The generation, propagation, and energetic transport of ocean waves play a crucial role in many technologies including near-shore watercraft navigation, the design of deep-sea oil-drilling rigs, and the generation and propagation of tsunamis. However, the fundamental technologies for the remote detection and measurement of these nonlinear waves are based upon the scattering of linear acoustic or electromagnetic radiation from the surface or bulk of the fluid. Not surprisingly, the state-of-the-art in theory, approximation, and numerical simulation of these two disparate wave phenomena is quite different. In this proposal the Principal Investigator (PI) advocates the development of a unified computational approach to these two problems 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 of water. Over the past ten years the PI has developed a set of stable, high-accuracy Boundary Perturbation (BP) algorithms for the study of both traveling ocean waves and scattering returns from irregularly shaped obstacles. These BP algorithms each involve an independent perturbation parameter and the PI proposes to investigate a strategy of coupling these two disparate algorithms by linking these parameters, thus realizing huge computational savings. Additionally, the PI proposes to significantly extend and improve existing BP algorithms for the simulation of linear waves in the high-frequency regime. In short, the Geometric Optics solution points the way to a solution Ansatz which can be represented with a frequency-independent number of unknowns. The PI, in collaboration with F. Reitich, has demonstrated how this Ansatz coupled to classical BP methods can be used to compute high-frequency scattering returns from shallow surfaces with enormous computational savings. In this proposal the PI advocates the further development of these algorithms to incorporate the scenario of multiple reflections.
The generation and propagation properties of ocean waves are crucial in many technologies including watercraft navigation, design of deep-sea oil-drilling rigs, and the study of tsunamis. However, the methods for the measurement of these nonlinear ocean waves are based upon the scattering of linear acoustic or electromagnetic radiation. The state-of-the-art in both the theory and practice of these two disparate wave phenomena is quite different. In this proposal the Principal Investigator (PI) advocates the development of a unified approach to the computation of these two problems with the goal of producing a set of reliable and accurate numerical algorithms for the detection and simulation of ocean waves. This will involve the PI bringing together two separate threads of his research program to implement this unified approach. Additionally, he will need to expand the capabilities of his algorithms to account for the highly oscillatory character of the linear waves which arise in these applications. The latter task will be particularly challenging and is the current focus of a great deal of research by both civilian and military agencies.
