Two-timing, variational principles and waves
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In this article, the author's theory of slowly varying wave trains was derived as the first term in a formal perturbation expansion, which was applied directly to the governing variational principle.Abstract:
In this paper, it is shown how the author's general theory of slowly varying wave trains may be derived as the first term in a formal perturbation expansion. In its most effective form, the perturbation procedure is applied directly to the governing variational principle and an averaged variational principle is established directly. This novel use of a perturbation method may have value outside the class of wave problems considered here. Various useful manipulations of the average Lagrangian are shown to be similar to the transformations leading to Hamilton's equations in mechanics. The methods developed here for waves may also be used on the older problems of adiabatic invariants in mechanics, and they provide a different treatment; the typical problem of central orbits is included in the examples.read more
Citations
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The soliton: A new concept in applied science
TL;DR: The term soliton has been coined to describe a pulselike nonlinear wave (solitary wave) which emerges from a collision with a similar pulse having unchanged shape and speed.
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On three-dimensional packets of surface waves
A. Davey,Keith Stewartson +1 more
TL;DR: In this article, the authors used the method of multiple scales to derive the two coupled nonlinear partial differential equations which describe the evolution of a three-dimensional wavepacket of wavenumber k on water of finite depth.
Book ChapterDOI
Nonlinear Dynamics of Deep-Water Gravity Waves
Henry C. Yuen,Bruce M. Lake +1 more
TL;DR: In this paper, a review of recent progress in the nonlinear dynamics of deep-water gravity waves is presented, highlighting the major developments in theory and experiment commencing with the finding by Lighthill (1965) that a nonlinear, deepwater gravity wave train is unstable to modulational perturbation, up to the present investigations of various aspects of nonlinear phenomena, including three-dimensional instabilities, bifurcations into new steady solutions, statistical properties of random wave fields, and chaotic behavior in time evolution.
Journal ArticleDOI
On wave-action and its relatives
TL;DR: In this article, the wave-action equation is derived in a simple but very general form which does not depend on the approximations of slow amplitude modulation, linearization, or conservative motion.
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Nonlinear deep water waves: Theory and experiment
Henry C. Yuen,Bruce M. Lake +1 more
TL;DR: In this article, the evolution and interaction of nonlinear wavepackets on deep water is studied both theoretically and experimentally, and the exact solution to this equation predicts the existence of stable envelope solitons, which is indeed verified by laboratory experiments.
References
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Book
Perturbation Methods in Applied Mathematics
TL;DR: In this paper, limit process expansions applied to Ordinary Differential Equations (ODE) are applied to partial differential equations (PDE) in the context of Fluid Mechanics.
Journal ArticleDOI
Perturbation Methods in Applied Mathematics
H. Weitzner,Julian D. Cole +1 more
TL;DR: In this article, limit process expansions applied to Ordinary Differential Equations (ODE) are applied to partial differential equations (PDE) in the context of Fluid Mechanics.