A close one-term approximation to the highest Stokes wave on deep water

A remarkably accurate approximation to the profile of a limiting progressive gravity wave in water of infinite depth is given by the expression (0.1) y / L = A cosh ( x / L ) , where L is the wavelength, x and y are horizontal and vertical coordinates and A is a constant given by (0.2) A = 1 / ( 3 s...

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Bibliographic Details
Published in:Ocean engineering 2006-10, Vol.33 (14), p.2012-2024
Main Authors: Rainey, R.C.T., Longuet-Higgins, Michael S.
Format: Article
Language:English
Subjects:
Approximation
Dynamics of the ocean (upper and deep oceans)
Earth, ocean, space
Exact sciences and technology
External geophysics
Highest wave
Marine
Physics of the oceans
Stokes wave
Waves in deep water
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Summary:A remarkably accurate approximation to the profile of a limiting progressive gravity wave in water of infinite depth is given by the expression (0.1) y / L = A cosh ( x / L ) , where L is the wavelength, x and y are horizontal and vertical coordinates and A is a constant given by (0.2) A = 1 / ( 3 sinh 1 2 ) = 1.1080 . This determines the wave steepness ( H / L ) as 0.14140 a proportional error of less than 0.3% (about 10 times closer than previous approximations) and the phase speed c / ( gL ) 1 / 2 as 0.43511, which is accurate to within 0.2%. The entire surface profile is accurate to less than 0.7%. The corresponding particle velocities are found by a straightforward numerical integration. It is shown that this type of approximation cannot be made exact by the introduction of further parameters.
ISSN:0029-8018
1873-5258
DOI:10.1016/j.oceaneng.2005.09.014