Статті в журналах: "Nonlinear water waves"

1

Horikawa, K., H. Maruo, and A. D. D. Craik. "Nonlinear Water Waves." Journal of Applied Mechanics 56, no. 2 (June 1, 1989): 487. http://dx.doi.org/10.1115/1.3176115.

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2

Constantin, Adrian. "Nonlinear water waves." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 370, no. 1964 (April 13, 2012): 1501–4. http://dx.doi.org/10.1098/rsta.2011.0594.

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3

Zhang, Yao, Andrew Brian Kennedy, Joannes Westerink, Nishant Panda, and Client Dawson. "NEW BOUSSINESQ SYSTEM FOR NONLINEAR WATER WAVES." Coastal Engineering Proceedings 1, no. 33 (October 12, 2012): 4. http://dx.doi.org/10.9753/icce.v33.waves.4.

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In this paper, a new Boussinesq water wave theory is derived which can simulate highly dispersive nonlinear waves, their depth-varying velocities, and wave-induced currents, from very deep, but still finite, depths through the surf zone to the shoreline.. Boussinesq scaling is employed. We removed the irrotationality assumption by using polynomial basis functions for velocity profile which are inserted into basic equations of motion. Keep terms up to the desired approximation level and solve the coupled weighted residual system together with vertically integrated mass equation. The computational cost is similar to normal Boussinesq theories although there are more unknown variables to be solved than that in normal Boussinesq models. Because we can reduce the number of the coupled equations by multiplying some coefficients and subtracting from each other which means the matix to be solved is in similar size as normal Boussinesq models. The models show rapid convergence to exact solutions for linear dispersion, shoaling, and orbital velocities; however, properties may be simultaneously and substantially improved for a given order of approximation using asymptotic rearrangements. This improvement is accomplished using the large numbers of degrees of freedom inherent in the definitions of the polynomial basis functions either to match additional terms in a Taylor series, or to minimize errors over a range. Future work will be focused on rotational performance in 2D model by including viscosity,breaking and turbulence.

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4

Kartashova, Elena. "Nonlinear resonances of water waves." Discrete & Continuous Dynamical Systems - B 12, no. 3 (2009): 607–21. http://dx.doi.org/10.3934/dcdsb.2009.12.607.

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5

Wu, Theodore Yaotsu. "Modeling Nonlinear Dispersive Water Waves." Journal of Engineering Mechanics 125, no. 7 (July 1999): 747–55. http://dx.doi.org/10.1061/(asce)0733-9399(1999)125:7(747).

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6

Scheffner, Norman W. "BIPERIODIC WAVES IN SHALLOW WATER." Coastal Engineering Proceedings 1, no. 20 (January 29, 1986): 55. http://dx.doi.org/10.9753/icce.v20.55.

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The propagation of waves in shallow water is a phenomenon of significant practical importance. The ability to realistically predict the complex wave characteristics occurring in shallow water regions has always been an engineering goal which would make the development of solutions to practical engineering problems a reality. The difficulty in making such predictions stems from the fact that the equations governing the complex three-dimensional flow regime can not be solved without linearizing the problem. The linear equations are solvable; however, their solutions do not reflect the nonlinear features of naturally occurring waves. A recent advance (1984) in nonlinear mathematics has resulted in an explicit solution to a nonlinear equation relevant to water waves in shallow water. This solution possesses features found in observed nonlinear three-dimensional wave fields. The nonlinear mathematical formulation referred to above has never been compared with actual waves, so that its practical value is unknown. The purpose of the present investigation was to physically generate three-dimensional nonlinear waves and compare these with exact mathematical solutions. The goals were successfully completed by first generating the necessary wave patterns with the new U.S. Army Engineer Waterways Experiment Station, Coastal Engineering Research Center's (CERC) directional spectral wave generation facility. The theoretical solutions were then formed through the determination of a unique correspondence between the free parameters of the solution and the physical characteristics of the generated wave.

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7

Peregrine, D. H., D. Skyner, M. Stiassnie, and N. Dodd. "NONLINEAR EFFECTS ON FOCUSSED WATER WAVES." Coastal Engineering Proceedings 1, no. 21 (January 29, 1988): 54. http://dx.doi.org/10.9753/icce.v21.54.

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A brief account is given of theory and experiments for water wave focussing. The theory uses weakly nonlinear wave modulation theory, that is the nonlinear Schrodinger equation, summarises earlier theoretical papers and augments them with numerical results. Experiments were performed to compare with theory. The limited comparison shown here indicates that the theory gives satisfactory results even for waves close to breaking. Both the numerical and experimental results indicate the importance of linear diffraction when waves are focussed. The relevance of diffraction is easily assessed, and is likely to dominate in many coastal examples of weak focussing.

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8

Constantin, A. "Nonlinear water waves: introduction and overview." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2111 (December 11, 2017): 20170310. http://dx.doi.org/10.1098/rsta.2017.0310.

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For more than two centuries progress in the study of water waves proved to be interdependent with innovative and deep developments in theoretical and experimental directions of investigation. In recent years, considerable progress has been achieved towards the understanding of waves of large amplitude. Within this setting one cannot rely on linear theory as nonlinearity becomes an essential feature. Various analytic methods have been developed and adapted to come to terms with the challenges encountered in settings where approximations (such as those provided by linear or weakly nonlinear theory) are ineffective. Without relying on simpler models, progress becomes contingent upon the discovery of structural properties, the exploitation of which requires a combination of creative ideas and state-of-the-art technical tools. The successful quest for structure often reveals unexpected patterns and confers aesthetic value on some of these studies. The topics covered in this issue are both multi-disciplinary and interdisciplinary: there is a strong interplay between mathematical analysis, numerical computation and experimental/field data, interacting with each other via mutual stimulation and feedback. This theme issue reflects some of the new important developments that were discussed during the programme ‘Nonlinear water waves’ that took place at the Isaac Newton Institute for Mathematical Sciences (Cambridge, UK) from 31st July to 25th August 2017. A cross-section of the experts in the study of water waves who participated in the programme authored the collected papers. These papers illustrate the diversity, intensity and interconnectivity of the current research activity in this area. They offer new insight, present emerging theoretical methodologies and computational approaches, and describe sophisticated experimental results. This article is part of the theme issue ‘Nonlinear water waves’.

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9

Wu, Theodore Yaotsu. "Nonlinear waves and solitons in water." Physica D: Nonlinear Phenomena 123, no. 1-4 (November 1998): 48–63. http://dx.doi.org/10.1016/s0167-2789(98)00111-0.

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10

London, Steven D. "Weakly nonlinear shallow water magnetohydrodynamic waves." Geophysical & Astrophysical Fluid Dynamics 108, no. 3 (April 10, 2014): 323–32. http://dx.doi.org/10.1080/03091929.2013.877133.

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11

Zhang, Hong-sheng, Hua-wei Zhou, Guang-wen Hong, and Jian-min Yang. "A FULLY NONLINEAR BOUSSINESQ MODEL FOR WATER WAVE PROPAGATION." Coastal Engineering Proceedings 1, no. 32 (January 31, 2011): 12. http://dx.doi.org/10.9753/icce.v32.waves.12.

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A set of high-order fully nonlinear Boussinesq-type equations is derived from the Laplace equation and the nonlinear boundary conditions. The derived equations include the dissipation terms and fully satisfy the sea bed boundary condition. The equations with the linear dispersion accurate up to [2,2] padé approximation is qualitatively and quantitatively studied in details. A numerical model for wave propagation is developed with the use of iterative Crank-Nicolson scheme, and the two-dimensional fourth-order filter formula is also derived. With two test cases numerically simulated, the modeled results of the fully nonlinear version of the numerical model are compared to those of the weakly nonlinear version.

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12

Fenton, John D. "POLYNOMIAL APPROXIMATION AND WATER WAVES." Coastal Engineering Proceedings 1, no. 20 (January 29, 1986): 15. http://dx.doi.org/10.9753/icce.v20.15.

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A different approach to the solution of water wave problems is considered. Instead of using an approximate wave theory combined with highly accurate global spatial approximation methods, as for example in many applications of linear wave theory, a method is developed which uses local polynomial approximation combined with the full nonlinear equations. The method is applied to the problem of inferring wave properties from the record of a pressure transducer, and is found to be capable of high accuracy for waves which are not too short, even for large amplitude waves. The general approach of polynomial approximation is well suited to problems of a rather more general nature, especially where the geometry is at all complicated. It may prove useful in other areas, such as the nonlinear interaction of long waves, shoaling of waves, and in three dimensional problems, such as nonlinear wave refraction and diffraction.

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13

Lin, Lihwa, Zeki Demirbilek, Jinhai Zheng, and Hajime Mase. "RAPID CALCULATION OF NONLINEAR WAVE-WAVE INTERACTIONS." Coastal Engineering Proceedings 1, no. 32 (January 27, 2011): 36. http://dx.doi.org/10.9753/icce.v32.waves.36.

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This paper presents an efficient numerical algorithm for the nonlinear wave-wave interactions that can be important in the evolution of coastal waves. Indeed, ocean waves truly interact with each others. However, because ocean waves can also interact with the atmosphere such as under variable wind and pressure fields, and waves will deform from deep to shallow water, it is generally difficult to differentiate the actual amount of the nonlinear energy transfer among spectral waves mixed with the atmospheric input and wave breaking. The classical derivation of the nonlinear wave energy transfer has involved tedious numerical calculation that appears impractical to the engineering application. The present study proposed a theoretically based formulation to efficiently calculate nonlinear wave-wave interactions in the spectral wave transformation equation. It is approved to perform well in both idealized and real application examples. This rapid calculation algorithm indicates the nonlinear energy transfer is more significant in the intermediate depth than in deep and shallow water conditions.

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14

Zhang, Huichen, and Markus BrÃ¼hl. "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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15

VORONOVICH, VYACHESLAV V., DMITRY E. PELINOVSKY, and VICTOR I. SHRIRA. "On internal wave–shear flow resonance in shallow water." Journal of Fluid Mechanics 354 (January 10, 1998): 209–37. http://dx.doi.org/10.1017/s0022112097007593.

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The work is concerned with long nonlinear internal waves interacting with a shear flow localized near the sea surface. The study is focused on the most intense resonant interaction occurring when the phase velocity of internal waves matches the flow velocity at the surface. The perturbations of the shear flow are considered as ‘vorticity waves’, which enables us to treat the wave–flow resonance as the resonant wave–wave interaction between an internal gravity mode and the vorticity mode. Within the weakly nonlinear long-wave approximation a system of evolution equations governing the nonlinear dynamics of the waves in resonance is derived and an asymptotic solution to the basic equations is constructed. At resonance the nonlinearity of the internal wave dynamics is due to the interaction with the vorticity mode, while the wave's own nonlinearity proves to be negligible. The equations derived are found to possess solitary wave solutions of different polarities propagating slightly faster or slower than the surface velocity of the shear flow. The amplitudes of the ‘fast’ solitary waves are limited from above; the crest of the limiting wave forms a sharp corner. The solitary waves of amplitude smaller than a certain threshold are shown to be stable; ‘subcritical’ localized pulses tend to such solutions. The localized pulses of amplitude exceeding this threshold form infinite slopes in finite time, which indicates wave breaking.

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16

Henry, D. "On three-dimensional Gerstner-like equatorial water waves." Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 376, no. 2111 (December 11, 2017): 20170088. http://dx.doi.org/10.1098/rsta.2017.0088.

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This paper reviews some recent mathematical research activity in the field of nonlinear geophysical water waves. In particular, we survey a number of exact Gerstner-like solutions which have been derived to model various geophysical oceanic waves, and wave–current interactions, in the equatorial region. These solutions are nonlinear, three-dimensional and explicit in terms of Lagrangian variables. This article is part of the theme issue ‘Nonlinear water waves’.

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17

Groves, M. D. "Steady Water Waves." Journal of Nonlinear Mathematical Physics 11, no. 4 (January 2004): 435–60. http://dx.doi.org/10.2991/jnmp.2004.11.4.2.

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18

Zeytounian, R. Kh. "Nonlinear long waves on water and solitons." Uspekhi Fizicheskih Nauk 165, no. 12 (1995): 1403. http://dx.doi.org/10.3367/ufnr.0165.199512f.1403.

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19

Kozlov, V. A., and N. G. Kuznetsov. "Qualitative theory of nonlinear steady water waves." Vestnik St. Petersburg University: Mathematics 41, no. 2 (June 2008): 113–24. http://dx.doi.org/10.3103/s1063454108020052.

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20

Yakimov, A. Yu. "Equations for nonlinear waves on shallow water." Fluid Dynamics 47, no. 6 (November 2012): 789–92. http://dx.doi.org/10.1134/s0015462812060117.

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21

Zeytounian, R. Kh. "Nonlinear long waves on water and solitons." Physics-Uspekhi 38, no. 12 (December 31, 1995): 1333–81. http://dx.doi.org/10.1070/pu1995v038n12abeh000124.

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22

Vanhille, Christian, and Cleofé Campos‐Pozuelo. "Nonlinear ultrasonic waves in water‐air mixtures." Journal of the Acoustical Society of America 123, no. 5 (May 2008): 3704. http://dx.doi.org/10.1121/1.2935121.

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23

Barth�lemy, Eric. "Nonlinear Shallow Water Theories for Coastal Waves." Surveys In Geophysics 25, no. 3-4 (July 2004): 315–37. http://dx.doi.org/10.1007/s10712-003-1281-7.

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24

Debnath, L. "A variational principle for nonlinear water waves." Acta Mechanica 72, no. 1-2 (May 1988): 155–60. http://dx.doi.org/10.1007/bf01176549.

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25

Tuah, Hang, and Robert T. Hudspeth. "Finite Water Depth Effects on Nonlinear Waves." Journal of Waterway, Port, Coastal, and Ocean Engineering 111, no. 2 (March 1985): 401–16. http://dx.doi.org/10.1061/(asce)0733-950x(1985)111:2(401).

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26

Liapidevskii, V. Yu, M. V. Turbin, F. F. Khrapchenkov, and V. F. Kukarin. "Nonlinear Internal Waves in Multilayer Shallow Water." Journal of Applied Mechanics and Technical Physics 61, no. 1 (January 2020): 45–53. http://dx.doi.org/10.1134/s0021894420010058.

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27

Debnath, Lokenath. "Some Nonlinear Evolution Equations in Water Waves." Journal of Mathematical Analysis and Applications 251, no. 2 (November 2000): 488–503. http://dx.doi.org/10.1006/jmaa.2000.7023.

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28

Zhao, Xi-zeng, Zhao-chen Sun, Shu-xiu Liang, and Chang-hong Hu. "A Numerical Method for Nonlinear Water Waves." Journal of Hydrodynamics 21, no. 3 (June 2009): 401–7. http://dx.doi.org/10.1016/s1001-6058(08)60163-8.

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29

Yamashita, Kei, Taro Kakinuma, and Keisuke Nakayama. "NUMERICAL ANALYSES ON PROPAGATION OF NONLINEAR INTERNAL WAVES." Coastal Engineering Proceedings 1, no. 32 (February 1, 2011): 24. http://dx.doi.org/10.9753/icce.v32.waves.24.

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A set of nonlinear surface/internal-wave equations, which have been derived on the basis of the variational principle without any assumptions concerning wave nonlinearity and dispersion, is applied to compare numerical results with experimental data of surface/internal waves propagating through a shallow- or a deep-water region in a tank. Internal waves propagating over a submerged breakwater or a uniformly sloping beach are also simulated. The internal progressive wave shows remarkable shoaling when the interface reaches the critical level, after which physical variables including wave celerity become unstable near the wave-breaking point. In the case of the internal-wave trough reflecting at the vertical wall, the vertical velocities of water particles in the vicinity of the interface are different from that of the moving interface at the wall near the wave breaking, which means that the kinematic boundary condition on the interface of trough has been unsatisfied.

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30

Volobuev A.N. "Some features of the solving of hydrodynamic equations for solitary waves in the open water channel." Technical Physics 92, no. 9 (2022): 1111. http://dx.doi.org/10.21883/tp.2022.09.54673.56-22.

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The opportunity of use of an impulse equation special form for the solving of a problem of solitary waves (solitons) occurrence in the open water channel is considered. It is shown that the used of an impulse equation allows take into account a role of surface tension and gravitational forces in formation of waves. Using of the continuity equation expansion into series on Rayleigh's method the system of the differential equations is received, one of which is nonlinear. Application of Dalembert's method for running waves for the solving of the nonlinear differential equation in a hydrodynamic problem of solitary waves spreading in the open water channel is considered. It is shown that as against Dalembert's theory for the linear hyperbolic equations where initial conditions completely determine the form of arising waves, for the nonlinear equations the form of waves is determined by character of the equation nonlinearity. Thus during the solution of equations the sum of the functions describing linear waves extending in opposite directions, in the Dalembert's method for nonlinear waves is replaced with the sum of the nonlinear differential equations. Keywords: soliton, open water channel, surface tension, gravitational forces, nonlinear differential equation, Dalembert's method.

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31

Qi, Yusheng, Guangyu Wu, Yuming Liu, Moo-Hyun Kim, and Dick K. P. Yue. "Nonlinear phase-resolved reconstruction of irregular water waves." Journal of Fluid Mechanics 838 (January 25, 2018): 544–72. http://dx.doi.org/10.1017/jfm.2017.904.

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We develop and validate a high-order reconstruction (HOR) method for the phase-resolved reconstruction of a nonlinear wave field given a set of wave measurements. HOR optimizes the amplitude and phase of $L$ free wave components of the wave field, accounting for nonlinear wave interactions up to order $M$ in the evolution, to obtain a wave field that minimizes the reconstruction error between the reconstructed wave field and the given measurements. For a given reconstruction tolerance, $L$ and $M$ are provided in the HOR scheme itself. To demonstrate the validity and efficacy of HOR, we perform extensive tests of general two- and three-dimensional wave fields specified by theoretical Stokes waves, nonlinear simulations and physical wave fields in tank experiments which we conduct. The necessary $L$, for general broad-banded wave fields, is shown to be substantially less than the free and locked modes needed for the nonlinear evolution. We find that, even for relatively small wave steepness, the inclusion of high-order effects in HOR is important for prediction of wave kinematics not in the measurements. For all the cases we consider, HOR converges to the underlying wave field within a nonlinear spatial-temporal predictable zone ${\mathcal{P}}_{NL}$ which depends on the measurements and wave nonlinearity. For infinitesimal waves, ${\mathcal{P}}_{NL}$ matches the linear predictable zone ${\mathcal{P}}_{L}$, verifying the analytic solution presented in Qi et al. (Wave Motion, vol. 77, 2018, pp. 195–213). With increasing wave nonlinearity, we find that ${\mathcal{P}}_{NL}$ contains and is generally greater than ${\mathcal{P}}_{L}$. Thus ${\mathcal{P}}_{L}$ provides a (conservative) estimate of ${\mathcal{P}}_{NL}$ when the underlying wave field is not known.

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32

MILEWSKI, PAUL A., J. M. VANDEN-BROECK, and ZHAN WANG. "Hydroelastic solitary waves in deep water." Journal of Fluid Mechanics 679 (May 19, 2011): 628–40. http://dx.doi.org/10.1017/jfm.2011.163.

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The problem of waves propagating on the surface of a two-dimensional ideal fluid of infinite depth bounded above by an elastic sheet is studied with asymptotic and numerical methods. We use a nonlinear elastic model that has been used to describe the dynamics of ice sheets. Particular attention is paid to forced and unforced dynamics of waves having near-minimum phase speed. For the unforced problem, we find that wavepacket solitary waves bifurcate from nonlinear periodic waves of minimum speed. When the problem is forced by a moving load, we find that, for small-amplitude forcing, steady responses are possible at all subcritical speeds, but for larger loads there is a transcritical range of forcing speeds for which there are no steady solutions. In unsteady computations, we find that if the problem is forced at a speed in this range, very large unsteady responses are obtained, and that when the forcing is released, a solitary wave is generated. These solitary waves appear stable, and can coexist within a sea of small-amplitude waves.

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33

Vengayil, Padmaraj, and James T. Kirby. "SHOALING AND REFLECTION OF NONLINEAR SHALLOW WATER WAVES." Coastal Engineering Proceedings 1, no. 20 (January 29, 1986): 60. http://dx.doi.org/10.9753/icce.v20.60.

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The formulation for shallow water wave shoaling and refraction diffraction given by Liu et al (1985) is extended to include reflected waves. The model is given in the form of coupled K-P equations for forward and backward propagation. Shoaling on a plane beach is studied using the forward-propagating model alone. Non-resonant reflection of a solitary wave from a slope and resonant reflection of periodic waves by sinusoidal bars are then studied.

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34

Teng, Michelle H., and Theodore Y. Wu. "Nonlinear water waves in channels of arbitrary shape." Journal of Fluid Mechanics 242 (September 1992): 211–33. http://dx.doi.org/10.1017/s0022112092002349.

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The generalized channel Boussinesq (gcB) two-equation model and the forced channel Korteweg–de Vries (cKdV) one-equation model previously derived by the authors are further analysed and discussed in the present study. The gcB model describes the propagation and generation of weakly nonlinear, weakly dispersive and weakly forced long water waves in channels of arbitrary shape that may vary both in space and time, and the cKdV model is applicable to unidirectional motions of such waves, which may be sustained under forcing at resonance of the system. These two models are long-wave approximations of a hierarchy set of section-mean conservation equations of mass, momentum and energy, which are exact for inviscid fluids. Results of these models are demonstrated with four specific channel shapes, namely variable rectangular, triangular, parabolic and semicircular sections, in which case solutions are obtained in closed form. In particular, for uniform channels of equal mean water depth, different cross-sectional shapes have a leading-order effect only on the variations of a k-factor of the coefficient of the term bearing the dispersive effects in the model equations. For this case, the uniform-channel analogy theorem enunciated here shows that long waves of equal (mean) height in different uniform channels of equal mean depth but distinct k-shape factors will propagate with equal velocity and with their effective wavelengths appearing k times of that in the rectangular channel, for which k = 1. It also shows that the further channel shape departs from the rectangular, the greater the value of k. Based on this observation, the solitary and cnoidal waves in a k-shaped channel are compared with experiments on wave profiles and wave velocities. Finally, some three-dimensional features of these solitary waves are presented for a triangular channel.

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35

Bryant, Peter J., and Michael Stiassnie. "Different forms for nonlinear standing waves in deep water." Journal of Fluid Mechanics 272 (August 10, 1994): 135–56. http://dx.doi.org/10.1017/s0022112094004416.

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Multiple forms for standing waves in deep water periodic in both space and time are obtained analytically as solutions of Zakharov's equation and its modification, and investigated computationally as irrotational two-dimensional solutions of the full nonlinear boundary value problem. The different forms are based on weak nonlinear interactions between the fundamental harmonic and the resonating harmonics of 2, 3,…times the frequency and 4, 9,…respectively times the wavenumber. The new forms of standing waves have amplitudes with local maxima at the resonating harmonics, unlike the classical (Stokes) standing wave which is dominated by the fundamental harmonic. The stability of the new standing waves is investigated for small to moderate wave energies by numerical computation of their evolution, starting from the standing wave solution whose only initial disturbance is the numerical error. The instability of the Stokes standing wave to sideband disturbances is demonstrated first, by showing the evolution into cyclic recurrence that occurs when a set of nine equal Stokes standing waves is perturbed by a standing wave of a length equal to the total length of the nine waves. The cyclic recurrence is similar to that observed in the well-known linear instability and sideband modulation of Stokes progressive waves, and is also similar to that resulting from the evolution of the new standing waves in which the first and ninth harmonics are dominant. The new standing waves are only marginally unstable at small to moderate wave energies, with harmonics which remain near their initial amplitudes and phases for typically 100–1000 wave periods before evolving into slowly modulated oscillations or diverging.

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36

Moreira, R. M., and D. H. Peregrine. "Nonlinear interactions between deep-water waves and currents." Journal of Fluid Mechanics 691 (December 6, 2011): 1–25. http://dx.doi.org/10.1017/jfm.2011.436.

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AbstractThe effects of nonlinearity on a train of linear water waves in deep water interacting with underlying currents are investigated numerically via a boundary-integral method. The current is assumed to be two-dimensional and stationary, being induced by a distribution of singularities located beneath the free surface, which impose sharp and gentle surface velocity gradients. For ‘slowly’ varying currents, the fully nonlinear results confirm that opposing currents induce wave steepening and breaking within the region where a high convergence of rays occurs. For ‘rapidly’ varying currents, wave blocking and breaking are more prominent. In this case reflection was observed when sufficiently strong adverse currents are imposed, confirming that at least part of the wave energy that builds up within the caustic can be released in the form of partial reflection and wave breaking. For bichromatic waves, the fully nonlinear results show that partial wave blocking occurs at the individual wave components in the wave groups and that waves become almost monochromatic upstream of the blocking region.

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37

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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38

Stamp, Andrew P., and Marcus Jacka. "Deep-water internal solitaty waves." Journal of Fluid Mechanics 305 (December 25, 1995): 347–71. http://dx.doi.org/10.1017/s0022112095004654.

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An experimental investigation of mode-2 (’lump-Like’) Solitary waves propagaling on a thin interface between two deep layers of different densities is presented. Small-and large-amplitude waves behaved differently: small waves carried energy and momentum, whereas sufficiently large waves also carried mass. Weakly nonlinear theory anticipated the result for amplitudes a/h [les ] 0.5 but did not provide even a qualitative description of the large-amplitude waves. In particular, the prediction that for waves to maintain permanent form their wavelength must decrease with increasing amplitude failed; instead the wavelength of large waves was observed to increase with increasing amplitude. Furthermore, whilst the waves were expected to emerge from interactions along their precollision trajectories, the large waves actually suffered a backward shift.

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39

Guyenne, Philippe, and Emilian I. Pǎrǎu. "Computations of fully nonlinear hydroelastic solitary waves on deep water." Journal of Fluid Mechanics 713 (October 17, 2012): 307–29. http://dx.doi.org/10.1017/jfm.2012.458.

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AbstractThis paper is concerned with the two-dimensional problem of nonlinear gravity waves travelling at the interface between a thin ice sheet and an ideal fluid of infinite depth. The ice-sheet model is based on the special Cosserat theory of hyperelastic shells satisfying Kirchhoff’s hypothesis, which yields a conservative and nonlinear expression for the bending force. A Hamiltonian formulation for this hydroelastic problem is proposed in terms of quantities evaluated at the fluid–ice interface. For small-amplitude waves, a nonlinear Schrödinger equation is derived and its analysis shows that no solitary wavepackets exist in this case. For larger amplitudes, both forced and free steady waves are computed by direct numerical simulations using a boundary-integral method. In the unforced case, solitary waves of depression as well as of elevation are found, including overhanging waves with a bubble-shaped profile for wave speeds $c$ much lower than the minimum phase speed ${c}_{\mathit{min}} $. It is also shown that the energy of depression solitary waves has a minimum at a wave speed ${c}_{m} $ slightly less than ${c}_{\mathit{min}} $, which suggests that such waves are stable for $c\lt {c}_{m} $ and unstable for $c\gt {c}_{m} $. This observation is verified by time-dependent computations using a high-order spectral method. These computations also indicate that solitary waves of elevation are likely to be unstable.

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40

Shugan, Igor, Sergey Kuznetsov, Yana Saprykina, and Yang-Yih Chen. "Physics of Traveling Waves in Shallow Water Environment." Water 13, no. 21 (October 22, 2021): 2990. http://dx.doi.org/10.3390/w13212990.

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We present a study of the physical characteristics of traveling waves at shallow and intermediate water depths. The main subject of study is to the influence of nonlinearity on the dispersion properties of waves, their limiting heights and steepness, the shape of solitary waves, etc. A fully nonlinear Serre–Green–Naghdi-type model, a classical weakly nonlinear Boussinesq model and fifth-order Stokes wave solutions were chosen as models for comparison. The analysis showed significant, if not critical, differences in the effect of nonlinearity on the properties of traveling waves for these models. A comparison with experiments was carried out on the basis of the results of a joint Russian–Taiwanese experiment, which was carried out in 2015 at the Tainan Hydraulic Laboratory, and on available experimental data. A comparison with the experimental results confirms the applicability of a completely nonlinear model for calculating traveling waves over the entire range of applicability of the model in contrast to the Boussinesq model, which shows contradictory and unrealistic wave properties for moderate wavelengths.

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41

Garnier, Erell-Isis, Zhenhua Huang, and Chiang C. Mei. "Nonlinear long waves over a muddy beach." Journal of Fluid Mechanics 718 (February 8, 2013): 371–97. http://dx.doi.org/10.1017/jfm.2012.617.

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

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42

Kaihatu, James M., and James T. Kirby. "Nonlinear transformation of waves in finite water depth." Physics of Fluids 7, no. 8 (August 1995): 1903–14. http://dx.doi.org/10.1063/1.868504.

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43

Langtangen, Hans Petter, and Geir Pedersen. "Computational models for weakly dispersive nonlinear water waves." Computer Methods in Applied Mechanics and Engineering 160, no. 3-4 (July 1998): 337–58. http://dx.doi.org/10.1016/s0045-7825(98)00293-x.

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44

Liu, Philip L. F., Sung B. Yoon, and James T. Kirby. "Nonlinear refraction–diffraction of waves in shallow water." Journal of Fluid Mechanics 153, no. -1 (April 1985): 185. http://dx.doi.org/10.1017/s0022112085001203.

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45

Kudryashov, N. A. "Nonlinear waves on water and theory of solitons." Journal of Engineering Physics and Thermophysics 72, no. 6 (November 1999): 1224–35. http://dx.doi.org/10.1007/bf02699470.

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46

Wang, Yingguang. "Nonlinear crest distribution for shallow water Stokes waves." Applied Ocean Research 57 (April 2016): 152–61. http://dx.doi.org/10.1016/j.apor.2016.03.006.

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47

Christou, M., C. H. Hague, and C. Swan. "The reflection of nonlinear irregular surface water waves." Engineering Analysis with Boundary Elements 33, no. 5 (May 2009): 644–53. http://dx.doi.org/10.1016/j.enganabound.2008.10.005.

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48

Arshad, Sarmad, Ayesha Sohail, and Khadija Maqbool. "Nonlinear shallow water waves: A fractional order approach." Alexandria Engineering Journal 55, no. 1 (March 2016): 525–32. http://dx.doi.org/10.1016/j.aej.2015.10.014.

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49

Mostafa, Attia. "Nonlinear water waves (KdV) equation and Painleve technique." International Journal of Basic and Applied Sciences 4, no. 2 (May 4, 2015): 216. http://dx.doi.org/10.14419/ijbas.v4i2.2708.

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<p>The Korteweg-de Vries (KdV) equation which is the third order nonlinear PDE has been of interest since Scott Russell (1844) . In this paper we study this kind of equation by Painleve equation and through this study, we find that KdV equation satisfies Painleve property, but we could not find a solution directly, so we transformed the KdV equation to the like-KdV equation, therefore, we were able to find four exact solutions to the original KdV equation.</p>

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50

Hsiao, Shih-Chun, Philip L. F. Liu, and Yongze Chen. "Nonlinear water waves propagating over a permeable bed." Proceedings of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences 458, no. 2022 (June 8, 2002): 1291–322. http://dx.doi.org/10.1098/rspa.2001.0903.

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