Standing-Traveling Water Waves: Stability, Singularity Formation,
Jon Wilkening (Dept. of Mathematics, UC Berkeley)
We develop an overdetermined shooting algorithm to compute new families of time-periodic and quasi-periodic solutions of the free-surface Euler equations involving extreme standing waves and collisions of traveling waves of various types. A Floquet analysis shows that many of the new solutions are linearly stable to harmonic perturbations. Evolving such perturbations (nonlinearly) 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.