On the behavior near the crest of waves of extreme form
Authors:
C. J. Amick and L. E. Fraenkel
Journal:
Trans. Amer. Math. Soc. 299 (1987), 273298
MSC:
Primary 76B15
MathSciNet review:
869412
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Abstract: The angle which the free boundary of an extreme wave makes with the horizontal is the solution of a singular, nonlinear integral equation that does not fit (as far as we know) into the theory of compact operators on Banach spaces. It has been proved only recently that solutions exist and that (as Stokes suggested in 1880) these solutions represent waves with sharp crests of included angle . In this paper we use the integral equation, known properties of solutions and the technique of the Mellin transform to obtain the asymptotic expansion , to arbitrary order; the coordinate is related to distance from the crest as measured by the velocity potential rather than by length. The first few (and probably all) of the exponents are transcendental numbers. We are unable to evaluate the coefficients explicitly, but define some in terms of global properties of , and the others in terms of earlier coefficients. It is proved in [8] that , and follows here that . The derivation of (*) includes an assumption about a question in number theory; if that assumption should be false, logarithmic terms would enter the series at very large values of .
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 C. J. Amick, L. E. Fraenkel and J. F. Toland, On the Stokes conjecture for the wave of extreme form, Acta Math. 148 (1982), 193214. MR 666110 (83m:35147)
 [2]
 C. J. Amick and J. F. Toland, On solitary waterwaves of finite amplitude, Arch. Rational Mech. Anal., 76 (1981), 995. MR 629699 (83b:76017)
 [3]
 , On periodic waterwaves and their convergence to solitary waves in the longwave limit, Philos. Trans. Roy. Soc. London A 303 (1981), 633669. MR 647410 (83b:76009)
 [4]
 M. A. Grant, The singularity at the crest of a finite amplitude progressive Stokes wave, J. Fluid Mech., 59 (1973), 257262.
 [5]
 M. S. LonguetHiggins and M. J. H. Fox, Theory of the almosthighest wave: the inner solution, J. Fluid Mech. 80 (1977), 721742. MR 0452143 (56:10424)
 [6]
 , Theory of the almosthighest wave. Part 2. Matching and analytic extension, J. Fluid Mech. 85 (1978), 769786. MR 0502858 (58:19766)
 [7]
 J. B. McLeod, The Stokes and Krasovskii conjectures for the wave of greatest height, Math. Res. Center Tech. Summary Rep. #2041, Univ. of WisconsinMadison, 1979: Math. Proc. Cambridge Philos. Soc. (to appear). MR 1446239 (98b:76012)
 [8]
 , The asymptotic behavior near the crest of waves of extreme form, Trans. Amer. Math. Soc. 299 (1987), 299302. MR 869413 (88c:76014b)
 [9]
 A. C. Norman, Expansions for the shape of maximum amplitude Stokes waves, J. Fluid Mech. 66 (1974), 261265.
 [10]
 C. L. Siegel, Transcendental numbers, Princeton Univ. Press, Princeton, N. J., 1949. MR 0032684 (11:330c)
 [11]
 E. C. Titchmarsh, The theory of functions, Clarendon Press, Oxford, 1932.
 [12]
 , Introduction to the theory of Fourier integrals, Clarendon Press, Oxford, 1948.
 [13]
 J. F. Toland, On the existence of a wave of greatest height and Stokes's conjecture, Proc. Roy. Soc. London A 363 (1978), 469485. MR 513927 (81d:76017)
 [14]
 J. M. Williams, Limiting gravity waves in water of finite depth, Philos. Trans. Roy. Soc. London A 302 (1981), 139188. MR 633482 (83h:76017)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198708694124
PII:
S 00029947(1987)08694124
Keywords:
Water waves,
nonlinear integral equations,
asymptotic analysis
Article copyright:
© Copyright 1987 American Mathematical Society
