Past Event: Oden Institute Seminar

Standing-Traveling Water Waves: Stability, Singularity Formation, and Microseisms - NOTE TIME

John Wilkening, Mathematics Department, Berkeley

Abstract

We develop an overdetermined shooting algorithm to compute new families of time-periodic and quasi-periodic solutions of the free-surface Euler equations involving standing-traveling waves and collisions of solitary waves of various types. The wave amplitudes are too large to be well-approximated by weakly nonlinear theory, yet we observe soliton-like behavior. A Floquet analysis shows that many of the new solutions are stable to harmonic perturbations. Evolving such perturbations over tens of thousands of cycles suggests that the solutions remain nearly time-periodic forever. We also discuss resonance and re-visit a long-standing conjecture of Penney and Price that the standing water wave of greatest height should form wave crests with sharp, 90 degree interior corner angles. We conclude with a geophysical application in which nearly-coherent standing waves at the ocean surface can lead to rapidly-moving pressure zones at the sea floor. These pressure zones can generate resonant elastic waves believed to be partially responsible for microseisms, the background noise observed in earthquake seismographs.