Various Aspects of the Water-Wave Problem

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The research outlined in this proposal addresses two separate aspects of the water-wave problem within the framework of Euler's Equations. First, the spectral stability of both two- and three-dimensional periodic traveling waves will be examined. Results will be obtained numerically and analytically using a nonlocal formulation of the water-wave problem due to Ablowitz, Fokas, and Musslimani. Second, this proposal will develop methods for the reconstruction of surface waves from pressure measurements. Building upon previous work with collaborators, this proposal will generalize the relationship between pressure and the surface elevation to allow for pressure measurements to be taken at arbitrary depths (not just the bottom). In addition, physical parameters that are difficult to physically measure a priori will be formally eliminated from the model without approximation. Asymptotic models will also be derived and compared with numerical experiments. Understanding the physical conditions necessary to generate rogue waves or tsunamis aids in predicting when these events could occur. This proposal seeks to better understand these physical conditions by investigating the mathematical models that describe surface water waves. Due to the complicated nature of these equations, techniques such as stability analysis increase our understanding of how waves behave over some period of time. This project will investigate the stability of traveling wave solutions in order to gain better insight into the physical parameters and types of perturbations that could give rise to rogue waves or tsunamis. Alone, this information is not sufficient in practical use. One must also rely on real-time measurements the current ocean state as part of any prediction or forecasting system. As part of this proposal, fast and accurate methods of reconstructing the current wave-height from pressure data will be identified and compared with numerical and field data.