O Wave-Wave Interactions on the Ocean Surface.

Part 1: Bragg scattering of sound from a line source by surface waves. The strong scattering of plane sound waves from a line source in a waveguide is analysed for a hard bottom and a soft undulatory surface representing one or two uniform surface waves. In the presence of one surface wave, sound is radiated and scattered in the same two directions. When two surface waves are present, each outgoing sound wave is scattered in two new directions in the horizontal plane. If the two surface waves satisfy one of two geometrical criteria, multiple scattering occurs and sound is resonantly scattered in a third new direction. Evolution equations are deduced for the sound wave envelopes, and the corresponding far field is asymptotically matched with the near field of the source. These envelopes are strongly dispersive and propagate in many directions. Part 2: Interactions of short and long waves on the sea surface. The evolution of a weakly nonlinear short wave interacting with a long gravity wave is investigated using a formulation based on Lagrangian variables. In Chapter I, the long wave is a weakly nonlinear irrotational Stokes wave. In Chapter II, we allow the long wave to assume a finite amplitude. Analytical results on the modulation of short waves agree fairly well with existing theories which rely on numerically obtained long waves. The evolution of short waves is described by a nonlinear Schrodinger equation with explicit time-periodic coefficients. Analysis of the stability of uniform short waves to sideband disturbances shows the appearance of additional bands of instability. Numerical results from both the nonlinear evolution and a lower order dynamical system suggest that the evolution of a uniform short wave disturbed by its most unstable sideband can become chaotic when the short wave slope increases and (or) when the long to short wave frequency ratio decreases. In Chapter III, the short waves are incident at an angle theta relative to the steep Gerstner wave. For small theta, the nonlinear evolution of the short wave envelope is described by a two-dimensional Schrodinger equation with explicit time-periodic coefficients. The Benjamin-Feir stability of uniform short waves to two-dimensional sidebands shows the proliferation of instability bands in addition to the single instability strip found in the absence of long waves. (Copies available exclusively from MIT Libraries, Rm. 14-0551, Cambridge, MA 02139-4307. Ph. 617-253-5668; Fax 617-253 -1690.) (Abstract shortened with permission of school.).