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Periodic travelling interfacial hydroelastic waves with or without mass II: Multiple bifurcations and ripples
- BENJAMIN F. AKERS, DAVID M. AMBROSE, DAVID W. SULON
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- Journal:
- European Journal of Applied Mathematics / Volume 30 / Issue 4 / August 2019
- Published online by Cambridge University Press:
- 10 July 2018, pp. 756-790
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In a prior work, the authors proved a global bifurcation theorem for spatially periodic interfacial hydroelastic travelling waves on infinite depth, and computed such travelling waves. The formulation of the travelling wave problem used both analytically and numerically allows for waves with multi-valued height. The global bifurcation theorem required a one-dimensional kernel in the linearization of the relevant mapping, but for some parameter values, the kernel is instead two-dimensional. In the present work, we study these cases with two-dimensional kernels, which occur in resonant and non-resonant variants. We apply an implicit function theorem argument to prove existence of travelling waves in both of these situations. We compute the waves numerically as well, in both the resonant and non-resonant cases.
4 - High-Order Perturbation of Surfaces Short Course: Stability of Traveling Water Waves
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- By Benjamin F. Akers, Air Force Institute of Technology
- Edited by 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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- Book:
- Lectures on the Theory of Water Waves
- Published online:
- 05 February 2016
- Print publication:
- 04 February 2016, pp 51-62
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Summary
Abstract
In this contribution we present High-Order Perturbation of Surfaces (HOPS) methods as applied to the spectral stability problem for traveling water waves. The Transformed Field Expansion method (TFE) is used for both the traveling wave and its spectral data. The Lyapunov-Schmidt reductions for simple and repeated eigenvalues are compared. The asymptotics of modulational instabilities are discussed.
Introduction
The water wave stability problem has a rich history, with great strides made in the late sixties in the work of Benjamin and Feir [1] and in the ensuing development of Resonant Interaction Theory (RIT) [2–5]. The predictions of RIT have since been leveraged heavily by numerical methods; the influential works of MacKay and Saffman [6] and McLean [7] led to a taxonomy of water wave instabilities based on RIT (Class I and Class II instabilities). The most recent review article is that of Dias & Kharif [8]; since the publication of this review a number of modern numerical stability studies have been conducted [9–13].
In these lecture notes, we explain how the spectral data of traveling water waves may be computed using a High-Order Perturbation of Surfaces (HOPS) approach, which numerically computes the coefficients in amplitude-based series expansions [14]. For the water wave problem, a crucial aspect of any numerical approach is the method used to handle the unknown fluid domain. Just as in the traveling waves lecture of this short course, numerical results will be presented from the Transformed Field Expansion (TFE) method, whose development for the spectral stability problem appears in [13, 15–17].
The TFE method computes the spectral data as a series in wave slope/ amplitude, and thus relies on analyticity of the spectral data in amplitude. A large number of studies of the spectrum have been made that do not make such an assumption [9, 10, 18, 19]. On the other hand, it is known that the spectrum is analytic for all Bloch parameters at which eigenvalues are simple in the zero amplitude limit [20]. Numerically it has been observed that the spectrum is analytic in amplitude at Bloch parameters for which there are eigenvalue collisions, but that the disc of analyticity is discontinuous in Bloch parameter. This discontinuity in radius is due to modulational instabilities, as explained in [21].
2 - High-Order Perturbation of Surfaces Short Course: Traveling Water Waves
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- By Benjamin F. Akers, Air Force Institute of Technology
- Edited by 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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- Book:
- Lectures on the Theory of Water Waves
- Published online:
- 05 February 2016
- Print publication:
- 04 February 2016, pp 19-31
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Summary
Abstract
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.
Introduction
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].