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1
Johnson, R. S. "Edge waves: theories past and present." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 365, no. 1858 (March 13, 2007): 2359–76. http://dx.doi.org/10.1098/rsta.2007.2013.
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The problem of edge waves as an example within classical water-wave theory is described by presenting an overview of some of the theories that have been offered for this phenomenon. The appropriate governing equations and boundary conditions are formulated, and then the important discoveries of Stokes and Ursell, concerning the travelling edge wave, are presented. (We do not address the corresponding problem of standing waves.) Thus, the linear problem and its spectrum are constructed; in addition, we also present the linear long-wave approximation to the problem, as well as Whitham's weakly nonlinear extension to Stokes' original theory. All these discussions are based on the same formulation of the problem, allowing an immediate comparison of the results, whether this be in terms of different approximations or whether the theory be for an irrotational flow or not. Gerstner's exact solution of the water-wave problem is then briefly described, together with a transformation that produces an exact solution of the full equations for the edge wave. The form of this solution is then used as the basis for a multiple-scale description of the edge wave over a slowly varying depth; this leads to a version of the shallow-water equations which has an exact solution that corresponds to the edge wave. Some examples of the theoretical predictions for the run-up pattern are presented. We conclude with three variants of nonlinear model equations that may prove useful in the study of edge waves and, particularly, the interaction of different modes.
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Hardy, Thomas A., and Nicholas C. Kraus. "COUPLING STOKES AND CNOIDAL WAVE THEORIES IN A NONLINEAR REFRACTION MODEL." Coastal Engineering Proceedings 1, no. 21 (January 29, 1988): 42. http://dx.doi.org/10.9753/icce.v21.42.
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An efficient numerical model is presented for calculating the refraction and shoaling of finite-amplitude waves over an irregular sea bottom. The model uses third-order Stokes wave theory in relatively deep water and second-order cnoidal wave theory in relatively shallow water. It can also be run using combinations of lower-order wave theories, including a pure linear wave mode. The problem of the connection of Stokes and cnoidal theories is investigated, and it is found that the use of second-order rather than first-order cnoidal theory greatly reduces the connection discontinuity. Calculations are compared with physical model measurements of the height and direction of waves passing over an elliptical shoal. The finite-amplitude wave model gives better qualitative and quantitative agreement with the measurements than the linear model.
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Zhang, Huichen, and Markus Brühl. "GENERATION OF EXTREME TRANSIENT WAVES IN EXPERIMENTAL MODELS." Coastal Engineering Proceedings, no. 36 (December 30, 2018): 51. http://dx.doi.org/10.9753/icce.v36.waves.51.
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The transfer of natural waves and sea states into small- and large-scale model teste contributes to the proper design of offshore and coastal structure. Such shallow-water ocean surface waves are highly nonlinear and subject to wave transformation and nonlinear wave-wave interactions. However, the standard methods of wave generation according to conventional wave theories and wave analysis methods are limited to simple regular waves, simple sea states and low-order wave generation without considering the nonlinear wave-wave interactions. The research project Generation of Extreme Transient Waves in Experimental Models (ExTraWaG) aims to accurately generate target transient wave profile at a pre-defined position in the wave flume (transfer point) under shallow water conditions. For this purpose, the KdV-based nonlinear Fourier transform is introduced as a continuative wave analysis method and is applied to investigate the nonlinear spectral character of experimental wave data. Furthermore, the method is applied to generate transient nonlinear waves as specific locations in the wave flume, considering the nonlinear transformation and interactions of the propagating waves.
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Eldrup, Mads Røge, and Thomas Lykke Andersen. "Applicability of Nonlinear Wavemaker Theory." Journal of Marine Science and Engineering 7, no. 1 (January 14, 2019): 14. http://dx.doi.org/10.3390/jmse7010014.
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Generation of high-quality waves is essential when making numerical or physically model tests. When using a wavemaker theory outside the validity area, spurious waves are generated. In order to investigate the validity of different wave generation methods, new model test results are presented where linear and nonlinear wave generation theories are tested on regular and irregular waves. A simple modification to the second-order wavemaker theory is presented, which significantly reduces the generation of spurious waves when used outside its range of applicability. For highly nonlinear regular waves, only the ad-hoc unified wave generation based on stream function wave theory was found acceptable. For irregular waves, similar conclusions are drawn, but the modified second-order wavemaker method is more relevant. This is because the ad-hoc unified generation method for irregular waves requires the wave kinematics to be calculated by a numerical model, which might be quite time-consuming. Finally, a table is presented with the range of applicability for each wavemaker method for regular and irregular waves.
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RAVAL, ASHISH, XIANYUN WEN, and MICHAEL H. SMITH. "Numerical simulation of viscous, nonlinear and progressive water waves." Journal of Fluid Mechanics 637 (September 23, 2009): 443–73. http://dx.doi.org/10.1017/s002211200999070x.
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A numerical simulation is performed to study the velocity, streamlines, vorticity and shear stress distributions in viscous water waves with different wave steepness in intermediate and deep water depth when the average wind velocity is zero. The numerical results present evidence of ‘clockwise’ and ‘anticlockwise’ rotation of the fluid at the trough and crest of the water waves. These results show thicker vorticity layers near the surface of water wave than that predicted by the theories of inviscid rotational flow and the low Reynolds number viscous flow. Moreover, the magnitude of vorticity near the free surface is much larger than that predicted by these theories. The analysis of the shear stress under water waves show a thick shear layer near the water surface where large shear stress exists. Negative and positive shear stresses are observed near the surface below the crest and trough of the waves, while the maximum positive shear stress is inside the water and below the crest of the water wave. Comparison of wave energy decay rate in intermediate depth and deep water waves with laboratory and theoretical results are also presented.
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Vakakis, A. F. "Scattering of Structural Waves by Nonlinear Elastic Joints." Journal of Vibration and Acoustics 115, no. 4 (October 1, 1993): 403–10. http://dx.doi.org/10.1115/1.2930364.
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An analytic study of the scattering of structural waves by nonlinear elastic joints is presented. Under the assumption of small nonlinearities and/or amplitudes of motion, an averaging methodology is implemented for analyzing the interaction between an incident wave and a nonlinear joint with symmetric stiffness. It is found that, contrary to the predictions of existing linear theories, a single incident wave gives rise to an infinity of reflected waves with frequencies equal to odd multiples of the frequency of the incident wave. The orders of magnitude of the amplitudes of the various reflected waves are considered, and an application of the theory is made by considering the wave scattering from a joint with cubic stiffness nonlinearity. In addition, it is shown that the wave propagation approach presented in this work can be effectively used for predicting nonlinear free oscillations (standing waves) in finite waveguides with nonlinear joints.
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Liu, C. M., H. H. Hwung, and R. Y. Yang. "The Consistence Between the Stokes Wave Theory and General Wave Theory." Journal of Mechanics 25, no. 3 (September 2009): N17—N20. http://dx.doi.org/10.1017/s172771910000280x.
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AbstractThe consistence between the Stokes wave theory and general wave theory is examined in this study. As well known, the nonlinear terms of Stokes wave are generated by the self-interaction of first-order wave. On the other side, using the general wave theory one can also obtain the nonlinear solutions according to the interaction of n waves with the same amplitude, frequency and phase. It is found that the inconsistence between these two wave trains arises due to the subharmonic effects included in general wave theory but not considered in the Stokes theory. In conclusion, these two theories are substantially different unless the Bernoulli constants are properly chosen for mathematical equivalence.
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Engelbrecht, J., and M. Braun. "Nonlinear Waves in Nonlocal Media." Applied Mechanics Reviews 51, no. 8 (August 1, 1998): 475–88. http://dx.doi.org/10.1115/1.3099016.
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This review article gives a brief overview on nonlocal theories in solid mechanics from the viewpoint of wave motion. The influence of two essential qualities of solids—nonlocality and nonlinearity—is discussed. The effects of microstructure are analyzed in order to understand their role in nonlocal theories. The various models are specified on the level of one-dimensional unidirectional motion in order to achieve mathematical clarity of interpreting physical phenomena. Three main types of evolution equations are shown to govern deformation waves under such assumptions. Based on the dispersion analysis, weak, true, and strong nonlocalities are distinguished. There are 75 references included with this article.
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Elfouhaily, Tanos, D. R. Thompson, D. Vandemark, and B. Chapron. "Truncated Hamiltonian versus surface perturbation in nonlinear wave theories." Waves in Random Media 10, no. 1 (January 2000): 103–16. http://dx.doi.org/10.1088/0959-7174/10/1/308.
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Noonan, Julie, and Roger Smith. "Linear and weakly nonlinear internal wave theories applied to "morning glory" waves." Geophysical & Astrophysical Fluid Dynamics 33, no. 1 (1985): 123–43. http://dx.doi.org/10.1080/03091928508240749.
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Noonan, Julie A., and Roger K. Smith. "Linear and weakly nonlinear internal wave theories applied to “morning glory” waves." Geophysical & Astrophysical Fluid Dynamics 33, no. 1-4 (September 1985): 123–43. http://dx.doi.org/10.1080/03091928508245426.
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Zhu, Ling, and Qin Chen. "Attenuation of Nonlinear Waves by Rigid Vegetation: Comparison of Different Wave Theories." Journal of Waterway, Port, Coastal, and Ocean Engineering 143, no. 5 (September 2017): 04017029. http://dx.doi.org/10.1061/(asce)ww.1943-5460.0000415.
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Miesen, R. H. M. "Applying weakly nonlinear internal wave theories to solitary waves in the atmosphere." Journal of Atmospheric and Terrestrial Physics 54, no. 3-4 (March 1992): 363–72. http://dx.doi.org/10.1016/0021-9169(92)90016-e.
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Zimmerman, W. B., and J. M. Rees. "Long solitary internal waves in stable stratifications." Nonlinear Processes in Geophysics 11, no. 2 (April 14, 2004): 165–80. http://dx.doi.org/10.5194/npg-11-165-2004.
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Abstract. Observations of internal solitary waves over an antarctic ice shelf (Rees and Rottman, 1994) demonstrate that even large amplitude disturbances have wavelengths that are bounded by simple heuristic arguments following from the Scorer parameter based on linear theory for wave trapping. Classical weak nonlinear theories that have been applied to stable stratifications all begin with perturbations of simple long waves, with corrections for weak nonlinearity and dispersion resulting in nonlinear wave equations (Korteweg-deVries (KdV) or Benjamin-Davis-Ono) that admit localized propagating solutions. It is shown that these theories are apparently inappropriate when the Scorer parameter, which gives the lowest wavenumber that does not radiate vertically, is positive. In this paper, a new nonlinear evolution equation is derived for an arbitrary wave packet thus including one bounded below by the Scorer parameter. The new theory shows that solitary internal waves excited in high Richardson number waveguides are predicted to have a halfwidth inversely proportional to the Scorer parameter, in agreement with atmospheric observations. A localized analytic solution for the new wave equation is demonstrated, and its soliton-like properties are demonstrated by numerical simulation.
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Sánchez-Garrido, J. C., and V. Vlasenko. "Long-term evolution of strongly nonlinear internal solitary waves in a rotating channel." Nonlinear Processes in Geophysics 16, no. 5 (September 25, 2009): 587–98. http://dx.doi.org/10.5194/npg-16-587-2009.
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Abstract. The evolution of internal solitary waves (ISWs) propagating in a rotating channel is studied numerically in the framework of a fully-nonlinear, nonhydrostatic numerical model. The aim of modelling efforts was the investigation of strongly-nonlinear effects, which are beyond the applicability of weakly nonlinear theories. Results reveal that small-amplitude waves and sufficiently strong ISWs evolve differently under the action of rotation. At the first stage of evolution an initially two-dimensional ISW transforms according to the scenario described by the rotation modified Kadomtsev-Petviashvili equation, namely, it starts to evolve into a Kelvin wave (with exponential decay of the wave amplitude across the channel) with front curved backwards. This transition is accompanied by a permanent radiation of secondary Poincaré waves attached to the leading wave. However, in a strongly-nonlinear limit not all the energy is transmitted to secondary radiated waves. Part of it returns to the leading wave as a result of nonlinear interactions with secondary Kelvin waves generated in the course of time. This leads to the formation of a slowly attenuating quasi-stationary system of leading Kelvin waves, capable of propagating for several hundreds hours as a localized wave packet.
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Rahman, Matiur, and Lokenath Debnath. "Nonlinear diffraction of water waves by offshore stuctures." International Journal of Mathematics and Mathematical Sciences 9, no. 4 (1986): 625–52. http://dx.doi.org/10.1155/s0161171286000807.
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This paper is concerned with a variational formulation of a nonaxisymmetric water wave problem. The full set of equations of motion for the problem in cylindrical polar coordinates is derived. This is followed by a review of the current knowledge on analytical theories and numerical treatments of nonlinear diffraction of water waves by offshore cylindrical structures. A brief discussion is made on water waves incident on a circular harbor with a narrow gap. Special emphasis is given to the resonance phenomenon associated with this problem. A new theoretical analysis is also presented to estimate the wave forces on large conical structures. Second-order (nonlinear) effects are included in the calculation of the wave forces on the conical structures. A list of important references is also given.
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Yoon, Sung B., and Philip L. F. Liu. "WAVE AND CURRENT INTERACTIONS IN SHALLOW WATER." Coastal Engineering Proceedings 1, no. 20 (January 29, 1986): 125. http://dx.doi.org/10.9753/icce.v20.125.
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Interactions between waves and currents are common and important phenomena in the coastal zone. Coastal currents, such as longshore currents, rip currents, and river flows, can significantly change wave heights and directions of wave propagation. Consequently, the design for shoreline protection measures must be adjusted accordingly. Various theories for wave-current interactions exist and have been reviewed by Peregrine and Jonsson (1983). Most of these theories are developed for large-scale currents where the length-scale for the current variation is much greater than the typical wavelength. These theories cannot be applied to the coastal currents which are small-scale currents. In this paper, the interactions between currents and nonlinear shallow water waves are investigated. Boussinesq equations are used to derive evolution equations for spectral wave components. The current intensity is assumed to be larger than the leading wave orbital velocity and smaller than the group velocity. The length-scale of the current is much shorter than those assumed in the existing large-scale theories. To facilitate numerical computations, the parabolic approximation is applied and a simplified model is developed. A numerical example is given for the refraction and diffraction of cnoidal waves over a rip current on a sloping topography. Both normal and oblique incident cases are examined.
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Willcocks, B. T., and J. G. Esler. "Nonlinear Baroclinic Equilibration in the Presence of Ekman Friction." Journal of Physical Oceanography 42, no. 2 (February 1, 2012): 225–42. http://dx.doi.org/10.1175/jpo-d-11-0112.1.
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Abstract Two theories for the nonlinear equilibration of baroclinic waves in a two-layer fluid in a β channel are tested by comparison with high-resolution numerical simulations. Predictions are tested for a range of parameters (β, κ), where the inverse criticality β measures the degree of instability and the quasigeostrophic Ekman number κ measures the strength of Ekman friction. The first theory, from Warn, Gauthier, and Pedlosky (WGP), is formally valid for marginally unstable waves at κ = 0. The second, from Romea, is formally valid for nonzero κ and for waves that are marginally stable with respect to a different criterion, which enters because of the dissipative destabilization of otherwise stable waves by Ekman friction. The predictions of the two theories are in conflict in the limit κ → 0. When κ is slightly greater than zero, it is found that the WGP accurately predicts the maximum wave amplitude attained during a baroclinic life cycle across a significant range of parameter space. By contrast, accurate predictions of the long-time asymptotic wave amplitude are obtained only from Romea’s theory, even for those cases where WGP describes the initial behavior during the life cycle accurately. The results first indicate the importance of understanding the nonlinear equilibration mechanism of dissipatively destabilized waves. Second, it follows that baroclinic adjustment theories formulated from inviscid and frictionless stability criterion make demonstrably incorrect predictions for the equilibrated state, even in the limit of vanishing Ekman friction.
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Itoi, C., and M. Kato. "Equivalence in spin wave theories." Journal of Physics A: Mathematical and General 27, no. 8 (April 21, 1994): 2915–22. http://dx.doi.org/10.1088/0305-4470/27/8/027.
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Smit, P. B., T. T. Janssen, and T. H. C. Herbers. "Nonlinear Wave Kinematics near the Ocean Surface." Journal of Physical Oceanography 47, no. 7 (July 2017): 1657–73. http://dx.doi.org/10.1175/jpo-d-16-0281.1.
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AbstractEstimation of second-order, near-surface wave kinematics is important for interpretation of ocean surface remote sensing and surface-following instruments, determining loading on offshore structures, and understanding of upper-ocean transport processes. Unfortunately, conventional wave theories based on Stokes-type expansions do not consider fluid motions at levels above the unperturbed fluid level. The usual practice of extrapolating the fluid kinematics from the unperturbed free surface to higher points in the fluid is generally reasonable for narrowband waves, but for broadband ocean waves this results in dramatic (and nonphysical) overestimation of surface velocities. Consequently, practical approximations for random waves are at best empirical and are often only loosely constrained by physical principles. In the present work, the authors formulate the governing equations for water waves in an incompressible and inviscid fluid, using a boundary-fitted coordinate system (i.e., sigma or s coordinates) to derive expressions for near-surface kinematics in nonlinear random waves from first principles. Comparison to a numerical model valid for highly nonlinear waves shows that the new results 1) are consistent with second-order Stokes theory, 2) are similar to extrapolation methods in narrowband waves, and 3) greatly improve estimates of surface kinematics in random seas.
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Engelbrecht, J. "Qualitative Aspects of Nonlinear Wave Motion: Complexity and Simplicity." Applied Mechanics Reviews 46, no. 12 (December 1, 1993): 509–18. http://dx.doi.org/10.1115/1.3120312.
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The nonlinear wave processes possess many qualitative properties which cannot be described by linear theories. In this presentation, an attempt is made to systematize the main aspects of this fascinating area. The sources of nonlinearities are analyzed in order to understand why and how the nonlinear mathematical models are formulated. The technique of evolution equations is discussed then as a main mathematical tool to separate multiwave processes into single waves. The evolution equations give concise but in many cases sufficient description of wave processes in solids permitting to analyze spectral changes, phase changes and velocities, coupling of waves, and interaction of nonlinearities with other physical effects of the same order. Several new problems are listed. Knowing the reasons, the seemingly complex problems can be effectively analyzed.
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Gavrilov, N. M., and S. P. Kshevetskii. "Verifications of the nonlinear numerical model and polarization relations of atmospheric acoustic-gravity waves." Geoscientific Model Development Discussions 7, no. 6 (November 18, 2014): 7805–22. http://dx.doi.org/10.5194/gmdd-7-7805-2014.
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Abstract. Comparisons of amplitudes of wave variations of atmospheric characteristics simulated using direct numerical simulation models with polarization relations given by conventional theories of linear acoustic-gravity waves (AGWs) could be helpful for testing these numerical models. In this study, we performed high-resolution numerical simulations of nonlinear AGW propagation at altitudes 0–500 km from a plane wave forcing at the Earth's surface and compared them with analytical polarization relations of linear AGW theory. After some transition time te (increasing with altitude) subsequent to triggering the wave source, initial wave pulse disappear and the main spectral components of the wave source dominate. The numbers of numerically simulated and analytical pairs of AGW parameters, which are equal with confidence 95%, are largest at altitudes 30–60 km at t > te. At low and high altitudes and at t < te numbers of equal pairs are smaller, because of influence of the lower boundary conditions, strong dissipation and AGW transience making substantial inclinations from conditions, assumed in conventional theories of linear nondissipative stationary AGWs in the free atmosphere. Reasonable agreements between simulated and analytical wave parameters satisfying the scope the limitations of the AGW theory proof adequacy of the used nonlinear numerical model. Significant differences between numerical and analytical AGW parameters reveal circumstances, when analytical theories give substantial errors and numerical simulations of wave fields are required. In addition, direct numerical AGW simulations may be useful tools for testing simplified parameterizations of wave effects in the atmosphere.
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Hattori, Masataro. "EXPERIMENTAL STUDY ON THE VALIDITY RANGE OF VARIOUS WAVE THEORIES." Coastal Engineering Proceedings 1, no. 20 (January 29, 1986): 18. http://dx.doi.org/10.9753/icce.v20.18.
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In the design of coastal structures and the study of nearshore dynamics, rational predictions of the wave kinematics are very important but difficult procedures. Although a large number of nonlinear wave theories have been proposed and used for computing the wave kinematics, there are no theories applicable from the deep water to very shallow water. It is, therefore, very important for coastal researchers and engineers to know which of theories describe well a wave field specified by the wave characteristics and water depth, and to select a particular wave theory for a problem of interest. Many intensive efforts have been made to examine the validity as well as the applicability of various wave theories. However, there are still no well-accepted guidlines for the application range of the wave theories. The validity evaluation of a particular wave theory has been basically made by means of the following two versions: the analytical (mathematical) validity and the experimental (physical) validity. The analytical validity study has been conducted by various researchers (Dean, 1970; Komar, 1976; Horikawa et al., 1977; Swart, 1978) and revealed the degree of mathematical satisification to the governing equations and boundary conditions for each wave theory. The analytical validity study probably tends to show the relative applicability for various wave theories. It does not ensure that the theory describe well laboratory or field phenomena. Based on the analytical validity of various wave theories by Horikawa et al., Isobe (1985) proposed application ranges for the finite amplitude wave theories in terms of the relative water depth and relative wave height. The experimental validity refers to how well the prediction of various wave theories agrees with actual measurements (Dean & Dalrymple, 1984). As the wave shoals, wave form becomes more asymmetrical, especially under high wave conditions of interest to design. Such nonlinearity influences greatly the wave kinematics and it makes difficult to predict readily the wave kinematics by several theories. From a practical viewpoint, it is , therefore, requested to establish the application ranges of available wave theories for shoaling waves.
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Barreto, W., H. P. de Oliveira, and E. L. Rodrigues. "Nonlinear interaction between electromagnetic and gravitational waves: An appraisal." International Journal of Modern Physics D 26, no. 12 (October 2017): 1743017. http://dx.doi.org/10.1142/s0218271817430179.
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Wave propagation of field disturbances is ubiquitous. The electromagnetic and gravitational are cousin theories in which the corresponding waves play a relevant role to understand several related physical aspects. It has been established that small electromagnetic waves can generate gravitational waves and vice versa when scattered by a charged black hole. In the realm of cylindrical spacetimes, we present here a simple nonlinear effect of the conversion of electromagnetic to gravitational waves reflected by the amount of mass extracted from them.
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Ölmez, H. S., and J. H. Milgram. "Nonlinear energy transfer to short gravity waves in the presence of long waves." Journal of Fluid Mechanics 289 (April 25, 1995): 199–226. http://dx.doi.org/10.1017/s0022112095001303.
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Existing theories for calculating the energy transfer rates to gravity waves due to resonant nonlinear interactions among wave components whose lengths are long in comparison to wave elevations have been verified experimentally and are well accepted. There is uncertainty, however, about prediction of energy transfer rates within a set of waves having short to moderate lengths when these are present simultaneously with a long wave whose amplitude is not small in comparison to the short wavelengths. Here we implement both a direct numerical method that avoids small-amplitude approximations and a spectral method which includes perturbations of high order. These are applied to an interacting set of short- to intermediate-length waves with and without the presence of a large long wave. The same cases are also studied experimentally. Experimentally and numerical results are in reasonable agreement with the finding that the long wave does influence the energy transfer rates. The physical reason for this is identified and the implications for computations of energy transfer to short waves in a wave spectrum are discussed.
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Bhushan, S., F. Stern, and L. J. Doctors. "Verification and Validation of URANS Wave Resistance for Air Cushion Vehicles, and Comparison With Linear Theory." Journal of Ship Research 55, no. 04 (September 1, 2011): 249–67. http://dx.doi.org/10.5957/jsr.2011.55.4.249.
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Verification and validation of URANS wave-resistance predictions for straight-ahead and yawed air-cushion vehicles in calm deep and shallow water are performed. The nonlinear and linear theories are compared to explicate their trends for large cushion pressures, water depth, and cushion dimensions, and the nonlinear theory sinkage and trim trends are discussed. The grid-verification study shows monotonically converged solutions with averaged uncertainty of 4% and 10% for straight-ahead motion in deep and shallow water, respectively. URANS predictions agree with the experimental data to within 6% and 9% for straight-ahead deep and shallow water simulations, respectively. The smooth-edged cushion-pressure simulations predict lower resistance than the sharp-edged case, whereas no significant dependence is obtained for Reynolds number and turbulence modeling. URANS predicts attenuation in the resistance secondary hump as the cushion-pressure level increases. On the other hand, the linear theory does not account for the effect of cushion-pressure level. The linear and nonlinear theories compare within 4.5% for static cushion-pressure-to length ratios less than 0.025 and Froude number greater than 0.5 for both deep and shallow water. The nonlinear theory predicts the effect of water depth better than the linear theory, when compared with the experiments. Both the theories agree well in predicting the decrease in resistance with the decrease in cushion width. The nonlinear theory does not show unrealistically large resistance and side force for sharp-edged cushion pressure for yawed cases, as observed in the linear theory. However, both the theories compare well for the resistance and side-force predictions for the smooth-edged cushion pressure, where the results agree within 10% of the deep-water experimental data. The nonlinear theory predictions for the sinkage and trim are in good agreement with the experimental data, but sinkage is overpredicted and trim is underpredicted. URANS wave-elevation patterns display transverse and diverging waves, which compare well with the Kelvin waves for Froude number less than 0.6 and greater than 1.0, respectively. URANS predicts breaking waves for large cushion pressures for Froude number less than 0.6.
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Chian, Abraham C. L. "Nonlinear Temporal Model for Formation of Pulsar Microstructures." International Astronomical Union Colloquium 128 (1992): 356–61. http://dx.doi.org/10.1017/s0002731600155520.
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AbstractA nonlinear plasma model which may account for temporal modulation of pulsar radio pulses is presented. Envelope solitons and envelope nonlinear wave trains can result from the nonlinear interaction of the high-frequency coherent pulsar radiation with the pulsar magnetosphere. Theories of electromagnetic envelope solitons and electromagnetic envelope nonlinear wave trains in electronpositron plasmas are reviewed. The application of this model for observation of pulsar microstructures is discussed.
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Deguchi, Kengo, and Andrew Walton. "Bifurcation of nonlinear Tollmien–Schlichting waves in a high-speed channel flow." Journal of Fluid Mechanics 843 (March 16, 2018): 53–97. http://dx.doi.org/10.1017/jfm.2018.137.
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Plane Poiseuille flow has long served as the simplest testing ground for Tollmien–Schlichting wave instability. In this paper, we provide a comprehensive comparison of equilibrium Tollmien–Schlichting wave solutions arising from new high-resolution Navier–Stokes calculations and the corresponding predictions of various large-Reynolds-number asymptotic theories developed in the last century, such as double-deck theory, viscous nonlinear critical layer theory and strongly nonlinear critical layer theory. In the relatively small to moderate amplitude regime, the theories excellently predict the behaviour of the numerical solutions at Reynolds numbers of order $10^{6}$ and above, whilst for larger amplitudes our computations suggest the need for further asymptotic theories to be developed.
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Bensimon, D., Paul Kolodner, C. M. Surko, Hugh Williams, and V. Croquette. "Competing and coexisting dynamical states of travelling-wave convection in an annulus." Journal of Fluid Mechanics 217 (August 1990): 441–67. http://dx.doi.org/10.1017/s0022112090000799.
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We describe experiments on convection in binary fluid mixtures in a large-aspect-ratio annular container. In this geometry, the convective rolls align radially and travel azimuthally, providing a model of travelling waves in an extended one-dimensional nonlinear dynamical system. Several different stable non-equilibrium states can be produced in this experiment, and the competition between them leads to a wide variety of steady and time-dependent behaviour. The observed spatiotemporal behaviour may shed light on recent theories of the nature of stable nonlinear travelling-wave convection, the pinning of travelling waves, and the creation of spatiotemporal defects.
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Doak, A., T. Gao, J. M. Vanden-Broeck, and J. J. S. Kandola. "Capillary-gravity waves on the interface of two dielectric fluid layers under normal electric fields." Quarterly Journal of Mechanics and Applied Mathematics 73, no. 3 (June 5, 2020): 231–50. http://dx.doi.org/10.1093/qjmam/hbaa009.
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Summary In this article, we consider capillary-gravity waves propagating on the interface of two dielectric fluids under the influence of normal electric fields. The density of the upper fluid is assumed to be much smaller than the lower one. Linear and weakly nonlinear theories are studied. The connection to the results in other limit configurations is discussed. Fully nonlinear computations for travelling wave solutions are achieved via a boundary integral equation method. Periodic waves, solitary waves and generalised solitary waves are presented. The bifurcation of generalised solitary waves is discussed in detail.
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韩, 清鹏. "Comparative Analyses of PP Wave Intervals Based on Nonlinear Chaotic Theories." Applied Physics 02, no. 03 (2012): 72–76. http://dx.doi.org/10.12677/app.2012.23012.
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Pinel, J., and S. Lovejoy. "Atmospheric waves as scaling, turbulent phenomena." Atmospheric Chemistry and Physics Discussions 13, no. 6 (June 5, 2013): 14797–822. http://dx.doi.org/10.5194/acpd-13-14797-2013.
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Abstract. It is paradoxical that while atmospheric dynamics are highly nonlinear and turbulent that atmospheric waves are commonly modelled by linear or weakly nonlinear theories. We postulate that the laws governing atmospheric waves are on the contrary high Reynold's number (Re), emergent laws so that – in common with the emergent high Re turbulent laws – they are also constrained by scaling symmetries. We propose an effective turbulence – wave propagator which corresponds to a fractional and anisotropic extension of the classical wave equation propagator with dispersion relations similar to those of inertial gravity waves (and Kelvin waves) yet with an anomalous (fractional) order Hwav/2. Using geostationary IR radiances, we estimate the parameters finding that Hwav/2 ≈ 0.17 ± 0.04 (the classical value = 2).
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Pinel, J., and S. Lovejoy. "Atmospheric waves as scaling, turbulent phenomena." Atmospheric Chemistry and Physics 14, no. 7 (April 2, 2014): 3195–210. http://dx.doi.org/10.5194/acp-14-3195-2014.
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Abstract. It is paradoxical that, while atmospheric dynamics are highly nonlinear and turbulent, atmospheric waves are commonly modelled by linear or weakly nonlinear theories. We postulate that the laws governing atmospheric waves are in fact high-Reynolds-number (Re), emergent laws so that – in common with the emergent high-Re turbulent laws – they are also constrained by scaling symmetries. We propose an effective turbulence–wave propagator which corresponds to a fractional and anisotropic extension of the classical wave equation propagator, with dispersion relations similar to those of inertial gravity waves (and Kelvin waves) yet with an anomalous (fractional) order Hwav/2. Using geostationary IR radiances, we estimate the parameters, finding that Hwav &approx; 0.17 ± 0.04 (the classical value = 2).
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Leary, Matthew, Curtis Rusch, Zhe Zhang, and Bryson Robertson. "Comparison and Validation of Hydrodynamic Theories for Wave Energy Converter Modelling." Energies 14, no. 13 (July 1, 2021): 3959. http://dx.doi.org/10.3390/en14133959.
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Dynamic Wave Energy Converter (WEC) models utilize a wide variety of fundamental hydrodynamic theories. When incorporating novel hydrodynamic theories into numerical models, there are distinct impacts on WEC rigid body motions, cable dynamics, and final power production. This paper focuses on developing an understanding of the influence several refined hydrodynamic theories have on WEC dynamics, including weakly nonlinear Froude-Krylov and hydrostatic forces, body-to-body interactions, and dynamic cable modelling. All theories have evolved from simpler approaches and are of importance to a wide array of WEC archetypes. This study quantifies the impact these theories have on modelling accuracy through a WEC case study. Theoretical differences are first explored in a regular sea state. Subsequently, numerical validation efforts are performed against field data following wave reconstruction techniques. Comparisons of significance are WEC motion and cable tension. It is shown that weakly nonlinear Froude-Krylov and hydrostatic force calculations and dynamic cable modelling both significantly improve simulated WEC dynamics. However, body-to-body interactions are not found to impact simulated WEC dynamics.
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Rushchitsky, J. J. "Interaction of Waves in Solid Mixtures." Applied Mechanics Reviews 52, no. 2 (February 1, 1999): 35–74. http://dx.doi.org/10.1115/1.3098925.
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The focus of this review article is on analytical procedures and physical effects which are characteristic of the theory of nonlinear and simple waves in materials. Waves are supposed to propagate in composite materials, which are modeled as solid two-phase mixtures. It is shown how procedures of wave interaction investigations in nonlinear acoustics, optics and radiophysics are applied to nonlinear mechanics of materials with a microstructure. Main effects of the interactions of waves in composite materials: new harmonics generation, self-generation, evolution and distortion, synchronization, breakdown instability, etc are commented upon. This article is proposed not only for specialists in wave theories; therefore it contains some facts which are obvious for researchers working in the field of waves. Many portions of this review are described in more detail in a book (Rushchitsky and Tsurpal (1998), 377 pages). This review article contains 286 references.
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Barut, A. O., and P. Rusu. "On the wave-particle-like solutions of nonlinear equations." Canadian Journal of Physics 67, no. 2-3 (February 1, 1989): 100–105. http://dx.doi.org/10.1139/p89-015.
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The problem of interpreting moving soliton solutions of some nonlinear equations as quantum-like particles is critically analyzed. Purely classical equations (sine-Gordon) as well as nonlinear quantum equations (Schrödinger, Dirac, Kelin–Gordon) are investigated. Three different definitions of energy associated with these equations are considered, and the functional relations replacing the de Broglie postulate, [Formula: see text], are explicitly given. Our analysis shows that fundamental problems generated by the nonlinear approach to quantum mechanics are still unsolved, and these theories have to be considered cautiously.
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Kânoğlu, Utku, Vasily V. Titov, Baran Aydın, Christopher Moore, Themistoklis S. Stefanakis, Hongqiang Zhou, Michael Spillane, and Costas E. Synolakis. "Focusing of long waves with finite crest over constant depth." Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences 469, no. 2153 (May 8, 2013): 20130015. http://dx.doi.org/10.1098/rspa.2013.0015.
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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.
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Yao, L. S., and S. Ghosh Moulic. "Taylor-Couette Instability With a Continuous Spectrum." Journal of Applied Mechanics 62, no. 4 (December 1, 1995): 915–23. http://dx.doi.org/10.1115/1.2896022.
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Nonlinear evolution of a continuous spectrum of unstable waves near the first bifurcation point in circular Couette flow has been investigated. The disturbance is represented by a Fourier integral over all possible axial wave numbers, and an integrodif-ferential equation for the amplitude-density function of a continuous spectrum is derived. The equations describing the evolution of monochromatic waves and slowly varying wave packets of classical weakly nonlinear instability theories are shown to be special limiting cases. Numerical integration of the integrodifferential equation shows that the final equilibrium state depends on the initial disturbance, as observed experimentally, and it is not unique. In all cases, the final equilibrium state consists of a single dominant mode and its harmonics of smaller amplitudes. The predicted range of wave numbers for stable supercritical Taylor vortices is found to be narrower than the span of the neutral curve from linear theory. Taylor-vortex flows with wave numbers outside this range are found to be unstable and to decay, but to excite another wave inside the narrow band. This result is in agreement with the Eckhaus and Benjamin-Feir sideband instability. The results also show that a linearly stable long wave can excite a short unstable wave through nonlinear wave interaction. An important implication of the existence of nonunique equilibrium states is that the torque induced by the fluid motion cannot be determined uniquely. The numerical results show that the uncertainty, associated with nonuniqueness, of using any accurately measured Taylor-vortex torque slightly above the first bifurcation point in engineering practice can be as large as ten percent. The presence of multiple solutions at a fixed Reynolds number for a given geometry in Taylor-Couette flows has been known since Coles’ monumental contribution in 1965. A theoretical confirmation has come only 30 years later. It is worthwhile to point out that the existence of multiple solutions, found by Coles, differs from current popular bifurcation theories. The current study indicates that the state of flows on a stable bifurcation branch can involve any wave number within a finite band and can not be determined uniquely. The multiple solutions in Coles’ sense have also been found for mixed-convection flows (Yao and Ghosh Moulic, 1993, 1994) besides the Taylor-Couette flows. We believe that the nonuniqueness of Coles sense, which complements the bifurcation theories, is a generic property for all fluid flows.
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Lo, Edmond, and Chiang C. Mei. "A numerical study of water-wave modulation based on a higher-order nonlinear Schrödinger equation." Journal of Fluid Mechanics 150 (January 1985): 395–416. http://dx.doi.org/10.1017/s0022112085000180.
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In existing experiments it is known that the slow evolution of nonlinear deep-water waves exhibits certain asymmetric features. For example, an initially symmetric wave packet of sufficiently large wave slope will first lean forward and then split into new groups in an asymmetrical manner, and, in a long wavetrain, unstable sideband disturbances can grow unequally to cause an apparent downshift of carrier-wave frequency. These features lie beyond the realm of applicability of the celebrated cubic Schrödinger equation (CSE), but can be, and to some extent have been, predicted by weakly nonlinear theories that are not limited to slowly modulated waves (i.e. waves with a narrow spectral band). Alternatively, one may employ the fourth-order equations of Dysthe (1979), which are limited to narrow-banded waves but can nevertheless be solved more easily by a pseudospectral numerical method. Here we report the numerical simulation of three cases with a view to comparing with certain recent experiments and to complement the numerical results obtained by others from the more general equations.
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Guedes Soares, C., and T. E. Schellin. "Nonlinear Effects on Long-Term Distributions of Wave-Induced Loads for Tankers." Journal of Offshore Mechanics and Arctic Engineering 120, no. 2 (May 1, 1998): 65–70. http://dx.doi.org/10.1115/1.2829525.
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A method of long-term formulation of the nonlinear wave-induced vertical load effects on ships was applied to three tanker hulls of different sizes. For large tanker hulls, the nonlinear effect is not significant, and thus linear theories can continue to be used for earlier studies on these kind of ships, contrary to what was shown earlier for containership hulls. However, for smaller tankers, significant nonlinear values were obtained, with both sagging and hogging nonlinear results being larger than the linear ones.
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Hara, Tetsu, and Chiang C. Mei. "Frequency downshift in narrowbanded surface waves under the influence of wind." Journal of Fluid Mechanics 230 (September 1991): 429–77. http://dx.doi.org/10.1017/s002211209100085x.
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It is well known that the spectral peak of wind-induced gravity waves on the sea surface tends to shift to lower frequencies as the fetch increases. In past theories the nonlinear dynamics subsequent to Benjamin–Feir instability has been found to initiate the downshift in narrow-banded waves in the absence of wind. However, these weakly nonlinear theories all predict the downshift to be only the first phase of an almost cyclic process. Limited by the length of a wave tank, existing experiments are usually made with relatively steep waves which often break. Although there is a theory on how breaking adds dissipation to stop the reversal of the initial trend of downshift, the details of breaking must be crudely characterized by semi-empirical hypotheses.Since the direct role of wind itself must be relevant to the entire development of wind-wave spectrum, we examine here the effect of wind on the nonlinear evolution of unstable sidebands in narrow-banded waves. We assume that the waves do not break and consider the case where the nonlinear effects that initiate the downshift, energy input by wind and damping by internal dissipation all occur on the same timescale. This means that not only must the waves be mild but the wind stress intensity must also lie within a certain narrow range. With these limitations we couple the air flow above the waves with Dysthe's extension of the cubic Schrödinger equation, and examine the initial as well as the long-time evolution of a mechanically generated wavetrain. For a variety of wind intensities, downshift is indeed found to be enhanced and rendered long lasting.
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Bertolotti, F. P., Th Herbert, and P. R. Spalart. "Linear and nonlinear stability of the Blasius boundary layer." Journal of Fluid Mechanics 242 (September 1992): 441–74. http://dx.doi.org/10.1017/s0022112092002453.
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Two new techniques for the study of the linear and nonlinear instability in growing boundary layers are presented. The first technique employs partial differential equations of parabolic type exploiting the slow change of the mean flow, disturbance velocity profiles, wavelengths, and growth rates in the streamwise direction. The second technique solves the Navier–Stokes equation for spatially evolving disturbances using buffer zones adjacent to the inflow and outflow boundaries. Results of both techniques are in excellent agreement. The linear and nonlinear development of Tollmien–Schlichting (TS) waves in the Blasius boundary layer is investigated with both techniques and with a local procedure based on a system of ordinary differential equations. The results are compared with previous work and the effects of non-parallelism and nonlinearly are clarified. The effect of nonparallelism is confirmed to be weak and, consequently, not responsible for the discrepancies between measurements and theoretical results for parallel flow. Experimental uncertainties, the adopted definition of the growth rate, and the transient initial evolution of the TS wave in vibrating-ribbon experiments probably cause the discrepancies. The effect of nonlinearity is consistent with previous weakly nonlinear theories. White nonlinear effects are small near branch I of the neutral curve, they are significant near branch II and delay or event prevent the decay of the wave.
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Jiao, Chongqing, and Jirun Luo. "Linear and nonlinear theories of a large-orbit gyrotron traveling wave amplifier." Physics of Plasmas 17, no. 5 (May 2010): 054503. http://dx.doi.org/10.1063/1.3400230.
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Malara, F., and M. Velli. "Wave-Based Heating Mechanisms for the Solar Corona." International Astronomical Union Colloquium 144 (1994): 443–51. http://dx.doi.org/10.1017/s025292110002577x.
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AbstractDissipation of MHD waves generated in the lower solar atmosphere has long been proposed as a means to heat the solar corona. Because of the extremely low dissipation coefficients of the coronal plasma large gradients are necessary to efficiently dissipate such waves. Interactions with the inhomogeneities of the background medium may represent a way to create small scale structures, phase-mixing and resonant absorption being important examples. The generalization of such ideas to propagation in complex geometries (e.g., containing X type neutral points) and the extension to nonlinear effects are paramount to the development of wave-heating theories.
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Mockutė, Agota, Enzo Marino, Claudio Lugni, and Claudio Borri. "Comparison of Nonlinear Wave-Loading Models on Rigid Cylinders in Regular Waves." Energies 12, no. 21 (October 23, 2019): 4022. http://dx.doi.org/10.3390/en12214022.
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Monopiles able to support very large offshore wind turbines are slender structures susceptible to nonlinear resonant phenomena. With the aim to better understand and model the wave-loading on these structures in very steep waves where ringing occurs and the numerical wave-loading models tend to lose validity, this study investigates the distinct influences of nonlinearities in the wave kinematics and in the hydrodynamic loading models. Six wave kinematics from linear to fully nonlinear are modelled in combination with four hydrodynamic loading models from three theories, assessing the effects of both types of nonlinearities and the wave conditions where each type has stronger influence. The main findings include that the nonlinearities in the wave kinematics have stronger influence in the intermediate water depth, while the choice of the hydrodynamic loading model has larger influence in deep water. Moreover, finite-depth FNV theory captures the loading in the widest range of wave and cylinder conditions. The areas of worst prediction by the numerical models were found to be the largest steepness and wave numbers for second harmonic, as well as the vicinity of the wave-breaking limit, especially for the third harmonic. The main cause is the non-monotonic growth of the experimental loading with increasing steepness due to flow separation, which leads to increasing numerical overpredictions since the numerical wave-loading models increase monotonically.
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Ardhuin, Fabrice, and T. H. C. Herbers. "Noise generation in the solid Earth, oceans and atmosphere, from nonlinear interacting surface gravity waves in finite depth." Journal of Fluid Mechanics 716 (January 25, 2013): 316–48. http://dx.doi.org/10.1017/jfm.2012.548.
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AbstractOceanic pressure measurements, even in very deep water, and atmospheric pressure or seismic records, from anywhere on Earth, contain noise with dominant periods between 3 and 10 s, which is believed to be excited by ocean surface gravity waves. Most of this noise is explained by a nonlinear wave–wave interaction mechanism, and takes the form of surface gravity waves, acoustic or seismic waves. Previous theoretical work on seismic noise focused on surface (Rayleigh) waves, and did not consider finite-depth effects on the generating wave kinematics. These finite-depth effects are introduced here, which requires the consideration of the direct wave-induced pressure at the ocean bottom, a contribution previously overlooked in the context of seismic noise. That contribution can lead to a considerable reduction of the seismic noise source, which is particularly relevant for noise periods larger than 10 s. The theory is applied to acoustic waves in the atmosphere, extending previous theories that were limited to vertical propagation only. Finally, the noise generation theory is also extended beyond the domain of Rayleigh waves, giving the first quantitative expression for sources of seismic body waves. In the limit of slow phase speeds in the ocean wave forcing, the known and well-verified gravity wave result is obtained, which was previously derived for an incompressible ocean. The noise source of acoustic, acoustic-gravity and seismic modes are given by a mode-specific amplification of the same wave-induced pressure field near zero wavenumber.
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Casciola, C. M., and M. Landrini. "Nonlinear long waves generated by a moving pressure disturbance." Journal of Fluid Mechanics 325 (October 25, 1996): 399–418. http://dx.doi.org/10.1017/s0022112096008178.
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The evolution of long waves generated by a pressure disturbance acting on an initially unperturbed free surface in a channel of finite depth is analysed. Both off-critical and transcritical conditions are considered in the context of the fully nonlinear inviscid problem. The solution is achieved by using an accurate boundary integral approach and a time-stepping procedure for the free-surface dynamics.The discussion emphasizes the comparison between the present results and those provided by both the Boussinesq and the related Korteweg–de Vries model. For small amplitudes of the forcing, the predictions of the asymptotic theories are essentially confirmed. However, for finite intensities of the disturbance, several new features significantly affect the physical results. In particular, the interaction among different wave components, neglected in the Korteweg–de Vries approximation, is crucial in determining the evolution of the wave system. A substantial difference is indeed observed between the solutions of the Korteweg–de Vries equation and those of both the fully nonlinear and the Boussinesq model. For increasing dispersion and fixed nonlinearity the agreement between the Boussinesq and fully nonlinear description is lost, indicating a regime where dispersion becomes dominant.Consistently with the long-wave modelling, the transcritical regime is characterized by an unsteady flow and a periodic emission of forward-running waves. However, also in this case, quantitative differences are observed between the three models. For larger amplitudes, wave steepening is almost invariably observed as a precursor of the localized breaking commonly detected in the experiments. The process occurs at the crests of either the trailing or the upstream-emitted wave system for Froude numbers slightly sub- and super-critical respectively.
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Marchant, T. R., and A. J. Roberts. "Reflection of nonlinear deep-water waves incident onto a wedge of arbitrary angle." Journal of the Australian Mathematical Society. Series B. Applied Mathematics 32, no. 1 (July 1990): 61–96. http://dx.doi.org/10.1017/s0334270000008213.
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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.
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SHUGAN, IGOR, and KONSTANTIN VOLIAK. "On phase kinks, negative frequencies, and other third-order peculiarities of modulated surface waves." Journal of Fluid Mechanics 368 (August 10, 1998): 321–38. http://dx.doi.org/10.1017/s0022112098001803.
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Numerous laboratory and field experiments on nonlinear surface wave trains propagating in deep water (Lake & Yuen 1978; Ramamonjiarisoa & Mollo-Christensen 1979; Mollo-Christensen & Ramamonjiarisoa 1982; Melville 1983) have showed a specific wave modulation that so far has not been explained by nonlinear theories. Typical effects were the so-called wave phase reversals, negative frequencies, and crest pairing, experimentally observed in some portions of the modulated wave train. In the present paper, in order to explain these modulation manifestations, the equations for wavenumber, frequency, and velocity potential amplitude are derived consistently in the third-order approximation related to the wave steepness. The resulting model generalizes, for instance, the well-known nonlinear Schrödinger equation theory, to which it transforms at certain values of the governing parameters.The stationary solutions to the derived set of equations are found in quadrature and then analysed. Within well-defined ranges of the model parameters, these solutions explicitly manifest the above-mentioned wave modulation effects. In particular, they show the wave phase kinks to arise on areas of relatively small free-surface displacement in complete accordance with the experiments.The model with deeply modulated wavenumber and frequency permits one also to analyse the appropriately short surface wavepackets and modulation periods. In this case, a variety of new interesting wave solutions arises revealing complicated alteration of smooth and rough portions of the free surface. Of special importance are solitary waves, naturally generalizing envelope solitons of the nonlinear Schrödinger equation, but having a varying frequency (as a principle of the proposed theory) and a non-zero wave ‘pedestal’ at infinity. These new types of modulated surface waves should be also observable in laboratory tanks and under field conditions, because the relevant free parameters of theory are not extreme.
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Gavrilov, N. M., S. P. Kshevetskii, and A. V. Koval. "Verifications of the high-resolution numerical model and polarization relations of atmospheric acoustic-gravity waves." Geoscientific Model Development 8, no. 6 (June 22, 2015): 1831–38. http://dx.doi.org/10.5194/gmd-8-1831-2015.
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Abstract. Comparisons of amplitudes of wave variations of atmospheric characteristics obtained using direct numerical simulation models with polarization relations given by conventional theories of linear acoustic-gravity waves (AGWs) could be helpful for testing these numerical models. In this study, we performed high-resolution numerical simulations of nonlinear AGW propagation at altitudes 0–500 km from a plane wave forcing at the Earth's surface and compared them with analytical polarization relations of linear AGW theory. After some transition time te (increasing with altitude) subsequent to triggering the wave source, the initial wave pulse disappears and the main spectral components of the wave source dominate. The numbers of numerically simulated and analytical pairs of AGW parameters, which are equal with confidence of 95 %, are largest at altitudes 30–60 km at t > te. At low and high altitudes and at t < te, numbers of equal pairs are smaller, because of the influence of the lower boundary conditions, strong dissipation and AGW transience making substantial inclinations from conditions, assumed in conventional theories of linear nondissipative stationary AGWs in the free atmosphere. Reasonable agreements between simulated and analytical wave parameters satisfying the scope of the limitations of the AGW theory prove the adequacy of the used wave numerical model. Significant differences between numerical and analytical AGW parameters reveal circumstances when analytical theories give substantial errors and numerical simulations of wave fields are required. In addition, direct numerical AGW simulations may be useful tools for testing simplified parameterizations of wave effects in the atmosphere.