Solitary Waves In Shallow Water Research Articles

Utmost travelling wave phenomena to the fractional type nonlinear evolution equation in mathematical physics

In applied physics and engineering, non-linear fractional types of partial differential equations become increasingly prominent as an estimation technique to explain a wide variety of non-linear phenomena. Throughout this study, we choose to use an adaptable extended tanh-function scheme over a conformable derivative in order to construct a comprehensive closed-form traveling wave solution of the time fractional Cahn–Hilliard and modified Kawahara equations. The mentioned equations, have been used as frameworks for various phenomena such as quantum theory, geosciences, water wave mechanics, and solitary waves in shallow water in the porous medium. The different values of the fractional order of the model can be employed successfully to organize the wave velocity. We recognized numerous forms of solutions by using the computational software, namely solitons type, bell types, kink types, periodic types, multiple periodic, single soliton, singular kink, and other types of solutions that are illustrated using 3D, contour, and spherical. The attained results have effectively handled the previously stated phenomena in various fields of engineering and mathematical physics. The results originating in this study were confirmed by the computational software, which was used to rewrite them as non-linear fractional partial differential equations. We ensure that the technique is revised to make it more effective, pragmatic, and credible, and we pursue more comprehensive exact results for traveling waves, and then for solitary waves.

Reconstructions of time-evolving sound-speed fields perturbed by deformed and dispersive internal solitary waves in shallow water

The high-fidelity reconstruction of sound speeds is crucial for predicting acoustic propagation in shallow water where internal solitary waves (ISWs) are prevalent. Mapping temperatures from time series to spatial fields is an approach widely used to reproduce the sound speed perturbed by deformed internal waves. However, wave-shape distortions are inherent in the modeling results. This paper analyzes the formation mechanism and dynamic behavior of the distorted waveform that is shown to arise from the mismatch between the modeled and real propagation speeds of individual solitons within an ISW packet. To mitigate distortions, a reconstruction method incorporating the dispersion property of an ISW train is proposed here. The principle is to assign each soliton a real speed observed in the experiment. Then, the modeled solitons propagate at their intrinsic speeds, and the packet disperses naturally with time. The method is applied to reconstruct the sound speed perturbed by ISWs in the South China Sea. The mean and median of the root-mean-square error between the reconstructed and measured sound speeds are below 2 m/s. The modeled shape deformations and packet dispersion agree well with observations, and the waveform distortion is reduced compared with the original method. This work ensures the high fidelity of waveguide-environment reconstructions and facilitates the investigation of sound propagation in the future.

Wind-induced changes to shoaling surface gravity wave shape

Unforced shoaling waves experience growth and changes to wave shape. Similarly, wind-forced waves on a flat-bottom likewise experience growth/decay and changes to wave shape. However, the combined effect of shoaling and wind-forcing on wave shape, particularly relevant in the near-shore environment, has not yet been investigated theoretically. Here, we consider small-amplitude, shallow-water solitary waves propagating up a gentle, planar bathymetry forced by a weak, Jeffreys-type wind-induced surface pressure. We derive a variable-coefficient Korteweg-de Vries-Burgers (vKdV-B) equation governing the wave profile's evolution and solve it numerically using a Runge-Kutta third-order finite difference solver. The simulations run until convective prebreaking -- a Froude number limit appropriate to the order of the vKdV-B equation. Offshore winds weakly enhance the ratio of prebreaking height to depth as well as prebreaking wave slope. Onshore winds have a strong impact on narrowing the wave peak, and wind also modulates the rear shelf formed behind the wave. Furthermore, wind strongly affects the width of the prebreaking zone, with larger effects for smaller beach slopes. After converting our pressure magnitudes to physically realistic wind speeds, we observe qualitative agreement with prior laboratory and numerical experiments on breakpoint location. Additionally, our numerical results have qualitatively similar temporal wave shape to shoaling and wind-forced laboratory observations, suggesting that the vKdV-B equation captures the essential aspects of wind-induced effects on shoaling wave shape. Finally, we isolate the wind's effect by comparing the wave profiles to the unforced case. This reveals that the numerical results are approximately a superposition of a solitary wave, a shoaling-induced shelf, and a wind-induced, bound, dispersive, and decaying tail.

Modal intensity fluctuation during dynamic propagation of internal solitary waves in shallow water

Internal solitary wave (ISW) is a kind of nonlinear internal wave commonly observed in the shallow water, which has the characteristics of large amplitude, short period and strong current velocity. With the distribution of the temperature and the salinity in the water column perturbed by ISWs, the sound speed profile becomes range-dependent, and thus affecting the characteristics of the underwater acoustic propagation. The ISWs usually propagate at a speed of the order of 1 m/s , and moving internal waves cause the energy in each acoustic mode to fluctuate dramatically. In this paper, the modal intensity is defined as the squared modulus of the modal coefficient, and is used to measure the sound energy in each mode. Based on the coupled mode theory, the expression of the acoustic modal intensity during the propagation of internal waves is derived in this paper, and the modal intensity is taken as the linear superposition of the oscillating term and the trend term. Most of previous researches were limited to the study of the time-varying characteristics of the acoustic modal intensity during the propagation of internal waves in the time domain or frequency domain. In this paper, the mechanism of modal intensity fluctuations is studied simultaneously in the time domain and the frequency domain with the aid of the short-time Fourier transform. Both the theoretical derivation and the numerical simulation show that the internal solitary wave gives rise to the energy transfer among acoustic modes, i.e., the mode coupling. The dynamic propagation of internal waves further leads to the modal interference, which behaves as an oscillating term in the modal intensity, and causes the modal intensity to fluctuate rapidly with time. The amplitude of the trend term changes with time due to the mode stripping (the difference in attenuation coefficients between different modes), which in turn adds a time-varying offset to the oscillations induced by the modal interference. The trend of the modal intensity and the time-varying characteristics of the amplitude of each frequency component in the oscillating term are closely associated with the modal attenuation. Meanwhile, the depth-integrated intensity is chosen as the measure of the total received acoustic intensity, and the influences of modal intensity fluctuations on the acoustic energy at the receivers during the propagation of internal waves are studied. It is demonstrated that the modal intensity with high energy which oscillates most dramatically will dominate the temporal variation of the received acoustic energy.

Inertial energy dissipation in shallow-water breaking waves

We present direct numerical simulations of breaking solitary waves in shallow water to quantify the energy dissipation during the active breaking time. We find that this dissipation can be predicted by an inertial model based on Taylor’s hypothesis as a function of the local wave height, depth and the beach slope. We obtain a relationship that gives the dissipation rate of a breaking wave on a shallow slope as a function of local breaking parameters. Next, we use empirical relations to relate the local wave parameters to the offshore conditions. This enables the energy dissipation to be predicted in terms of the initial conditions. We obtain good collapse of the numerical data with respect to the theoretical scaling.

Head-on Collision Between Two Hydroelastic Solitary Waves in Shallow Water

A comprehensive theoretical study on the head-on collision between two solitary waves in a thin elastic plate floating on an inviscid fluid of finite depth is investigated analytically by means of a singular perturbation method. The effects of plate compression are also taken into account. The Poincare–Lighthill–Kuo method has been used to derive the solution up to the fourth order of the resulting nonlinear differential equation, which in principle gives the asymptotic series solution. It is found that after collision, both the hydroelastic solitary waves preserves their original shape and positions. However a collision does have imprints on colliding waves with non-uniform phase shift up to the third order which creates tilting in the wave profile. Maximum run-up amplitude, wave speed, phase shift and distortion profile have also been calculated and plotted for two colliding solitary waves.

Comparison of Analytical and Numerical Simulations of Long Nonlinear Internal Solitary Waves in Shallow Water

ABSTRACT Mohapatra, S.C.; Fonseca, R.B., and Guedes Soares, C., 2018. Comparison of analytical and numerical simulations of long nonlinear internal solitary waves in shallow water. This study compares the results of long nonlinear internal waves between analytical and numerical wave models in shallow water. To present the analytical solutions, the (G′/G) expansion method is applied to obtain the exact and explicit nonlinear solitary wave solutions of the generalized Gardner equation of any order in the presence of dispersive coefficient. The internal waves are demonstrated by analysing the solutions of the classical Gardner equation, modified Korteweg-de Vries equation, and Korteweg-de Vries equations from the generalized solution. The shape and characteristics of the three-dimensional solitary waves are studied by analysing different numerical results of the nonlinear analytical solutions. The Gardner equation basically behaves with the same properties as the Boussinesq equations of travelling wave solut...

A fractional Whitham-Broer-Kaup equation and its possible application to Tsunami prevention

A fractional Whitham-Broer-Kaup equation is suggested using He?s fractional derivative to model solitary waves in shallow water in porous medium near a dam. A modification of the exp-function method, the generalized exponential rational function method, is adopted to elucidate the basic solution properties of the equation, revealing that the value of the fractional order can be used effectively to control the wave velocity, the wave height, and the wave morphology. This theoretical result can be used for possible tsunami prevention.

On head-on collision between two solitary waves in shallow water: the use of the extended PLK method

In the present work, we examined the head-on collision of solitary waves in shallow water theory, through the use of extended Poincare–Lighthill–Kuo (PLK) method based on the combination of reductive perturbation method with strained coordinates. Motivated with the result obtained by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), we introduced a set of stretched coordinates that include some unknown functions which are to be determined so as to remove secularities that might occur in the solution. By expanding these unknown functions and the field variables into power series in the smallness parameter \(\epsilon \), introducing them into the field equations and imposing the conditions to remove the secularities, we obtained some evolution equations. By seeking a progressive wave solution to these evolution equations, we determined the speed correction terms and the phase-shift functions. The result obtained here is exactly the same with found by Ozden and Demiray (Int J Nonlinear Mech 69:66–70, 2015), wherein the analysis employed by Su and Mirie (J Fluid Mech 98:509–525, 1980) is utilized.

Dynamics of strongly nonlinear internal long waves in a three-layer fluid system

Dynamics of internal solitary waves in shallow water were studied with a strongly nonlinear internal long wave model in a three-layer fluid system. It is an extension of the two-layer MCC (Miyata, Choi, and Camassa) model, which is derived under the assumption that the characteristic wavelength is long in comparison to the thickness of each fluid layer. Phase velocity was assessed to understand the stability of solitary waves when they propagate. The instability mechanism turns out to be a Kelvin-Helmholtz type instability that is similar to the twolayer MCC model. We numerically investigated the three-layer model in terms of the effects of the middle layer and non-uniform bottom topography, and examined the generation of solitary waves to demonstrate the rich dynamics of the three-layer model.

Re-visiting the head-on collision problem between two solitary waves in shallow water

Upon discovering the wrongness of the statement “although this term does not cause any secularity for this order it will cause secularity at higher order expansion, therefore, that term must vanish” by Su and Mirie [4], in the present work, we studied the head-on collision of two solitary waves propagating in shallow water by introducing a set of stretched coordinates in which the trajectory functions are of order of ϵ2, where ϵ is the smallness parameter measuring non-linearity. Expanding the field variables and trajectory functions into power series in ϵ, we obtained a set of differential equations governing various terms in the perturbation expansion. By solving them under non-secularity condition we obtained the evolution equations and also the expressions for phase functions. By seeking a progressive wave solution to these evolution equations we have determined the speed correction terms and the phase shifts. As opposed to the result of Su and Mirie [4] and similar works, our calculations show that the phase shifts depend on both amplitudes of the colliding waves.

On cusped solitary waves in finite water depth

It is well-known that the Camassa–Holm (CH) equation admits both of the peaked and cusped solitary waves in shallow water. However, it was an open question whether or not the exact wave equations can admit them in finite water depth. Besides, it was traditionally believed that cusped solitary waves, whose 1st-derivative tends to infinity at crest, are essentially different from peaked solitary ones with finite 1st-derivative. Currently, based on the symmetry and the exact water wave equations, Liao [1] proposed a unified wave model (UWM) for progressive gravity waves in finite water depth. The UWM admits not only all traditional smooth progressive waves but also the peaked solitary waves in finite water depth: in other words, the peaked solitary progressive waves are consistent with the traditional smooth ones. In this paper, in the frame of the linearized UWM, we give, for the first time, some explicit expressions of cusped solitary waves in finite water depth, and besides reveal a close relationship between the cusped and peaked solitary waves: a cusped solitary wave is consist of an infinite number of peaked solitary ones with the same phase speed, so that it can be regarded as a special peaked solitary wave. This also well explains why and how a cuspon has an infinite 1st-derivative at crest. Besides, it is found that, when wave height is small enough, the effect of nonlinearity is negligible for the interaction of peaked waves so that these explicit expressions are good enough approximations of peaked/cusped solitary waves in finite water depth. In addition, like peaked solitary waves, the vertical velocity of a cusped solitary wave in finite water depth is also discontinuous at crest (x=0), and especially its phase speed has nothing to do with wave height, too. All of these would deepen and enrich our understandings about the cusped solitary waves.

Two new standing solitary waves in shallow water

In this paper, the closed-form analytic solutions of two new Faraday’s standing solitary waves due to the parametric resonance of liquid in a vessel vibrating vertically with a constant frequency are given for the first time. Using a model based on the symmetry of wave elevation and the linearized Boussinesq equation, we gain the closed-form wave elevations of the two kinds of non-monotonically decaying standing solitary waves with smooth crest and the even or odd symmetry. All of them have never been reported, to the best of our knowledge. Besides, they can explain some experimental phenomena well. All of these are helpful to deepen and enrich our understanding about standing solitary waves and Faraday’s wave.

Bell-polynomial construction of Bäcklund transformations with auxiliary independent variable for some soliton equations with one Tau-function

The Korteweg–de Vries (KdV)-type models are of significance in describing many physical situations in fluid flows (particularly for surface and internal waves), plasma physics, and solid state physics. In fluid dynamics, for example, the shallow water wave equation is utilized as a mathematical description of regular and generalized solitary waves in shallow water. Further, higher-order dispersive (e.g., the Lax fifth-order KdV equation) and higher-dimensional [e.g., the (2+1)- and (3+1)-dimensional breaking soliton equations] generalized nonlinear models are useful in analyzing and obtaining modulation theory, existence and stability of solitary waves, bores, and shocks, as well as other integrable properties. With symbolic computation, Bell-polynomial-typed Bäcklund transformations (BTs) are constructed for some single-field bilinearizable nonlinear evolution equations including the shallow water wave equation, Lax fifth-order KdV equation, and (2+1)- and (3+1)-dimensional breaking soliton equations. Bell-polynomial expressions are derived, which can be cast into the bilinear equations with one Tau-function. Key point lies in the introduction of certain auxiliary independent variable in the Bell-polynomial expression. With one auxiliary independent variable, the Bell-polynomial-typed BTs are then constructed according to the coupled two-field conditions between the primary and replica fields with both the fields satisfying the Bell-polynomial-expression equations. Auxiliary-independent-variable-involved Bell-polynomial-typed BTs are changed into their bilinear forms. Aforementioned equations turn out to be integrable in the sense of possessing the Bell-polynomial-typed BTs.

Higher order corrections for shallow-water solitary waves: elementary derivation and experiments

We present an elementary method to obtain the equations of the shallow-water solitary waves in different orders of approximation. The first two of these equations are solved to get the shapes and propagation velocities of the corresponding solitary waves. The first-order equation is shown to be equivalent to the Korteweg–de Vries (KdV) equation, while the second-order equation is solved numerically. The propagation velocity found for the solitary waves of the second-order equation coincides with a known expression, but it is obtained in a simpler way. By measuring the propagation velocity of solitary waves in the laboratory, we demonstrate that the second-order theory gives a considerably improved fit to experimental results.

Symmetry and Uniqueness of the Solitary-Wave Solution for the Ostrovsky Equation

We consider herein the Ostrovsky equation which arises in modeling the propagation of the surface and internal solitary waves in shallow water, or the capillary waves in a plasma with the effects of rotation. Using the modified sliding method, we prove that the solitary wave moving to the left to the Ostrovsky equation is symmetric about the origin and unique up to translations. We also establish the regularity and decay properties of solitary waves and obtain some results of the nonexistence of solitary wave solutions depending on the wave speed, weak rotation, and dispersive parameter.

Temporal sound field fluctuations in the presence of internal solitary waves in shallow water

Temporal variations of intensity fluctuations are presented from the SWARM95 experiment. It is hypothesized that specific features of these fluctuations can be explained by mode coupling due to the presence of an internal soliton moving approximately along the acoustic track. Estimates are presented in conjunction with theoretical consideration of the shallow water waveguide.

Discussion of “A function to determine wavelength from deep in to shallow water based on the length of the cnoidal wave at breaking” by J.P. Le Roux

An empirical modification to the Airy equation for wave celerity reduces to the expression for solitary waves in shallow water whilst retaining its usual form for deep water. The equation yields celerities in reasonable agreement with those for cnoidal waves in intermediate water depths. In this aspect, it is similar to the work described by Le Roux [Le Roux, J.P., 2007. A function to determine wavelength from deep into shallow water based on the length of the cnoidal wave at breaking. Coastal Engineering 54, 770–774]. The empirical modification has been widely applied in computer programs over the past 30 years.

Stability of solitary waves for the Ostrovsky equation

Considered herein is the Ostrovsky equation which is widely used to describe the effect of rotation on the surface and internal solitary waves in shallow water or the capillary waves in a plasma. It is shown that the solitary-wave solutions are orbitally stable for certain wave speeds.

Laboratory study of sand bed forms induced by solitary waves in shallow water

The surface profile of water waves propagating in shoaling water approaches the solitary waveform before wave breaking. The effect of the high non‐linearity of solitary waves may be very significant on bed forms induced in the nearshore zone. In this study, experiments on bed form generation beneath solitary waves are carried out in a 10‐m‐long flume used in resonant mode. Solitary waves are generated in shallow water on the background of a standing harmonic wave. One solitary wave (soliton) propagates in each direction of the flume on the time period of the flow, above an initially flat sandy bed. Ripples form rapidly on the bed and a strong interaction with the free surface occurs. The amplitude of the soliton and the phase shift between the soliton and the harmonic wave decrease with time, while the ripple amplitude increases. The amplitude of the harmonic wave is not affected by the ripples. The final ripple wavelength is about 1000 times the sand median diameter. Bars with superimposed ripples appear, with bar crests being positioned beneath the nodes of the standing wave, when bars form with crests beneath the antinodes of surface wave for standing waves without solitons. The Eulerian drift distribution in the flume is not affected by the propagation of solitons. We propose an energy balance for solitons propagating in shallow water above flat beds in which a term for the dissipation due to sand ripples is introduced, which defines a coefficient of interaction between solitons and ripples.