Weakly nonlinear long waves in a prestretched Blatz–Ko cylinder: Solitary, kink and periodic waves

Wave Motion ◽  
2011 ◽  
Vol 48 (8) ◽  
pp. 761-772 ◽  
Author(s):  
Hui-Hui Dai ◽  
Xiaochun Peng
Keyword(s):  
Long Waves ◽  
Periodic Waves ◽  
Weakly Nonlinear

scholarly journals Focusing of long waves with finite crest over constant depth

2013 ◽  
Vol 469 (2153) ◽  
pp. 20130015 ◽  
Author(s):  
Utku Kânoğlu ◽  
Vasily V. Titov ◽  
Baran Aydın ◽  
Christopher Moore ◽  
Themistoklis S. Stefanakis ◽  
...  
Keyword(s):  
Source Region ◽  
Wave Theory ◽  
Long Waves ◽  
Constant Depth ◽  
Large Wave ◽  
Weakly Nonlinear ◽  
Scaling Relationships ◽  
2011 Japan Tsunami ◽  
Wave Heights ◽  
Run Up

Tsunamis are long waves that evolve substantially, through spatial and temporal spreading from their source region. Here, we introduce a new analytical solution to study the propagation of a finite strip source over constant depth using linear shallow-water wave theory. This solution is not only exact, but also general and allows the use of realistic initial waveforms such as N -waves. We show the existence of focusing points for N -wave-type initial displacements, i.e. points where unexpectedly large wave heights may be observed. We explain the effect of focusing from a strip source analytically, and explore it numerically. We observe focusing points using linear non-dispersive and linear dispersive theories, analytically; and nonlinear non-dispersive and weakly nonlinear weakly dispersive theories, numerically. We discuss geophysical implications of our solutions using the 17 July 1998 Papua New Guinea and the 17 July 2006 Java tsunamis as examples. Our results may also help to explain high run-up values observed during the 11 March 2011 Japan tsunami, which are otherwise not consistent with existing scaling relationships. We conclude that N -waves generated by tectonic displacements feature focusing points, which may significantly amplify run-up beyond what is often assumed from widely used scaling relationships.

scholarly journals Nonlinear long waves over a muddy beach

2013 ◽  
Vol 718 ◽  
pp. 371-397 ◽  
Author(s):  
Erell-Isis Garnier ◽  
Zhenhua Huang ◽  
Chiang C. Mei
Keyword(s):  
Boundary Layer ◽  
Thin Layer ◽  
Surface Waves ◽  
Nonlinear Waves ◽  
Water Waves ◽  
Long Waves ◽  
Fluid Mud ◽  
Weakly Nonlinear ◽  
Sea Bottom ◽  
Oscillatory Boundary Layer

AbstractWe analyse theoretically the interaction between water waves and a thin layer of fluid mud on a sloping seabed. Under the assumption of long waves in shallow water, weakly nonlinear and dispersive effects in water are considered. The fluid mud is modelled as a thin layer of viscoelastic continuum. Using the constitutive coefficients of mud samples from two field sites, we examine the interaction of nonlinear waves and the mud motion. The effects of attenuation on harmonic evolution of surface waves are compared for two types of mud with distinct rheological properties. In general mud dissipation is found to damp out surface waves before they reach the shore, as is known in past observations. Similar to the Eulerian current in an oscillatory boundary layer in a Newtonian fluid, a mean displacement in mud is predicted which may lead to local rise of the sea bottom.

The growth of periodic waves in gas-fluidized beds

1996 ◽  
Vol 325 ◽  
pp. 261-282 ◽  
Author(s):  
S. E. Harris
Keyword(s):  
Nonlinear Waves ◽  
Fluidized Beds ◽  
Kdv Equation ◽  
Periodic Waves ◽  
Weakly Nonlinear ◽  
Strongly Nonlinear ◽  
The Kdv Equation ◽  
Weakly Nonlinear Waves ◽  
Gas Fluidized Beds ◽  
The One

In this paper, we analyse the development of initially small, periodic, voidage disturbances in gas-fluidized beds. The one-dimensional model was proposed by Needham & Merkin (1983), and Crighton (1991) showed that weakly nonlinear waves satisfied a perturbed Korteweg–de Vries or KdV equation. Here, we take periodic cnoidal wave solutions of the KdV equation and follow their evolution when the perturbation terms are amplifying. Initially, all such waves grow, but at a later stage a rescaling shows that shorter wavelengths are stabilized in a weakly nonlinear state. Longer wavelengths continue to develop and eventually strongly nonlinear solutions are required. Necessary conditions for periodic waves are found and matching back onto the growing cnoidal waves is possible. It is shown further that these fully nonlinear waves also reach an equilibrium state. A comparison with numerical results from Needham & Merkin (1986) and Anderson, Sundaresan & Jackson (1995) is then carried out.

Oblique interactions of weakly nonlinear long waves in dispersive systems

2006 ◽  
Vol 38 (12) ◽  
pp. 868-898 ◽  
Author(s):  
Masayuki Oikawa ◽  
Hidekazu Tsuji
Keyword(s):  
Long Waves ◽  
Weakly Nonlinear ◽  
Dispersive Systems

On periodic and solitary wavelike solutions of the intermediate long-wave equation

1990 ◽  
Vol 211 ◽  
pp. 617-627 ◽  
Author(s):  
Touvia Miloh
Keyword(s):  
Periodic Solutions ◽  
Finite Depth ◽  
Periodic Waves ◽  
Solitary Wave Solutions ◽  
Remarkable Property ◽  
Weakly Nonlinear ◽  
Wave Solutions ◽  
Long Wave ◽  
Analytic Expressions ◽  
Infinite Sum

The intermediate long-wave (ILW) equation is a weakly nonlinear integrodifferential equation which governs the evolution of long internal waves in a stratified fluid of finite depth. It reduces to the Korteweg–de Vries (KdV) and to the Benjamin–Ono (BO) equations for shallow and large depths respectively. Solitary wave solutions of the ILW equation are well known, however analytic expressions for periodic solutions of the same equation do not seem to exist. Such expressions are derived in this paper and a remarkable property discovered for these periodic waves is that they can be represented as an infinite sum of spatially repeated solitons. Thus, nonlinear periodic solutions of the ILW equation are obtained by linear superposition of solitons.

Ray tracing in weakly nonlinear moving media

1976 ◽  
Vol 16 (3) ◽  
pp. 415-426 ◽  
Author(s):  
Dan Censor
Keyword(s):  
Velocity Field ◽  
Ray Tracing ◽  
Complete System ◽  
Dispersive Media ◽  
Periodic Waves ◽  
Weakly Nonlinear ◽  
Optical Media ◽  
Moving Media ◽  
Ray Path ◽  
Excitation Model

Electromagnetic ray propagation is discussed for weakly nonlinear dispersive media. Specialization to periodic waves justifies the use of a self-excitation model and facilitates the construction of a complete system of ray equations. The solution describes the evolution of frequencies, propagation vectors and amplitudes along the ray path. The formalism may be extended to include the case where a velocity field exists. This is based on special relativity and the Minkowski model for the electrodynamics of moving media. It is assumed that the velocity field varies slowly in space and time. The results are relevant to propagation problems in weakly nonlinear plasmas and optical media.

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