Numerical Algorithms for the Detection and Simulation of Surface Water Waves

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The generation, propagation, and energetic transport of ocean wavesplay a crucial role in many technologies including near-shorewatercraft navigation, the design of deep-sea oil-drilling rigs, andthe generation and propagation of tsunamis. However, the fundamentaltechnologies for the remote detection and measurement of thesenonlinear waves are based upon the scattering of linear acoustic orelectromagnetic radiation from the surface or bulk of the fluid. Notsurprisingly, the state-of-the-art in theory, approximation, andnumerical simulation of these two disparate wave phenomena is quitedifferent. In this proposal the Principal Investigator (PI) advocatesthe development of a unified computational approach to these twoproblems with the goal of producing a set of highly accurate, stable,and error-controllable numerical algorithms for the detection,simulation, and evaluation of waves on the surface of large bodies ofwater. Over the past ten years the PI has developed a set of stable,high-accuracy Boundary Perturbation (BP) algorithms for the study ofboth traveling ocean waves and scattering returns from irregularlyshaped obstacles. These BP algorithms each involve an independentperturbation parameter and the PI proposes to investigate a strategyof coupling these two disparate algorithms by linking theseparameters, thus realizing huge computational savings. Additionally,the PI proposes to significantly extend and improve existing BPalgorithms for the simulation of linear waves in the high-frequencyregime. In short, the Geometric Optics solution points the way to asolution Ansatz which can be represented with a frequency-independentnumber of unknowns. The PI, in collaboration with F. Reitich, hasdemonstrated how this Ansatz coupled to classical BP methods can beused to compute high-frequency scattering returns from shallowsurfaces with enormous computational savings. In this proposal the PIadvocates the further development of these algorithms to incorporatethe scenario of multiple reflections.The generation and propagation properties of ocean waves are crucialin many technologies including watercraft navigation, design ofdeep-sea oil-drilling rigs, and the study of tsunamis. However, themethods for the measurement of these nonlinear ocean waves are basedupon the scattering of linear acoustic or electromagnetic radiation.The state-of-the-art in both the theory and practice of these twodisparate wave phenomena is quite different. In this proposal thePrincipal Investigator (PI) advocates the development of a unifiedapproach to the computation of these two problems with the goal ofproducing a set of reliable and accurate numerical algorithms for thedetection and simulation of ocean waves. This will involve the PIbringing together two separate threads of his research program toimplement this unified approach. Additionally, he will need to expandthe capabilities of his algorithms to account for the highlyoscillatory character of the linear waves which arise in theseapplications. The latter task will be particularly challenging and isthe current focus of a great deal of research by both civilian andmilitary agencies.