A unified intrinsic functional expansion theory for solitary waves

Abstract

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, Stokes’s formula, F2μπ= tan μπ, relating the wave speed (the Froude number F) and the logarithmic decrement μ 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 Stokes’s basic term (singular in μ), such that 2Mμ is just somewhat beyond unity, i.e. 2Mμ≃1. This fundamental criterion is fully validated by solutions for waves of various amplitude-to-water depth ratio α=a/h, especially about α≃0.01, at which M=10 by the criterion. In this pursuit, the class of dwarf solitary waves, defined for waves with α≤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 αhst=0.8331990, and speed Fhst=1.290890, accurate to the last significant figure, which seems to be a new record.