Quasi-periodic Water Waves and Their Stability

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Understanding the complex dynamics of water waves is essential in many engineering applications, including power generation from ocean waves, early detection of tsunamis, and safeguarding coastal power plants. While pure traveling waves and pure standing waves have been studied extensively in the past, real ocean waves generally contain more than two dominant frequencies and wavelengths. This project will develop mathematical and numerical techniques for studying quasi-periodic water waves. Traveling and standing waves are special cases with one and two quasi-periods, respectively. The stability of these waves will also be investigated, including stability transitions. Of course, only stable solutions will be seen in the ocean or laboratory, but unstable solutions can be computed numerically, often completing the picture of how the stable solutions fit together. These methods will also be used to compute microseisms, which are geophysical elastic waves in the sea bed generated by nearly coherent standing waves at the ocean surface, and Faraday waves, which are surface waves on water or oil in a wave tank that self-organize into various standing wave patterns when the container is driven to oscillate with a prescribed motion. The numerical results of this project will be compared with wave tank experiments done by Diane Henderson's group at Penn State in the William G. Pritchard Fluid Mechanics Laboratory. Additional broader impacts include course and curriculum development, organization of seminars and minisymposia, advising of graduate students, and development of new computational tools with many applications beyond water waves.The first technical goal of the project is to devise and implement a generalized shooting method for computing quasi-periodic solutions of differential equations. Several new types of solutions of the free-surface Euler equations are expected to be found, including traveling-standing waves, KdV-like elastic collisions, and NLS-like breathers. Harmonic and subharmonic stability of standing waves and other relative periodic solutions will also be determined, with the goal of studying orbital stability, long-time dynamics, and quasi-periodic perturbations of relative-periodic solutions. Subharmonic stability will be investigated using Bloch theory in space and Floquet theory in time, with solutions of the linearized Euler equations computed in parallel batches to compute the monodromy operator. Standing-wave analogues of the Benjamin-Feir instability will also be studied. A new approach to computing cyclic steady states and stability transitions of parametrically driven Faraday wave systems will also be developed, along with a new algorithm for computing the Dirichlet-Neumann operator in a cylindrical geometry using tensor products of orthogonal polynomials tailored to the cylindrical geometry and a variant of the Transformed Field Expansion technique. The Arnoldi algorithm will be used to compute the eigenvalues of the monodromy operator for the Faraday wave problem to obtain the largest Floquet multipliers. The method can track families of solutions through unstable branches to yield a more complete picture of the stability transitions that affect pattern formation. Quasi-periodic forcing of Faraday waves will also be investigated.