problems is discussed using the Wiener‐Hopf method. A key attribute of this approach is that it is not fundamentally numerical in nature and allows additional insight into the mathematical and physical structure of the acoustic field due to range discontinuities. The problem discussed here is a canonical one: A plane wave is incident upon a planar surface where the boundary condition changes from Dirichlet (free surface) to Neumann (hard). The boundary conditions addressed in this problem are highly idealized renditions of what happens, when, say, a plane wave is incident upon a water‐to‐ice canopy surface; nevertheless, important features of the diffraction process are produced here, and the solution gives considerable insight into the process. The solution of the diffracted potential is partitioned into two components: a field consisting of cylindrical waves weighted by a polar gain function, resulting from the artificial source created by the discontinuity in the boundary; and a field containing a residue contribution which restores field continuity along the line corresponding to specular reflection. Contour plots of equal pressure amplitude show how the component fields superimpose such that the boundary conditions are maintained, and how energy is redistributed across the angular spectrum in the diffraction process. The latter is related to mode coupling due to boundary changes in waveguide problems. [Work supported by ONR.]