Exact steady periodic water waves with vorticity

We consider the classical water wave problem described by the Euler equations with a free surface under the influence of gravity over a flat bottom. We construct two-dimensional inviscid periodic traveling waves with vorticity. They are symmetric waves whose profiles are monotone between each crest and trough. We use bifurcation and degree theory to construct a global connected set of such solutions. (C) 2004 Wiley Periodicals, Inc.

[1]  G. Keady,et al.  On the existence theory for irrotational water waves , 1978, Mathematical Proceedings of the Cambridge Philosophical Society.

[2]  J. Vanden-Broeck,et al.  Gravity-capillary waves in the presence of constant vorticity , 2000 .

[3]  H. Kielhöfer.  Multiple eigenvalue bifurcation for Fredholm operators. , 1985 .

[4]  C. Swan,et al.  An experimental study of two-dimensional surface water waves propagating on depth-varying currents. Part 1. Regular waves , 2001, Journal of Fluid Mechanics.

[5]  S. Agmon.  On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems , 1962 .

[6]  D. J. Struik.  Détermination rigoureuse des ondes irrotationelles périodiques dans un canal à profondeur finie , 1926 .

[7]  M. Dubreil-Jacotin,et al.  Sur la détermination rigoureuse des ondes permanentes périodiques d'ampleur finie. , 1934 .

[8]  W. Strauss,et al.  Exact periodic traveling water waves with vorticity , 2002 .

[9]  James Lighthill,et al.  Waves In Fluids , 1966 .

[10]  R. S. Johnson,et al.  A Modern Introduction to the Mathematical Theory of Water Waves: Bibliography , 1997 .

[11]  J. Toland,et al.  On the stokes conjecture for the wave of extreme form , 1982 .

[12]  Walter Craig,et al.  Traveling Two and Three Dimensional Capillary Gravity Water Waves , 2000, SIAM J. Math. Anal..

[13]  T. Healey,et al.  Symmetry and nodal properties in the global bifurcation analysis of quasi-linear elliptic equations , 1991 .

[14]  T. Benjamin.  The solitary wave on a stream with an arbitrary distribution of vorticity , 1962, Journal of Fluid Mechanics.

[15]  R. E. Baddour,et al.  The rotational flow of finite amplitude periodic water waves on shear currents , 1998 .

[16]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[17]  Walter Craig,et al.  Traveling gravity water waves in two and three dimensions , 2002 .

[18]  M. Crandall,et al.  Bifurcation from simple eigenvalues , 1971 .

[19]  R. Gouyon,et al.  Contributions à la théorie des houles , 1958 .

[20]  Hsien-Kuo Chang,et al.  Highly nonlinear interactions between a gravity wave and a current of constant vorticity , 1995 .

[21]  E. Cokelet,et al.  SLAR and In-Situ Observations of Wave-Current Interaction on the Columbia River Bar , 1985 .

[22]  Gary M. Lieberman,et al.  Nonlinear oblique boundary value problems for nonlinear elliptic equations , 1986 .

[23]  F. Gerstner.  Theorie der Wellen , 1809 .

[24]  J. Toland.  Errata to "Stokes waves" , 1996 .

[25]  D. H. Peregrine,et al.  Steep, steady surface waves on water of finite depth with constant vorticity , 1988, Journal of Fluid Mechanics.

[26]  R. Simons,et al.  The interaction between waves and a turbulent current: waves propagating with the current , 1982, Journal of Fluid Mechanics.

[27]  G. P. Thomas,et al.  Wave–current interactions: an experimental and numerical study. Part 2. Nonlinear waves , 1981, Journal of Fluid Mechanics.

[28]  Paul H. Rabinowitz,et al.  Some global results for nonlinear eigenvalue problems , 1971 .

[29]  Adrian Constantin,et al.  Wave-current interactions , 2005 .

[30]  Adrian Constantin,et al.  On the deep water wave motion , 2001 .

[31]  Frédéric Dias,et al.  NONLINEAR GRAVITY AND CAPILLARY-GRAVITY WAVES , 1999 .

[32]  James Serrin,et al.  A symmetry problem in potential theory , 1971 .

[33]  Yu. P. Krasovskii.  On the theory of steady-state waves of finite amplitude☆ , 1962 .

[34]  E. Zeidler.  Existenzbeweis für permanente kapillar-schwerewellen mit allgemeinen wirbelverteilungen , 1973 .

[35]  H. Simpson,et al.  Global Continuation in Nonlinear Elasticity , 1998 .

[36]  J. Toland,et al.  On periodic water-waves and their convergence to solitary waves in the long-wave limit , 1981, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences.

[37]  Rodney J. Sobey,et al.  Stokes theory for waves on linear shear current , 1988 .

[38]  Hisashi Okamoto,et al.  The mathematical theory of permanent progressive water-waves , 2001 .

[39]  G. Crapper.  Introduction to Water Waves , 1984 .

[40]  J. Vanden-Broeck.  Periodic waves with constant vorticity in water of infinite depth , 1996 .

[41]  J. Toland.  On the existence of a wave of greatest height and Stokes’s conjecture , 1978, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences.