INTERNAL GRAVITY WAVES AND HYPERBOLIC BOUNDARY-VALUE PROBLEMS † P. A. Martin ‡,∗ , Stefan G. Llewellyn Smith †† ‡ Department of Mathematical and Computer Sciences, Colorado School of Mines, Golden, USA †† Department of Mechanical and Aerospace Engineering, UCSD, La Jolla, USA ∗ Email: pamartin@mines.edu Talk Abstract Three-dimensional time-harmonic internal gravity waves are generated by oscillating a bounded object in an unbounded stratified fluid. Energy is found in conical wave beams. The problem is to calculate the wave fields for an object of arbitrary shape. It can be formulated as a hyperbolic boundary-value problem. The following as- pects are discussed: reduction to boundary integral equa- tions; single-layer and double-layer potentials; estimation of far fields and radiation conditions. The problem is complicated because the group and phase velocities are orthogonal. In addition, singular boundary integrals arise: 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). Introduction Boundary-value problems (BVPs) for hyperbolic par- tial differential equations (PDEs) are unfamiliar to most mathematicians. However, they do arise in applications, as we shall see later. To see that they do differ from el- liptic BVPs, start with interior Dirichlet problems for a function u in a disc, r<a, with u =0 at r = a. For Laplace’s equation, u xx + u yy =0, the only solution is u ≡ 0, whereas for the “wave” equation, u xx − u yy =0, there is a simple non-trivial solution, namely u = a 2 − r 2 ; this example was noted by Bateman [1, p. 611] in 1929. Studies of u xx = u yy in rectangles were made by Bour- gin and Duffin [2] in 1939 and John [3] in 1941. Since then, the pure mathematical literature is sparse. In applications, interior hyperbolic BVPs arise with certain models of granular flow [4] and with internal waves [5]. We shall also consider internal waves, but our interest is with exterior problems. Governing equations Consider an inviscid unbounded fluid with a uniform density stratification, under gravity. There is a bounded 3D object (with boundary S ) in the fluid. Time-harmonic † Appeared as Proc. 10th International Conference on Mathematical and Numerical Aspects of Waves, Vancouver, Canada, 2011, 375–378. S a x, y z I II III IV III II V VI V Figure 1: The oscillating body S is located inside the sphere, S a . Regions III and V are the conical wave beams bounded by characteristic cones. The angle between the conical surfaces and the z -axis is θ c . internal waves are generated by oscillating S . It is found that the significant wave motion is confined to beams forming a “Saint Andrew’s cross” (in 2D), as shown in famous images obtained by Mowbray and Rarity [6]; in 3D, the beams are conical (see Figure 1). Under the Boussinesq approximation, the wave mot- ion can be found by calculating the pressure Re{pe −iωt }; p(x,y,z ) solves ∂ 2 p ∂x 2 + ∂ 2 p ∂y 2 +Υ ∂ 2 p ∂z 2 =0, where Υ= ω 2 ω 2 − N 2 (1) is a constant, z is the vertical coordinate and N is the constant Brunt–V¨ ais¨ al¨ a frequency. The two frequencies, ω and N , satisfy 0 <ω<N , so that Υ < 0 and the PDE (1) is hyperbolic. It is to be solved subject to boundary and far-field conditions. The boundary condition is natu- ral: v · n is prescribed on S , where n is a normal to S and the velocity v =(u,v,w) is given in terms of p by u = − i ω ∂p ∂x ,v = − i ω ∂p ∂y ,w = − iΥ ω ∂p ∂z . (2) The far-field conditions are discussed next.