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2 - High-Order Perturbation of Surfaces Short Course: Traveling Water Waves

Published online by Cambridge University Press:  05 February 2016

Benjamin F. Akers
Air Force Institute of Technology
Thomas J. Bridges
University of Surrey
Mark D. Groves
Universität des Saarlandes, Saarbrücken, Germany
David P. Nicholls
University of Illinois, Chicago
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In this contribution we discuss High-Order Perturbation of Surfaces (HOPS) methods with particular application to traveling water waves. The Transformed Field Expansion method (TFE) is discussed as a method for handling the unknown fluid domain. The procedures for computing Stokes waves and Wilton Ripples are compared. The Lyapunov-Schmidt procedure for the Wilton Ripple is presented explicitly in a simple, weakly nonlinear model equation.


Traveling water waves have been studied for over a century, most famously by Stokes, for whom weakly-nonlinear periodic waves are now named [1–3]. In his 1847 paper, Stokes expanded the wave profile as a power series in a small parameter, the wave slope, a technique that has since become commonplace. This classic perturbation expansion, which we will refer to as the Stokes’ expansion, has been applied to the water wave problem numerous times [4–9]. When the effect of surface tension is included, the expansion may be singular. This singularity, due to a resonance between a long and a short wave, was noted first by Wilton [10] and has been studied more recently in [11–15].

In these lecture notes, we explain how traveling water waves may be computed using a High-Order Perturbation of Surfaces (HOPS) approach, which numerically computes the coefficients in an amplitude-based series expansion of the free surface. For the water wave problem, a crucial aspect of any numerical approach is the method used to handle the unknown fluid domain. Popular examples include Boundary Integral Methods [16, 17], conformal mappings [18, 19], and series computations of the Dirichelet-to-Neumann operator [20, 21]. Here we discuss an alternative approach, in which the solution is expanded using the Transformed Field Expansion (TFE) method, developed in [22, 23].

The TFE method has been used to compute traveling waves on both two-dimensional (one horizontal and one vertical dimension) and three-dimensional fluids, both for planar and short-crested waves [23]. Short-crested wave solutions to the potential flow equations have been computed without surface tension [22, 24, 25] and with surface tension [26]. They have also been studied experimentally [27, 28].

Publisher: Cambridge University Press
Print publication year: 2016

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