# American Institute of Mathematical Sciences

August  2014, 34(8): 3135-3153. doi: 10.3934/dcds.2014.34.3135

## A boundary integral formulation for particle trajectories in Stokes waves

 1 Instituto Nacional de Matemática Pura e Aplicada/IMPA, Est. D. Castorina, 110, J. Botânico, Rio de Janeiro, RJ 22460-320, Brazil

Received  July 2013 Revised  September 2013 Published  January 2014

Recently important theorems have been established presenting qualitative results for particle trajectories below a Stokes wave. A diversity of orbit patterns were described, including the case of a closed orbit when a Stokes wave propagates in the presence of an adverse current. In this work these results are revisited in a quantitative fashion through a boundary integral formulation which leads to very accurate numerical simulations of particle trajectories. The boundary integral formulation allows the accurate evaluation of the vector field of the (particle's) dynamical system, without resorting to a series expansion and a small parameter. Accurate trajectories are benchmarked against well known expansions for weakly nonlinear waves. Simulations are then performed beyond this regime. Closed orbits are found in the presence of an adverse current, as well as non-smooth trajectories that have not been reported. These occur for both adverse and favorable currents.
Citation: André Nachbin, Roberto Ribeiro-Junior. A boundary integral formulation for particle trajectories in Stokes waves. Discrete & Continuous Dynamical Systems, 2014, 34 (8) : 3135-3153. doi: 10.3934/dcds.2014.34.3135
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##### References:
 [1] D. P. Bertsekas, Nonlinear Programming, $2^{nd}$ edition, Athena Scientific, Belmont, Massachusetts, 1999. doi: 10.1038/sj.jors.2600425.  Google Scholar [2] H.-K. Chang, Y.-Y. Chen and J.-C. Liou, Particle trajectories of nonlinear gravity waves in deep water, Ocean Engineering, 36 (2009), 324-329. doi: 10.1016/j.oceaneng.2008.12.007.  Google Scholar [3] A. Constantin, The trajectories of particles in Stokes waves, Invent. Math., 166 (2006), 523-535. doi: 10.1007/s00222-006-0002-5.  Google Scholar [4] A. Constantin, Nonlinear Water Waves with Applications to Wave-Current Interactions and Tsunamis, SIAM, 2011. doi: 10.1137/1.9781611971873.  Google Scholar [5] A. Constantin, Mean velocities in a Stokes wave, Arch. Ration. Mech. Anal., 207 (2013), 907-917. doi: 10.1007/s00205-012-0584-6.  Google Scholar [6] A. Constantin, M. Ehrnström and E. Wahlén, Symmetry of steady periodic gravity water waves with vorticity, Duke Math. J., 140 (2007), 591-603. doi: 10.1215/S0012-7094-07-14034-1.  Google Scholar [7] A. Constantin, M. Ehrnström and E. Wahlén, Particle trajectories in linear deep-water waves, Nonlinear Anal. Real World Appl., 9 (2008), 1-18. doi: 10.1016/j.nonrwa.2007.03.003.  Google Scholar [8] A. Constantin and W. A. Strauss, Pressure beneath a Stokes wave, Comm. Pure Appl. Math., 63 (2010), 533-557. doi: 10.1002/cpa.20299.  Google Scholar [9] A. Constantin and G. Villari, Particle trajectories in linear water waves, J. Math. Fulid Mech., 10 (2008), 1336-1344. doi: 10.1007/s00021-005-0214-2.  Google Scholar [10] M. W. Dingemans, Water Waves Propagation Over Uneven Bottoms, World Scientific, Singapore, 1997. doi: 10.1142/1241-part1.  Google Scholar [11] J. D. Fenton, A fifth-order Stokes theory for steady waves, Journal of Waterway, Port, Coastal and Ocean Engineering, 111 (1985), 216-234. doi: 10.1061/(ASCE)0733-950X(1985)111:2(216).  Google Scholar [12] J. D. Fenton, Nonlinear wave theories, The Sea: Ocean Engineering Science, 9 (1990), 1-18. Google Scholar [13] P. Guidotti, A new first-kind boundary integral formulation for the Dirichlet-to-Neumman map in 2D, J. Comput. Phy., 190 (2008), 325-345. doi: 10.1016/S0021-9991(03)00277-8.  Google Scholar [14] D. Henry, On the deep-water Stokes wave flow, IMRN, 2008 (2008), 1-7. doi: 10.1093/imrn/rnn071.  Google Scholar [15] M. Isobe, H. Nishimura and K. Horikawa, Expressions of Pertubation Solutions for Conservative Waves by Using Wave Height, Proceedings of 33rd annual conference of JSCE 1978, 760-761. Google Scholar [16] F. John, Partial Differential Equations, $4^{th}$ edition, Springer-Verlag, New York, 1982.  Google Scholar [17] I. G. Jonsson and L. Arneborg, Energy properties and shoaling of higher-order stokes waves on current, Ocean Engineering, 22 (1995), 819-857. doi: 10.1016/0029-8018(95)00008-9.  Google Scholar [18] H. Lamb, Hydrodynamics, Cambridge, Univ. Press, 1895.  Google Scholar [19] M. S. Longuet-Higgins, Eulerian and Lagrangian aspects of surface waves, J. Fluid Mech., 173 (1986), 683-707. doi: 10.1017/S0022112086001325.  Google Scholar [20] , Available at:, , ().   Google Scholar [21] H. Okamoto and M. Shōji, The Mathematical Theory of Permanent Progressive Water-Waves, World Scientific, River Edge, NJ, 2001.  Google Scholar [22] H. Okamoto and M. Shōji, Trajectories of fluid particles in a periodic water wave, Phil. Trans. R. Soc. A, 370 (2012), 1661-1676. doi: 10.1098/rsta.2011.0447.  Google Scholar [23] F. Ruellan and A. Wallet, Trajectoires internes das un clapotis partiel, La Houille Blanche, 5 (1950), 483-489. Google Scholar [24] L. Skjelbreia and J. Hendrinck, Fifth Order Gravity Wave Theory, JProceedings of 7th conference on coastal engineering, ASCE 1960, 184-196. Google Scholar [25] J. J. Stoker, Water Waves. The Mathematical Theory with Applications, Intersciene Publ. Inc., New York, 1957.  Google Scholar [26] G. G. Stokes, On the theory of oscillatory waves, Trans. Cambridge Phil. Soc., 8 (1847), 441-455. doi: 10.1017/CBO9780511702242.013.  Google Scholar [27] L. N. Trefethen, Spectral Methods in MATLAB, SIAM, Philadelphia, 2001. doi: 10.1137/1.9780898719598.  Google Scholar [28] M. Umeyama, Eulerian-Lagrangian analysis for particle velocities and trajectories in a pure wave motion using particle image velocimetry, Phil. Trans. R. Soc. A, 370 (2012), 1687-1702. doi: 10.1098/rsta.2011.0450.  Google Scholar [29] F. Ursell, Mass transport in gravity waves, Proc. Cambridge Phil. Soc., 40 (1953), 145-150. doi: 10.1017/S0305004100028140.  Google Scholar [30] G. B. Whitham, Linear and Nonlinear Waves, John Wiley, New York, 1974.  Google Scholar
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