Internal gravity waves, boundary integral equations and radiation conditions

Three-dimensional time-harmonic internal gravity waves are generated by oscillating a bounded object (or by scattering from a fixed object) in a stratified fluid. Energy is found in conical wave beams: the problem is to calculate the wave fields for an object of arbitrary shape. An integral formula for the pressure is derived, using a reciprocal theorem and a Green’s function. The boundary integrals are singular: their integrands are infinite along a certain curve (not just at a point) on the boundary, and this happens even when the field point is off the boundary (but within one of the conical wave beams). This is very different to the situation with classical potential theory (Laplace’s equation) or linear acoustics (Helmholtz’s equation), and is a consequence of the hyperbolic nature of the governing partial differential equation. The boundary integrals are identified as single-layer and double-layer potentials. A method is given for calculating the far field of these potentials. It is verified by comparing with known solutions for spherical objects. © 2012 Elsevier B.V. All rights reserved.