Physical mechanisms of the rogue wave phenomenon

A review of physical mechanisms of the rogue wave phenomenon is given. The data of marine observations as well as laboratory experiments are briefly discussed. They demonstrate that freak waves may appear in deep and shallow waters. Simple statistical analysis of the rogue wave probability based on the assumption of a Gaussian wave field is reproduced. In the context of water wave theories the probabilistic approach shows that numerical simulations of freak waves should be made for very long times on large spatial domains and large number of realizations. As linear models of freak waves the following mechanisms are considered: dispersion enhancement of transient wave groups, geometrical focusing in basins of variable depth, and wave-current interaction. Taking into account nonlinearity of the water waves, these mechanisms remain valid but should be modified. Also, the influence of the nonlinear modulational instability (Benjamin–Feir instability) on the rogue wave occurence is discussed. Specific numerical simulations were performed in the framework of classical nonlinear evolution equations: the nonlinear Schrödinger equation, the Davey–Stewartson system, the Korteweg–de Vries equation, the Kadomtsev–Petviashvili equation, the Zakharov equation, and the fully nonlinear potential equations. Their results show the main features of the physical mechanisms of rogue wave phenomenon.     

Nonlinear dispersive waves in a relaxing medium

The dispersion of nonlinear waves in a relaxing medium is analysed by making use of the evolution equations for longitudinal waves. The dispersion relations are obtained and the behaviour of the waves compared to those that arise when they are governed by the well-known Korteweg-de Vries (KdV) and Benjamin-Bona-Mahony (BBM) equations that describe unidirectional motion and also by the time regularized long wave (TRLW) equation that describes bi-directional motion. The nonlinear steady wave solutions are obtained. The general mathematical model used throughout this paper is obtained by the theory of nonlinear elasticity with weak relaxation effects (standard viscoelasticity). A further generalization using a four-element model is also discussed briefly.     

Soliton-like excitations in the one-dimensional electrical transmission line

The dynamics of modulated waves are studied in the one-dimensional discrete nonlinear electrical transmission line. The contribution of the linear dispersive capacitance is taken into account, and it is shown via the reductive perturbation method that the evolution of such waves in this system is governed by the higher-order nonlinear Schrödinger equation. Passing through the Stokes analysis, we establish a generalized criterion for the Benjamin-Feir instability in the network and determine the exact solutions of the obtained wave equation by using the Pathria-Morris approach.     

Are waves with negative spatial damping unstable?

Conventional plane harmonic waves decay in direction of propagation, but unconventional harmonic waves grow in the direction of propagation. While a single unconventional wave cannot be a solution to a physically meaningful boundary value problem, these waves may have an essential contribution to the overall solution of a problem as long as this is a superposition of unconventional and conventional waves. A fourth order diffusion equation with proper thermodynamic structure, and the Burnett equations of rarefied gas dynamics exhibit conventional and unconventional waves. Steady state oscillating boundary value problems are considered to discuss the interplay of conventional and unconventional waves. Results show that as long as the second law of thermodynamics is valid, unconventional waves may contribute to the overall solution, which, however is dominated by conventional waves, and behaves as these.     

Long time interaction of envelope solitons and freak wave formations

This paper concerns long time interaction of envelope solitary gravity waves propagating at the surface of a two-dimensional deep fluid in potential flow. Fully nonlinear numerical simulations show how an initially long wave group slowly splits into a number of solitary wave groups. In the example presented, three large wave events are formed during the evolution. They occur during a time scale that is beyond the time range of validity of simplified equations like the nonlinear Schrödinger (NLS) equation or modifications of this equation. A Fourier analysis shows that these large wave events are caused by significant transfer to side-band modes of the carrier waves. Temporary downshiftings of the dominant wavenumber of the spectrum coincide with the formation large wave events. The wave slope at maximal amplifications is about three times higher than the initial wave slope. The results show how interacting solitary wave groups that emerge from a long wave packet can produce freak wave events.Our reference numerical simulation are performed with the fully nonlinear model of Clamond and Grue [D. Clamond, J. Grue, A fast method for fully nonlinear water wave computations, J. Fluid Mech. 447 (2001) 337–355]. The results of this model are compared with that of two weakly nonlinear models, the NLS equation and its higher-order extension derived by Trulsen et al. [K. Trulsen, I. Kliakhandler, K.B. Dysthe, M.G. Velarde, On weakly nonlinear modulation of waves on deep water, Phys. Fluids 12 (10) (2000) 2432–2437]. They are also compared with the results obtained with a high-order spectral method (HOSM) based on the formulation of West et al. [B.J. West, K.A. Brueckner, R.S. Janda, A method of studying nonlinear random field of surface gravity waves by direct numerical simulation, J. Geophys. Res. 92 (C11) (1987) 11 803–11 824]. An important issue concerning the representation and the treatment of the vertical velocity in the HOSM formulation is highlighted here for the study of long-time evolutions.     

On the interaction of cubically nonlinear transverse plane waves in an elastic material

The paper presents theoretical results on the interaction of cubically nonlinear harmonic elastic plane waves in a nonlinear material described by the Murnaghan potential. The interaction of two harmonic transverse waves is studied using the method of slowly varying amplitude. Reduced and evolution equations and the Manley-Rowe relations are derived. An analysis is made of the mechanism of energy transfer from the strong pumping wave, which has frequency ω, to the weak signal wave, which has frequency 3ω because of this interaction. A switching mechanism for hypersonic waves in a nonlinear elastic material is described, which is similar to the switching mechanism observed in transistors __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 6, pp. 61–70, June 2006.     

Nonlinear waves in a viscous compressible fluid with relaxation

The propagation and stability of nonlinear waves in a viscous compressible fluid with relaxation that satisfies a Theological equation of state of Oldroyd type are investigated. An equation that describes the structure of the wave perturbations and its evolution is derived subject to the condition of balance of the nonlinear dissipative and relaxation effects, and its solutions of the solitary wave type are analyzed.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No. 3, pp. 31–35, May–June, 1993.     

An Initial-Value Problem for Fully Three-Dimensional Inflectional Boundary Layer flows

In a recent paper the present authors considered the effects of small cross-flow upon the nonlinear evolution of two oblique waves of unequal amplitude. The present analysis extends this work in two ways: (i) the cross-flow is increased to O(1) making it of comparable magnitude to the streamwise component and (ii) by taking the long-wavelength limit of Rayleigh's equation a whole spectrum of wave numbers can now be catered for. Thus (ii) allows a fairly general initial disturbance to be accommodated. Another significant effect due to the presence of a whole spectrum of wave numbers is that the critical layer jump is now forced by a quadratic, as opposed to the usual cubic, nonlinearity. Numerical solutions of the new nonlinear amplitude evolution equation are presented for a special class of initial disturbance. Received 7 January 1998 and accepted 15 September 1998     

Thermal Fatigue Damage Assessment in an Isotropic Pipe Using Nonlinear Ultrasonic Guided Waves

The use of nonlinear ultrasonic waves has been accepted as a potential technique to characterize the state of material micro-structure in solids. The typical nonlinear phenomenon is generation of second harmonics. Second harmonic generation of ultrasonic waves propagation has been vigorously studied for tracking material micro-damages in unbounded media and plate-like waveguides. However, there are few studies of launching second harmonic guided wave propagation in tube-like structures. Considering that second harmonics could provide useful information sensitive for material degradation condition, this research aims at developing a procedure for detecting second harmonics of ultrasonic guided wave in an isotropic pipe. The second harmonics generation of guided wave propagation in an isotropic and stress-free elastic pipe is investigated. Flexible polyvinylidene fluoride (PVDF) comb transducers are used to measure fundamental wave and second harmonic one. Experimental results show that nonlinear parameters increase monotonically with propagation distance. This work experimentally verifies that the second harmonics of guided waves in pipe have the cumulative effect with propagation distance. The proposed procedure is applied to assessing thermal fatigue damage indicated by nonlinearity in an aluminum pipe. The experimental observation verifies that nonlinear guided waves can be used to assess damage levels in early thermal fatigue state by correlating them with the acoustic nonlinearity.     

Multiple scales analysis of wave�Cwave interactions in?a?cubically nonlinear monoatomic chain

The interaction of waves in nonlinear media is of practical interest in the design of acoustic devices such as waveguides and filters. This investigation of the monoatomic mass?Cspring chain with a cubic nonlinearity demonstrates that the interaction of two waves results in different amplitude and frequency dependent dispersion branches for each wave, as opposed to a single amplitude-dependent branch when only a single wave is present. A theoretical development utilizing multiple time scales results in a set of evolution equations which are validated by numerical simulation. For the specific case where the wavenumber and frequency ratios are both close to 1:3 as in the long wavelength limit, the evolution equations suggest that small amplitude and frequency modulations may be present. Predictable dispersion behavior for weakly nonlinear materials provides additional latitude in tunable metamaterial design. The general results developed herein may be extended to three or more wave?Cwave interaction problems.     

Analysis and computation for nonlinear elastic surface waves of permanent form

Linear elastic surface waves are nondispersive. All wavelengths travel at the Rayleigh wave speed c _R. This absence of frequency dispersion means that nonlinear waves of permanent form cannot be determined as a small perturbation from a sinusoidal wavetrain. By representing the general Rayleigh wave of the linear theory in terms of a pair of conjugate harmonic functions, waves which propagate without distortion are characterized as those having surface elevation profiles which satisfy a certain nonlinear functional equation. In the small-strain limit, this reduces to a quadratic functional equation. Methods for the analysis of this equation are presented for both periodic and nonperiodic waveforms. For periodic waveforms, the infinite system of quadratic equations for the Fourier coefficients of the profile is solved numerically in the case of a certain harmonic elastic material. Two distinct families of profiles having phase speed differing from the linearized Rayleigh wave speed are found. Additionally, two families of exceptional waveforms are found, describing profiles which travel at the Rayleigh wave speed.     

Propagation of acoustic waves and shock waves radiated by a sinusoidal pulsation of a sphere during a single period

A spherical sound wave is emitted by a sphere which executes a small sinusoidal pulsation of a single period at high frequency in an inviscid fluid. Nonlinear propagation of the waves is formulated as an initial boundary value problem and is analysed in detail. The governing equation is linear near the sphere, while it is a nonlinear hyperbolic equation in a far field. The nonlinearity has a significant effect there, leading to the formation of two shocks. The exact solution to match the near field solution can easily be obtained for the far field equation. The nonlinear distortion of waveform and the shock formation distance are evaluated from the representation of the solution with strained coordinates. The evolution and nonlinear attenuation of the two shock discontinuities are also examined by making use of the equal-areas rule. In its asymptotic form the entire profile is an N wave with a long tail.     

Evolution of a random inhomogeneous field of nonlinear deep-water gravity waves

We have derived an equation governing the evolution of a random field of nonlinear, deep-water, gravity waves by extending the approach used by Zakharov [1] for describing the deterministic system. This equation accounts for both the effects of inhomogeneity and the energy transfer mechanism associated with the homogeneous spectrum. The narrow-band limit of this equation is used to study the stability of a random wavetrain to two-dimensional deterministic perturbations. The effect of randomness is found to reduce the growth rate and the extent of the instability.     

Doubly periodic patterns of modulated hydrodynamic waves: Exact solutions of the Davey-Stewartson system

Exact doubly periodic standing wave patterns of the Davey-Stewartson (DS) equations are derived in terms of rational expressions of elliptic functions.In fluid mechanics,DS equations govern the evolution of weakly nonlinear,free surface wave packets when long wavelength modulations in two mutually perpendicular,horizontal directions are incorporated.Elliptic functions with two different moduli (periods) are necessary in the two directions.The relation between the moduli and the wave numbers constitutes the dispersion relation of such waves.In the long wave limit,localized pulses are recovered.     

Two-dimensional modulation and instabilities of flexural waves of a thin plate on nonlinear elastic foundation

In the present paper we intend to examine in detail the formation of localized modes and waves mediated by modulational instability in an elastic structure. The elastic composite structure consists of a nonlinear foundation coated with an elastic thin plate. The problem deals with flexural waves traveling on the plate. The attention is devoted to the behavior of nonlinear waves in the small-amplitude limit in view of deducing criteria of instability which produce localized waves. It is shown that, in the small-amplitude limit, the basic equation which governs the plate deflection is approximated by a two-dimensional nonlinear Schrödinger equation. The latter equation allows us to study the modulational instability conditions leading to different zones of instability. The examination of the instability provides useful information about the possible selection mechanism of the modulus of the carrier wave vector and growth rate of the instabilities taking place in both (longitudinal and transverse) directions of the plate. The mechanism of the self-generated nonlinear waves on the plate beyond the birth of modulational instability is numerically investigated. The numerics show that an initial plane wave is then transformed, through the instability process, into nonlinear localized waves which turn out to be particularly stable. In addition, the influence of the prestress on the nature of localized structures is also examined. At length, in the conclusion some other wave problems and extensions of the work are evoked.     

Suppression of transients in wave-excited response testing

When a floating body in a wave tank has low hydrodynamic damping, for example in the heave mode, very long duration transient responses can arise if it is excited from a state of rest by sinusoidal waves. Such behaviour can be undesirable when steady state response characteristics are the object of investigation in a numerical tank, because of the consequential need for very long computations. The present paper develops a method for suppressing such transient behaviour in computational models. The success of the approach is demonstrated in the context of the heaving motion of a simple buoy. A linear model of such a buoy initially at rest in a wave tank, excited by propagating sinusoidal waves, is used here for a preliminary investigation of the removal of transients. The technique is then incorporated into a fully nonlinear potential flow simulation of the buoy, and the approach is shown to be effective.     

Modulation of an internal gravity wave packet in a stratified shear flow

Modulations of an internal gravity wave packet in a stratified shear flow are discussed in the weakly nonlinear and weakly dispersive context. It is shown that the modulations are described by a variable coefficient nonlinear Schrödinger equation when the modulations are confined to the direction of wave propagation. Transverse modulations couple the nonlinear Schrödinger equation to the mean flow equations. For long waves, it is shown that the modulation equations may be somewhat simplified. An Appendix describes the equations governing long wave resonance.     

Nonlinear Energy Transfer over the Wave Spectrum in Water Covered by Solid Ice

On the basis of the hydrodynamic equations for nonlinear elastic-gravity waves beneath a solid ice cover and their Hamiltonian representation, a three-wave kinetic equation for the time evolution of the wave spectrum is formulated. The properties of the kernel of the kinetic integral describing the nonlinear interactions between wave triplets are investigated. An algorithm for numerically calculating the kinetic integral is developed. The rate of nonlinear energy transfer over the wave spectrum is estimated quantitatively and its most important characteristics are found.     

Evolution of wide-spectrum unidirectional wave groups in a tank: an experimental and numerical study

Evolution of unidirectional nonlinear wave groups with wide spectra is studied experimentally and numerically. As an example of such an evolution, focusing of an initially wide wave train that is modulated both in amplitude and in frequency, to a single steep wave at a prescribed location along the laboratory wave tank is investigated. When numerous frequency harmonics arrive at the focusing location in phase, a very wave steep single emerges. The experimental study was carried out in two wave flumes that differ in size by an order of magnitude: a 330 m long Large Wave Channel in Hanover, and in 18 m long Tel-Aviv University wave tank. The spatial version of the Zakharov equation was applied in the numerical simulations. Detailed quantitative comparison is carried out between the experimental results and the numerical simulations. Spectra of the 2nd order bound waves are calculated using the theoretical model adopted. It is demonstrated that with the contribution of bound waves accounted for, a very good agreement between experiments and simulations is achieved.     

Intense bending waves in a bar

Summary In this paper, the work presented in [1] is extended to study higher-order approximations of nonlinear effects in a bar. It has been found that long bending waves, being the low-frequency modes involved in resonant triads, are stable against small perturbations. Consequently, a bending wave with group velocity which is less than that of longitudinal waves should behave as a linear quasi-harmonic wavetrain. On the other hand, one may expect self-modulation instability of intense bending wavetrains during the long-time evolution. This paper overcomes such a contradiction. To describe the nonlinear dynamics in detail, one should allow for higher-order approximation effects in the model. Such effects are associated with the diffusion of linear wave packets due to different group velocities, and amplitude dispersion caused by nonlinearity. Within the second-order approximation analysis, an amplitude modulation is indeed experienced for intense bending waves. As a result, envelope solitons can be formed from unstable bending wavetrains. The group matching of long longitudinal and short bending waves, being a particular case of the self-modulation, is of special interest as a limit case of the triple-wave resonant interactions. It demonstrates the relation between the first- and the second-order approximation effects. Accepted for publication 20 July 1996