Abstract
Our overall objective is to find mathematical models that describe accurately how waves in nature propagate and evolve. One process that affects evolution is dissipation (Segur et al., J Fluid Mech 539:229–271, 2005), so in this paper we explore several models in the literature that incorporate various dissipative physical mechanisms. In particular, we seek theoretical models that (1) agree with measured dissipation rates in laboratory and field experiments, and (2) have the mathematical properties required to be of use in weakly nonlinear models of the evolution of waves with narrow-banded spectra, as they propagate over long distances on deep water.
We dedicate this paper to our friend and colleague, Walter Craig.
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References
Benjamin, B., Feir, F.: The disintegration of wavetrains in deep water. Part 1. J. Fluid Mech. 27, 417–430 (1967)
Benney, D., Newell, A.: The propagation of nonlinear wave envelopes. J. Math. Phys. Stud. Appl. Math. 46, 133–139 (1967)
Bonnet, J.-P., Devesvre, L., Artaud, J., Moulin, P.: Dynamic viscosity of olive oil as a function of composition and temperature: a first approach. Eur. J. Lipid Sci. Technol. 113, 1019–1025 (2011)
Collard, F., Ardhuin, F., Chapron, B: Monitoring and analysis of ocean swell fields from space: new methods for routine observations. J. Geophys. Res. 114, C07023 (2009)
Davies, J.T., Vose, R.M.: On the damping of capillary waves by surface films. Proc. Roy. Soc. Lond. A286, 218 (1965)
Dias, F., Dyachenko, A., Zakharov, V.: Theory of weakly damped free-surface flows: a new formulation based on potential flow solutions. Phys. Lett. A372, 1297–1302 (2008)
Dore, B.: Some effects of the air-water interface on gravity waves. Geophys. Astophys. Fluid Dyn. 10, 213–230 (1978)
Dysthe, K.: Note on a modification to the nonlinear Schrödinger equation for application to deep water waves. Proc. Roy. Soc. Lond. A369, 105–114 (1979)
Gramstad, O., Trulsen, K.: Fourth-order coupled nonlinear Schrödinger equations for gravity waves on deep water. Phys. Fluids. 23, 062102–062111 (2011)
Henderson, D., Segur, H.: The role of dissipation in the evolution of ocean swell. J. Geo. Res: Oceans 118, 5074–5091 (2013)
Henderson, D., Segur, H., Carter, J.: Experimental evidence of stable wave patterns on deep water. J. Fluid Mech. 658, 247–278 (2010)
Henderson, D.M.: Effects of surfactants on Faraday-wave dynamics. J. Fluid Mech. 365, 89–107 (1998)
Huhnerfuss, H., Lange, P.A., Walters, W.: Relaxation effects in monolayers and their contribution to water wave damping. II. The Marangoni phenomenon and gravity wave attenuation. J. Coll. Int. Sci. 108, 442–450 (1985)
Jenkins, A.D., Jacobs, S.J.: Wave damping by a thin layer of viscous fluid. Phys. Fluids 9, 1256–1264 (1997)
Lamb, H.: Hydrodynamics, 6th edn. Dover, New York (1932)
Lo, E., Mei, C.C.: A numerical study of water-wave modulations based on a higher-order nonlinear Schrödinger equation. J. Fluid Mech. 150, 395–415 (1985)
Ma, Y., Dong, G., Perlin, M., Ma, X., Wang, G.: Experimental investigation on the evolution of the modulation instability with dissipation. J. Fluid Mech. 711, 101–121 (2012)
Miles, J.: Surface-wave damping in closed basins. Proc. Roy. Soc. Lond. A297, 459–475 (1967)
Ostrovsky, L.: Propagation of wave packets and space-time self-focussing in a nonlinear medium. Sov. J. Exp. Theor. Phys. 24, 797–800 (1967)
Segur, H., Henderson, D., Carter, J., Hammack, J., Li, C.-M., Pheiff, D., Socha, K.: Stabilizing the Benjamin-Feir instability. J. Fluid Mech. 539, 229–271 (2005)
Stokes, G.G.: On the theory of oscillatory waves. Trans. Camb. Phil. Soc 8, 441 (1847)
Snodgrass, F.E., Groves, G.W., Hasselmann, K., Miller, G.R., Munk, W.H., Powers, W.H.: Propagation of ocean swell across the Pacific. Phil. Trans. Roy. Soc. Lond. A259, 431–497 (1966)
Trulsen, K., Dysthe, K.: A modified nonlinear Schrödinger equation for broader bandwidth gravity waves on deep water. Wave Motion 24, 281–289 (1996)
van Dorn, W.G.: Boundary dissipation of oscillatory waves. J. Fluid Mech. 24, 769–779 (1966)
Wu, G., Liu, Y., Yue, D.: A note on stabilizing the Benjamin-Feir instability. J. Fluid Mech. 556, 45–54 (2006)
Zakharov, V.: The wave stability in nonlinear media. Sov. J. Exp. Theor. Phys. 24, 455–459 (1967)
Zakharov, V.: Stability of periodic waves of finite amplitude on the surface of a deep fluid. J. Appl. Mech. Tech. Phys. 2, 190–194 (1968)
Acknowledgements
This work was supported in part by the National Science Foundation, DMS-1107379 and DMS-1107354.
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Henderson, D., Rajan, G.K., Segur, H. (2015). Dissipation of Narrow-Banded Surface Water Waves. In: Guyenne, P., Nicholls, D., Sulem, C. (eds) Hamiltonian Partial Differential Equations and Applications. Fields Institute Communications, vol 75. Springer, New York, NY. https://doi.org/10.1007/978-1-4939-2950-4_6
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