Asymptotic Methods for Weakly Nonlinear and Other Water Waves

2016 ◽  
pp. 121-196
Author(s):  
Robin Stanley Johnson
Keyword(s):  
Water Waves ◽  
Asymptotic Methods ◽  
Weakly Nonlinear

scholarly journals “Extraordinary” modulation instability in optics and hydrodynamics

2021 ◽  
Vol 118 (14) ◽  
pp. e2019348118
Author(s):  
Guillaume Vanderhaegen ◽  
Corentin Naveau ◽  
Pascal Szriftgiser ◽  
Alexandre Kudlinski ◽  
Matteo Conforti ◽  
...  
Keyword(s):  
Wave Propagation ◽  
Stability Analysis ◽  
Linear Stability ◽  
Nonlinear Theory ◽  
Linear Stability Analysis ◽  
Water Waves ◽  
Modulation Instability ◽  
Nonlinear Effects ◽  
Weakly Nonlinear ◽  
Weakly Nonlinear Theory

The classical theory of modulation instability (MI) attributed to Bespalov–Talanov in optics and Benjamin–Feir for water waves is just a linear approximation of nonlinear effects and has limitations that have been corrected using the exact weakly nonlinear theory of wave propagation. We report results of experiments in both optics and hydrodynamics, which are in excellent agreement with nonlinear theory. These observations clearly demonstrate that MI has a wider band of unstable frequencies than predicted by the linear stability analysis. The range of areas where the nonlinear theory of MI can be applied is actually much larger than considered here.

scholarly journals Spectral analysis of vibration in weakly non-linear systems

2000 ◽  
Vol 22 (3) ◽  
pp. 181-192
Author(s):  
Nguyen Tien Khiem
Keyword(s):  
Nonlinear Systems ◽  
Spectral Density ◽  
Density Function ◽  
Asymptotic Methods ◽  
Random Excitation ◽  
Spectral Density Function ◽  
Kolmogorov Equation ◽  
Spectral Structure ◽  
Random Load ◽  
Weakly Nonlinear

The weakly nonlinear systems subjected to deterministic excitations have been fully and deeply studied by use of the well developed asymptotic methods [1-4]. The systems excited by a random load have been investigated mostly using the Fokker-Plank-Kolmogorov equation technique combined with the asymptotic methods [5-8]. However, the last approach in most successful cases allows to obtain only a stationary single point probability density function, that contains no information about the correlation nor' consequently, the spectral structure of the response. The linearization technique [9, 10] in general permits the spectral density of the response to be determined, but the spectral function obtained by this method because of the linearization eliminates the effect of the nonlinearity. Thus, spectral structure of response of weakly nonlinear systems to random excitation, to the author's knowledge, has not been studied enough. This paper deals with the above mentioned problem. The main idea of this work is the use of an analytical simulation of random excitation given by its spectral density function and afterward application of the well known procedure of the asymptotic method to obtain an asymptotic expression of the response spectral density function. The obtained spectral relationship covers the linear system case and especially emphasizes the nonlinear effect on the spectral density of response. The theory will be illustrated by an example and at the end of this paper there will be a discussion about the obtained results.  

Water Waves Generated by Thin Ships

1960 ◽  
Vol 4 (03) ◽  
pp. 25-36
Author(s):  
Milton S. Plesset ◽  
T. Yao-tsu Wu
Keyword(s):  
Velocity Field ◽  
Partial Derivative ◽  
Water Waves ◽  
Asymptotic Methods ◽  
Series Representation ◽  
Double Fourier Series ◽  
Tangent Plane ◽  
Field Calculation ◽  
Numerical Result ◽  
Longitudinal Slope

The problem of interest is that of the water waves in a body of water of infinite depth generaied by a thin ship of given hull form, moving with constant velocity U along a straight course on the otherwise undisturbed water surface. A particular method is evaluated for computing the velocity field at an arbitrary distance (not too near the ship) fixed in the fluid. A new proposal is made here that the hull profile be represented by a double Fourier series with its half-periods spanning over the region occupied by the longitudinal mid-section of the ship. The convergence of this series representation is found to be satisfactorily rapid, especially when the tangent plane of the hull is everywhere continuous. In the latter case the longitudinal slope of the hull, which is the only partial derivative appearing in the analysis, is found in a specific case to be well represented by the partial derivative of the series. With this series representation of the hull, the analysis of the velocity-field calculation is greatly reduced so that the final result can be expressed in terms of a combination of several single and double Fourier integrals which are susceptible to numerical methods. However, for large values of or, where r is the distance from the ship, a = gL/U2, with g being the acceleration of gravity and L the ship length, these integrals can be evaluated with good approximation by asymptotic methods. The method of stationary phase and other asymptotic methods are employed in different regions in the water and the final expression for the velocity field is given explicitly. The numerical result for a specific ship will be given elsewhere.

Three-Dimensional Weakly Nonlinear Shallow Water Waves Regime and its Traveling Wave Solutions

2018 ◽  
Vol 15 (03) ◽  
pp. 1850017 ◽  
Author(s):  
Aly R. Seadawy
Keyword(s):  
Free Surface ◽  
Solitary Wave ◽  
Shallow Water ◽  
Traveling Wave ◽  
Water Waves ◽  
Three Dimensional ◽  
Traveling Wave Solutions ◽  
Shallow Water Waves ◽  
Weakly Nonlinear ◽  
Wave Solutions

The problem formulations of models for three-dimensional weakly nonlinear shallow water waves regime in a stratified shear flow with a free surface are studied. Traveling wave solutions are generated by deriving the nonlinear higher order of nonlinear evaluation equations for the free surface displacement. We obtain the velocity potential and pressure fluid in the form of traveling wave solutions of the obtained nonlinear evaluation equation. The obtained solutions and the movement role of the waves of the exact solutions are new travelling wave solutions in different and explicit form such as solutions (bright and dark), solitary wave, periodic solitary wave elliptic function solutions of higher-order nonlinear evaluation equation.

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.

scholarly journals Reflection of nonlinear deep-water waves incident onto a wedge of arbitrary angle

1990 ◽  
Vol 32 (1) ◽  
pp. 61-96 ◽  
Author(s):  
T. R. Marchant ◽  
A. J. Roberts
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Wave Reflection ◽  
Mach Reflection ◽  
Wedge Angle ◽  
Arbitrary Angle ◽  
Weakly Nonlinear ◽  
Progressive Waves ◽  
The Waves ◽  
Wave Properties

AbstractWave reflection by a wedge in deep water is examined, where the wedge can represent a breakwater of finite length or the bow of a ship heading directly into the waves. In addition, the form of the solution allows the results to apply to ships heading at an angle into the waves. We consider a deep-water wavetrain approaching the wedge head on from infinity and being reflected. Far from the wedge there is a field of progressive waves (the incident wavetrain) while close to the wedge there is a short-crested wavefield (the incident and reflected wavetrains). A weakly-nonlinear slowly-varying averaged Lagrangian theory is used to describe the problem (see Whitham [16]) as the theory includes the nonlinear interaction between the incident and reflected wavetrains. This modelling of a short-crested wavefield allows the nonlinear wavefield to be found for broad wedges, as opposed to previous theories which are applicable to thin wedges only.It is shown that the governing partial differential equations are hyperbolic and that the solution comprises two regions, within which the wave properties are constant separated by a wave jump. Given the wedge angle and the incident wavefield, the jump angle and the wave steepness and wavenumber of the short-crested wave-field behind the wave jump can be determined. Two solution branches are found to exist: one corresponds to regular reflection, while for small amplitudes the other is similar to Mach-reflection and so it is called near Mach-reflection. Results are presented describing both solution branches and the transition between them.

On the weakly nonlinear development of Tollmien-Schlichting wavetrains in boundary layers

1996 ◽  
Vol 323 ◽  
pp. 133-171 ◽  
Author(s):  
Xuesong Wu ◽  
Philip A. Stewart ◽  
Stephen J. Cowley
Keyword(s):  
Numerical Solutions ◽  
Asymptotic Methods ◽  
Nonlinear Effects ◽  
Mean Flow ◽  
Flow Distortion ◽  
Weakly Nonlinear ◽  
Nonlinear Development ◽  
Finite Distance ◽  
Tollmien Schlichting ◽  
High Reynolds

The nonlinear development of a weakly modulated Tollmien-Schlichting wavetrain in a boundary layer is studied theoretically using high-Reynolds-number asymptotic methods. The ‘carrier’ wave is taken to be two-dimensional, and the envelope is assumed to be a slowly varying function of time and of the streamwise and spanwise variables. Attention is focused on the scalings appropriate to the so-called ‘upper branch’ and ‘high-frequency lower branch’. The dominant nonlinear effects are found to arise in the critical layer and the surrounding ‘diffusion layer’: nonlinear interactions in these regions can influence the development of the wavetrain by producing a spanwise-dependent mean-flow distortion. The amplitude evolution is governed by an integro-partial-differential equation, whose nonlinear term is history-dependent and involves the highest derivative with respect to the spanwise variable. Numerical solutions show that a localized singularity can develop at a finite distance downstream. This singularity seems consistent with the experimentally observed focusing of vorticity at certain spanwise locations, although quantitative comparisons have not been attempted.

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