Nonlinear Waves

A statical model of random wave is developed using Stokes wave theory of water wave dynamics. A new nonlinear probability distribution function of wave height is presented. The results indicate that wave steepness not only could be a parameter of the distribution function of wave height but also could reflect the degree of wave height distribution deviation from the Rayleigh distribution. The new wave height distribution overcomes the problem of Rayleigh distribution that the prediction of big wave is overestimated and the general wave is underestimated. The prediction of small probability wave height value of new distribution is also smaller than that of Rayleigh distribution. Wave height data taken from East China Normal University are used to verify the new distribution. The results indicate that the new distribution fits the measurements much better than the Rayleigh distribution.

PURPOSE: The purpose of this Coastal and Hydraulics Engineering Technical Note (CHETN) is to demonstrate the use of BOUSS-2D in three example applications. The BOUSS-2D is a Boussinesq wave model that is now part of the U.S. Army Corps of Engineers (USACE) Surface-Water Modeling System (SMS). Demirbilek et al. (2004) gives a detailed description of the SMS interface of BOUSS-2D. This second technical note in BOUSS-2D series is a step-by-step tutorial that shows the usage of the model through illustrative examples. The next CHETN in the series will describe the time-series and frequency-domain analysis utilities and techniques developed for post-processing BOUSS-2D model results. Details of the BOUSS-2D model theory, numerical implementation, and example applications are presented in a technical report (Nwogu and Demirbilek 2001). BACKGROUND: As a phase-resolving nonlinear wave model, BOUSS-2D can be used in the modeling of various wave phenomena including shoaling, refraction, diffr...

We study analytically and numerically the stability of the standing waves for a nonlinear Schrödinger equation with a point defect and a power type nonlinearity. A main difficulty is to compute the number of negative eigenvalues of the linearized operator around the standing waves, and it is overcome by a perturbation method and continuation arguments. Among others, in the case of a repulsive defect, we show that the standing wave solution is stable in and unstable in under subcritical nonlinearity. Further we investigate the nature of instability: under critical or supercritical nonlinear interaction, we prove the instability by blowup in the repulsive case by showing a virial theorem and using a minimization method involving two constraints. In the subcritical radial case, unstable bound states cannot collapse, but rather narrow down until they reach the stable regime (a finite-width instability). In the non-radial repulsive case, all bound states are unstable, and the instability...

A digital wave tank (DWT) simulation technique has been developed by authors to investigate the interactions of fully nonlinear waves with 3D marine structures. A finite-difference/volume method and a modified marker-and-cell (MAC) algorithm have been used, which are based on the Navier-Stokes (NS) and continuity equations. The fully nonlinear kinematic free-surface condition is implemented by the marker-density function (MDF) technique or the Level-Set (LS) technique developed for one or two fluid layers. In this paper, some applications for various engineering problems with free-surface are introduced and discussed. It includes numerical simulation of marine environments by simulation equipments, fully nonlinear wave motions around offshore structures, nonlinear ship waves, ship motions in waves and marine flow simulation with free-surface. From the presented simulations, it seems that the developed DWT simulation technique can handle various engineering problems with free-surface...

The nonlinear aspects of longitudinal motion of interacting point masses in a lattice are revisited, with emphasis on the paradigm of charged dust grains in a dusty plasma (DP) crystal. Different types of localized excitations, predicted by nonlinear wave theories, are reviewed and conditions for their occurrence (and characteristics) in DP crystals are discussed. Making use of a general formulation, allowing for an arbitrary (e.g. the Debye electrostatic or else) analytic potential form $\phi(r)$ and arbitrarily long site-to-site range of interactions, it is shown that dust-crystals support nonlinear kink-shaped localized excitations propagating at velocities above the characteristic DP lattice sound speed v 0. Both compressive and rarefactive kink-type excitations are predicted, depending on the physical parameter values, which represent pulse- (shock-)like coherent structures for the dust grain relative displacement. Furthermore, the existence of breather-type localized oscillations, envelope-modulated wavepackets and shocks is established. The relation to previous results on atomic chains as well as to experimental results on strongly-coupled dust layers in gas discharge plasmas is discussed.

Modern aerospace structures make increasing use of fibre reinforced plastic composites, due to their high specific mechanical properties. However, due to their brittleness, low velocity impact can cause delaminations beneath the surface, while the surface may appear to be undamaged upon visual inspection. Such damage is called barely visible impact damage (BVID). Such internal damages lead to significant reduction in local strengths and ultimately could lead to catastrophic failures. It is therefore important to detect and monitor damages in high loaded composite components to receive an early warning for a well timed maintenance of the aircraft. Non-linear ultrasonic spectroscopy methods are promising damage detection and material characterization tools. In this paper, two different non-linear elastic wave spectroscopy (NEWS) methods are presented: single mode nonlinear resonance ultrasound (NRUS) and nonlinear wave modulation technique (NWMS). The NEWS methods were applied to detect delamination damage due to low velocity impact (<12 J) on various composite plates. The results showed that the proposed methodology appear to be highly sensitive to the presence of damage with very promising future NDT and structural health monitoring applications.

 A nonlinear time-domain procedure is presented which is used to calculate the vertical responses of a container ship advancing in head waves. The method assumes linear radiation forces represented by time convolution of memory functions, infinite frequency added masses, and radiation restoring coefficients. The nonlinear hydrostatic restoring and Froude–Krilov forces are computed exactly over the instantaneous wetted surface of the ship's hull. Forces due to green water on deck are calculated using the momentum method. Nonlinear effects are identified on different vertical ship responses, namely on the heave and pitch motions, the vertical accelerations, and the vertical bending moment. These non-linear effects are expressed by the variation of the transfer function with the wave amplitude, the higher-order harmonics of the time signals, the offset of the time series, and the asymmetry of the peaks. The numerical results and the quantified nonlinear effects are compared with experimental results showing an ability to reproduce the main nonlinear effects.

We investigate bi-Hamiltonian structures and mKdV hierarchies of solitonic equations generated by (semi) Riemannian metrics and curve flows of non-stretching curves. There are applied methods of the geometry of nonholonomic manifolds enabled with metric-induced nonlinear connection (N-connection) structure. On spacetime manifolds, we consider a nonholonomic splitting of dimensions and define a new class of liner connections which are ‘N-adapted’, metric compatible and uniquely defined by the metric structure. We prove that for such a linear connection, one yields couples of generalized sine-Gordon equations when the corresponding geometric curve flows result in solitonic hierarchies described in explicit form by nonholonomic wave map equations and mKdV analogs of the Schrödinger map equation. All geometric constructions can be re-defined for the Levi-Civita connection but with “noholonomic mixing” of solitonic interactions. Finally, we speculate why certain methods and results from the geometry of nonholonmic manifolds and solitonic equations have general importance in various directions of modern mathematics, geometric mechanics, fundamental theories in physics and applications, and briefly analyze possible nonlinear wave configurations for modeling gravitational interactions by effective continuous media effects.

The influence of nonlinear wave theory on the sloshing natural periods and their modal pressure distributions are investigated for rectangular tanks under the assumption of two-dimensional behavior. Natural periods and mode shapes are computed and compared for both linear wave theory (LWT) and nonlinear wave theory (NLWT) models, using the finite element package ABAQUS. Linear wave theory is implemented in an acoustic model, whereas a plane strain problem with large displacements is used in NLWT. Pressure distributions acting on the tank walls are obtained for the first three sloshing modes using both linear and nonlinear wave theory. It is found that the nonlinearity does not have significant effects on the natural sloshing periods. For the sloshing pressures on the tank walls, different distributions were found using linear and nonlinear wave theory models. However, in all cases studied, the linear wave theory conservatively estimated the magnitude of the pressure distribution, whereas larger pressures resultant heights were obtained when using the nonlinear theory. It is concluded that the nonlinearity of the surface wave does not have major effects in the pressure distribution on the walls for rectangular tanks.

In the present paper we have studied the nonlinear dynamical equation of Landau damped kinetic Alfvén wave (KAW) to investigate the nonlinear evolution of KAW and the resulting turbulent spectra in solar wind plasmas. We have introduced a parameter g which governs the coupling between the amplitude of the pump KAW and the density perturbation. The numerical solution has been carried out to see the dependence on the parameter g in the nonlinear part of our equation. Our results reveal the formation of damped localized structures of KAWas well as steepening of the turbulent spectra by increasing  when damping is taken into account. The power spectra of magnetic field fluctuations indicate the redistribution of energy among the higher wave numbers. Each power spectrum with and without damping splits up into two different scaling ranges, Kolmogorov scaling followed by a steeper scaling. The steepening in the power spectra with Landau damping is more than without Landau damping case (for the same value of g). This type of steeper spectra has also been observed in the solar wind and is attributed to the Landau damping effects.

J. Fluid Mech. (2001), vol. 427, pp. 107–129. Printed in the United Kingdom c 2001 Cambridge University Press ... Evolution of a nonlinear wave field along ... By L. SHEMER1, HAIYING JIAO1, E. KIT1 ... 1Department of Fluid Mechanics and Heat Transfer, Faculty of ...

Nanobubbles are a mystery as to their stability and longevity. So far their uses are limited to high value applications, such as medical imaging and controlled drug delivery and release. The current methods of nanobubble generation and understanding of nanobubble interfacial structure are reviewed. The analysis of the potential benefits of nanobubbles leads to the conclusion that if energy efficient

In this chapter we analyze the existence and forms of invariants of the extended Korteweg-de Vries equation (KdV2). This equation appears when the Euler equations for shallow water are extended to the second order, beyond Korteweg-de Vries (KdV). We show that contrary to KdV for which there is an infinite number of invariants, for KdV2 there exists only one, connected to mass (volume) conservation of the fluid. For KdV2 we found only so-called adiabatic invariants, that is, functions of the solutions which are constants neglecting terms of higher order than the order of the equation. In this chapter we present two methods for construction of such invariants. The first method, a direct one, consists in using constructions of higher KdV invariants and eliminating non-integrable terms in an approximate way. The second method introduces a near-identity transformation (NIT) which transforms KdV2 into equation (asymptotically equivalent) which is integrable. For the equation obtained by NIT, exact invariants exist, but they become approximate (adiabatic) when the inverse NIT transformation is applied and original variables are restored. Numerical tests of the exactness of adiabatic invariants for KdV2 in several cases of initial conditions are presented. These tests confirm that relative changes in these approximate invariants are small indeed. The relations of KdV invariants and KdV2 adiabatic invariants to conservation laws are discussed, as well.