Computing Strategy in the Integral Equation Solution of Limiting Gravity Waves in Water
Progressive irrotational gravity waves in water have a limiting form in which, in their steady flow representation, the crest reaches the total energy line. The crest is then angled rather than rounded, with an included angle of 12O°, implying a singulari
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COMPUTING STRATEGY IN THE INTEGRAL EQUATION SOLUTION OF LIMITING GRAVITY WAVES IN WATER J.M. Williams Hydraulics Research Station Wallingford, Oxfordshire U.K.
ABSTRACT Progressive irrotational gravity waves in water have a limiting form in which, in their steady flow representation, the crest reaches the total energy line. The crest is then angled rather than rounded, with an included angle of 120°, implying a singularity of order 2/3 in the complex potential plane. An integral equation technique, employing two leading terms to represent the crest, has been successfully used by the author to generate definitive solutions of limiting waves over the whole range of depth : wavelength ratios. The paper presents a specimen solution and discusses various aspects of computing strategy, including the choice of number and position of nodal points. Conditions under which convergence fails are also briefly considered. INTRODUCTION Inviscid irrotational water waves of infinitesimal amplitude form part of classical hydrodynamics (Lamb, 1932). The solution depends on the depth : wavelength ratio which can vary from zero (the solitary wave) to infinity (the deep-water wave). The extension to finite wave amplitudes, when the nonlinear surface condition arising from Bernoulli's equation can no longer be linearised, was begun by Stokes (1847). Stokes developed a series expansion, with successive terms g1v1ng successive Fourier components of the wave form. Third order solutions by Stokes's method have been tabulated, for example by Skjelbreia (1959),and used extensively in engineering design. More recently, and with the aid of more advanced computers, Dean (1965) has developed his so-called stream function wave theory which has also been published in tabular form (Dean, 1974).
C. A. Brebbia (ed.), Boundary Element Methods © Springer-Verlag Berlin Heidelberg 1981
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Whereas in Stokes's method the coefficients of the first, second and third order solutions are found in sequence, each from its predecessors, Dean finds the coefficients of his component terms simultaneously by minimising the root mean square error in the free-surface condition; this is, in effect, a type of integral equation technique. Dean's tabulated solutions range from 3rd to 19th order; the higher orders extend the range of useful accuracy to larger wave amplitudes compared with the Stokes 3rd order tabulations. A uniform wave motion may be brought to a steady state by superimposing a velocity opposed to the wave celerity. This steady motion has a total energy line above the surface to which the Bernoulli condition at the surface is referred. The highest possible wave is that whose crest just reaches the total energy line to give a stagnation point there. Stokes (1880) showed that the crest of this limiting wave would be no longer rounded but angled, with an included angle of 1200. It follows that waves nearing the limiting form will need a large number of Fourier components to describe the profile accurately, and consequently a high-order solution whether by S
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