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On solitary *** 2 A unified perturbation theory for higher-order waves

Acta Mechanica Sinica 2005年第6期21卷 515-530页

作者： Theodore Yaotsu Wu Xinlong Wang Wendong Qu California Institute of Technology Pasadena CA 91125 U.S.A. Fellow Institute of Mechanics Chinese Academy of Sciences Beijing 100080 China State Key Laboratory of Modern Acoustics and Institute of Acoustics Nanjing University Nanjing 210093 China California Institute of Technology Pasadena CA 91125 U.S.A PIMCO Newport Beach CA 92660 U.S.A.

A unified perturbation theory is developed here for calculating solitary waves of all heights by series expansion of base flow variables in powers of a small base parameter to eighteenth order for the one-parameter family of solutions in exact form, with all the coefficients determined in rational numbers. Comparative studies are pursued to investigate the effects due to changes of base parameters on （i） the accuracy of the theoretically predicted wave properties and （ii） the rate of convergence of perturbation expansion. Two important results are found by comparisons between the theoretical predictions based on a set of parameters separately adopted for expansion in turn. First, the accuracy and the convergence of the perturbation expansions, appraised versus the exact solution provided by an earlier paper [1] as the standard reference, are found to depend, quite sensitively, on changes in base parameter. The resulting variations in the solution are physically displayed in various wave properties with differences found dependent on which property （e.g. the wave amplitude, speed, its profile, excess mass, momentum, and energy）, on what range in value of the base, and on the rank of the order n in the expansion being addressed. Secondly, regarding convergence, the present perturbation series is found definitely asymptotic in nature, with the relative error δ （n）（the relative mean-square difference between successive orders n of wave elevations） reaching a minimum, δm at a specific order, n = n both depending on the base adopted, e.g. nm,α= 11-12 based on parameter α （wave amplitude）, nm,δ = 15 on δ （amplitude-speed square ratio）, and nm.ε= 17 on ε （ wave number squared）. The asymptotic range is brought to completion by the highest order of n = 18 reached in this work.

关键词： solitary waves on water Unified perturbation theory Base functions Base parameters Asymptotic representation

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A unified intrinsic functional expansion theory for solitary waves

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Acta Mechanica Sinica 2005年第1期21卷 1-15页

作者： Theodore Yaotsu Wu John Kao Jin E.Zhang California Institute of Technology Pasadena.CA 91125U.S.A.FellowInstitute of MechanicsChinese Academy of SciencesBeijing 100080China Warner Brothers GlendaleCA 91203.U.S.A. The university of Hong Kong.Pokfulam Road.Hong Kong SAR.China

A new theory is developed here for evaluating solitary waves on water, with results of high accuracy uniformly valid for waves of all heights, from the highest wave with a corner crest of 120 down to very low ones of diminishing height. Solutions are sought for the Euler model by employing a unified expansion of the logarithmic hodograph in terms of a set of intrinsic component functions analytically determined to represent all the intrinsic properties of the wave entity from the wave crest to its outskirts. The unknown coefficients in the expansion are determined by minimization of the mean-square error of the solution, with the minimization optimized so as to take as few terms as needed to attain results as high in accuracy as attainable. In this regard, Stokess formula, F2= tan , relating the wave speed (the Froude number F) and the logarithmic decrement of its wave field in the outskirt, is generalized to establish a new criterion requiring (for minimizing solution error) the functional expansion to contain a finite power series in M terms of Stokess basic term (singular in ), such that 2M is just somewhat beyond unity, i.e. 2M1. This fundamental criterion is fully validated by solutions for waves of various amplitude-to-water depth ratio =a/h, especially about 0.01, at which M=10 by the criterion. In this pursuit, the class of dwarf solitary waves, defined for waves with 0.01, is discovered as a group of problems more challenging than even the highest wave. For the highest wave, a new solution is determined here to give the maximum height hst=0.8331990, and speed Fhst=1.290890, accurate to the last significant figure, which seems to be a new record.

关键词： solitary waves on water Unified intrinsic functional expansion theory Exact solutions High-accuracy computation of waves of arbitrary height Mass and energy transfer

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