The Department of Applied Mathematics is pleased to host this series of colloquium lectures, funded in part by a generous gift from the Boeing Company. This series will bring to campus prominent applied mathematicians from around the world.
Title: Understanding nonlinear surface water waves on deep water
Abstract: Oceanographers in the 60s conducted an ambitious experiment (1) in which they tracked waves that were generated by large storms near New Zealand across the Pacific Ocean until they hit the beaches at Alaska. Paradoxically, at about the same time, mathematicians in the Soviet Union, the U.S., and England (2) independently developed mathematical models that predicted such waves to be unstable, meaning that they could not survive to be tracked all the way across the Pacific. In the 70s experimentalists (3) conducted laboratory experiments on these types of waves. They generated waves with a given frequency that propagated down a wavetank, but at the end of the wavetank, the waves had a slightly lower frequency. The mathematical model did not explain this observation. In this talk, we consider these observations and our experiments on modulated wavetrains within the framework of the mathematical models: the scalar and vector nonlinear Schroedinger equations with and without dissipation and/or higher order terms. We examine the data within the context of conserved quantities of these equations to determine when the models are likely to be valid or not. We present recent results from our quest, motivated by recent stability analyses of Stokes waves (4), to observe subharmonic instabilities of waves in deep, finite and shallow water.
- 1) Snodgrass, F. E., G. W. Groves, K. F. Hasselmann, G. R. Miller, W. H. Munk, and W. H. Powers (1966), Propagation of ocean swell across the Pacific, Philos. Trans. R. Soc. London A, 259, 431–497.
- (2a) Benney, D. J. & Newell, A. C. 1967 The propagation of nonlinear wave envelopes. Stud. Appl.Maths 46, 133–139.
- (2b) Lighthill, M. J. 1965 Contribution to the theory of waves in nonlinear dispersive systems. J. Inst. Math. Applics. 1, 269–306.
- (2c) Benjamin, T. B. & Feir, J. E. 1967 The disintegration of wavetrains in deep water. Part 1. J. Fluid Mech. 27, 417–430.
- (2d) Ostrovsky, L. A. 1967 Propagation of wave packets and space-time self-focussing in a nonlinearmedium. Sov. Phys. J. Exp. Theor. Phys. 24, 797–800.
- (2e) Whitham, G. B. 1967 Nonlinear dispersion of water waves. J. Fluid Mech. 27, 399–412.
- (2f) Zakharov, V. E. 1967 Instability of self-focusing of light. Sov. Phys. J. Exp. Theor. Phys. 24, 455–459.
- (2g) Zakharov, V. E. 1968 Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194.
- (3a) Lake, B. M. & Yuen, H. C. 1977 A note on some water-wave experiments and the comparison ofdata with theory. J. Fluid Mech. 83, 75–81.
- (3b) Lake, B. M., Yuen, H. C., Rungaldier, H. & Ferguson, W. E. 1977 Nonlinear deep-water waves:theory and experiment. Part 2. Evolution of a continuous wave train. J. Fluid Mech. 83, 49–74.
- (4) B. Deconinck, S. A. Dyachenko, P. M. Lushnikov, A. Semenova, The instability of near-extreme Stokes waves, Proceedings of the National Academy of Sciences, 120(32):p.e2308935120 (2023)
Youtube: At the request of Dr. Henderson, no video was recorded.