Do peaked solitary water waves indeed exist?

Many models of shallow water waves admit peaked solitary waves. However, it is an open question whether or not the widely accepted peaked solitary waves can be derived from the fully nonlinear wave equations. In this paper, a unified wave model (UWM) based on the symmetry and the fully nonlinear wave equations is put forward for progressive waves with permanent form in finite water depth. Different from traditional wave models, the flows described by the UWM are not necessarily irrotational at crest, so that it is more general. The unified wave model admits not only the traditional progressive waves with smooth crest, but also a new kind of solitary waves with peaked crest that include the famous peaked solitary waves given by the Camassa-Holm equation. Besides, it is proved that Kelvin's theorem still holds everywhere for the newly found peaked solitary waves. Thus, the UWM unifies, for the first time, both of the traditional smooth waves and the peaked solitary waves. In other words, the peaked solitary waves are consistent with the traditional smooth ones. So, in the frame of inviscid fluid, the peaked solitary waves are as acceptable and reasonable as the traditional smooth ones. It is found that the peaked solitary waves have some unusual and unique characteristics. First of all, they have a peaked crest with a discontinuous vertical velocity at crest. Especially, the phase speed of the peaked solitary waves has nothing to do with wave height. In addition, the kinetic energy of the peaked solitary waves either increases or almost keeps the same from free surface to bottom. All of these unusual properties show the novelty of the peaked solitary waves, although it is still an open question whether or not they are reasonable in physics if the viscosity of fluid and surface tension are considered.Comment: 53 pages, 13 figures, 7 tables. Accepted by Communications in Nonlinear Science and Numerical Simulatio

Finite-amplitude interfacial waves in the presence of a current

Solutions for interfacial waves of permanent form in the presence of a current wcre obtained for small-to-moderate wave amplitudes. A weakly nonlinear approximation was used to give simple analytical solutions to second order in wave height. Numerical methods were usctl to obtain solutions for larger wave amplitudes, details are reported for a number of selected cases. A special class of finite-amplitude solutions, closely related to the well-known Stokes surface waves, were identified. Factors limiting the existence of steady solutions are examined

Nonlinear resonances of water waves

In the last fifteen years, a great progress has been made in the understanding of the nonlinear resonance dynamics of water waves. Notions of scale- and angle-resonances have been introduced, new type of energy cascade due to nonlinear resonances in the gravity water waves have been discovered, conception of a resonance cluster has been much and successful employed, a novel model of laminated wave turbulence has been developed, etc. etc. Two milestones in this area of research have to be mentioned: a) development of the $q$-class method which is effective for computing integer points on the resonance manifolds, and b) construction of the marked planar graphs, instead of classical resonance curves, representing simultaneously all resonance clusters in a finite spectral domain, together with their dynamical systems. Among them, new integrable dynamical systems have been found that can be used for explaining numerical and laboratory results. The aim of this paper is to give a brief overview of our current knowledge about nonlinear resonances among water waves, and formulate three most important open problems at the end.Comment: 14 pages, 3 figures, to appear in DCDS, final version (small changes in the text, type errors corrected, some additional bibliographic items added

Experimental studies on the tripping behavior of narrow T-stiffened flat plates subjected to hydrostatic pressure and underwater shock

An experimental investigation was conducted to determine the static and dynamic responses of a specific stiffened flat plate design. The air-backed rectangular flat plates of 6061-T6 aluminum with an externally machined longitudinal narrow-flanged T-stiffener and clamped boundary conditions were subjected to static loading by water hydropump pressure and shock loading from an eight pound TNT charge detonated underwater. The dynamic test plate was instrumented to measure transient strains and free field pressure. The static test plate was instrumented to measure transient strains, plate deflection, and pressure. Emphasis was placed upon forcing static and dynamic stiffener tripping, obtaining relevant strain and pressure data, and studying the associated plate-stiffener behavior

Steep sharp-crested gravity waves on deep water

A new type of steady steep two-dimensional irrotational symmetric periodic gravity waves on inviscid incompressible fluid of infinite depth is revealed. We demonstrate that these waves have sharper crests in comparison with the Stokes waves of the same wavelength and steepness. The speed of a fluid particle at the crest of new waves is greater than their phase speed.Comment: 4 pages, 2 figures, submitted to Phys. Rev. Let

Long waves of political contestation

This paper develops a wave theory of political contestation, and places the current economic and political turmoil in a historical perspective. Based on legitimacy, it serves as an alternative to the waves of democratization of Samuel Huntington (1991). The theoretical framework is based on two main theories: the theory of long waves in political economics and the theory about state-legitimacy and fiscal crisis. In the first section, this paper gives a short overview of the different economic dynamics which over time have been incorporated in long wave theories, predominantly based on the works of Kondratieff (1979) and Schumpeter (1939), and puts the current economic situation in this perspective. The second part analyzes the general interdependency between long waves and politics, and the original criticisms of the endogenous model by Trotsky (1923). The third section considers long waves theories in politics, in particular Samuel Huntington's theory, and discusses the main criticisms of his theory. The fourth section analyzes the influence of the long wave upswing and downturn on state-legitimacy, and is based on the work of O'Connor (2001) and Habermas (1975). The fifth section combines the long wave's concept with legitimacy and protest against a long wave theory of political contestation and gives the first elements of some empirical evidence, comparing the political contestation in the thirties and today. The sixth section draws conclusions and takes a look on the need for further research

How Hertzian solitary waves interact with boundaries in a 1-D granular medium

We perform measurements, numerical simulations, and quantitative comparisons with available theory on solitary wave propagation in a linear chain of beads without static preconstrain. By designing a nonintrusive force sensor to measure the impulse as it propagates along the chain, we study the solitary wave reflection at a wall. We show that the main features of solitary wave reflection depend on wall mechanical properties. Since previous studies on solitary waves have been performed at walls without these considerations, our experiment provides a more reliable tool to characterize solitary wave propagation. We find, for the first time, precise quantitative agreements.Comment: Proof corrections, ReVTeX, 11 pages, 3 eps (Focus and related papers on http://www.supmeca.fr/perso/jobs/

New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

Steady two-dimensional surface capillary-gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained