Propagation Of Shallow Water Waves Research Articles

A hybrid SPH model of the Navier-Stokes and Boussinesq equations for simulating wave propagation and deformation

A two-way hybrid model based on the smoothed particle hydrodynamics (SPH) method is proposed in this paper to simulate wave propagation and deformation effectively. The hybrid model is composed of two submodels: the first submodel is based on the Boussinesq equations (B-SPH) for simulating wave propagation, and the second submodel is based on the Navier-Stokes (N-S) equations (NS-SPH) for simulating wave deformation caused by interactions with structures. The exchange of information between the two submodels is realized through an open-boundary method: the results derived from the B-SPH submodel are transferred to the open boundaries of the NS-SPH submodel through a relaxation function; the information derived from the NS-SPH submodel is matched to the open boundaries of the B-SPH submodel through the continuity equation. To obtain a stable pressure field in the NS-SPH submodel, the particle shifting technique (PST) is implemented. The subsequent validation of the hybrid model includes four typical numerical examples related to solitary and regular waves. The agreement between the numerical results and the experimental data indicates that the proposed hybrid model is suitable for simulating the propagation and deformation of shallow water waves. Compared with the traditional NS-SPH model, which uses N-S equations to describe the whole fluid domain, the hybrid model has greater computational efficiency.

Diverse soliton wave profile analysis in ion-acoustic wave through an improved analytical approach

In engineering and applied sciences, several physical phenomena can be more precisely characterized by employing nonlinear fractional partial differential equations. The primary goal of this research is to examine the traveling wave solution in closed form for the nonlinear acoustic wave propagation model known as the time fractional simplified modified Camassa–Holm equation, which is used to explain the unidirectional propagation of shallow-water waves with non-hydrostatic pressure and explains the dispersion properties of numerous phenomena like fluid flow, control theory, liquid drop patterning in plasma, acoustics, fusion, and fission processes, etc. The utmost potential approach, namely the new auxiliary equation technique, is applied for analyzing the time nonlinear fractional simplified modified Camassa-Holm equation in the logic of the newest established truncated M-fractional derivative. The fractional partial differential equations have been transformed to the ordinary differential equation using the complex wave transformation in the sense of truncated M-fractional derivative. A variety of soliton solutions, including anti-kink, single soliton, anti-bell, bell, kink, multiple soliton, double soliton, singular-kink, compacton shape, periodic shape, and so many, are displayed in the diagram of 3D and contour plots by taking into account a number of various parameters. It is essential to point out that all derived outcomes are directly compared to the original solutions to certify their exactness. Results show that the used scheme is capable, simple, and straightforward and can be useful to a variety of complex phenomena. The acquired results are unique for the model equation and could be applied to the analysis of several nonlinear study fields.

Diverse soliton wave profile assessment to the fractional order nonlinear Landau-Ginzburg-Higgs and coupled Boussinesq-Burger equations

The space–time fractional Landau-Ginzburg-Higgs equation and coupled Boussinesq-Burger equation describe the behavior of nonlinear waves in the tropical and mid-latitude troposphere, exhibiting weak scattering, extended connections, arising from the interactions between equatorial and mid-latitude Rossby waves, fluid flow in dynamic systems, and depicting wave propagation in shallow water. The improved Bernoulli sub-equation function method has been used to achieve new and wide-ranging closed-form solitary wave solutions to the mentioned nonlinear fractional partial differential equations through beta-derivative. A wave transformation is applied to renovate the fractional-order equation into an ordinary differential equation. Some standard wave shapes of multiple soliton type, single soliton, kink shape, double soliton shape type, triple soliton shape, anti-kink shape, and other types of solitons have been established. The more updated software Python is used to display the solutions by using 3D and contour plotlines to describe the physical significances of attained solutions more clearly. The findings of this study are straightforward, adaptable, and quicker to simulate. It has been notable that the improved Bernoulli sub-equation function method is practical, effective, and offers more sophisticated solutions that can help togenerate a large number of wave solutions for various models.

Impact of north-west Bay of Bengal bathymetry along the off-coast of Digha on the propagation of shallow water waves using finite difference method

Numerical simulation of shallow water equations has been carried out using the Crank-Nicolson scheme of Finite Difference Method (FDM) to study the interaction of waves with different forms of bathymetry especially that of the off-coast of Digha, a tourist spot situated in the north-west coast of Bay of Bengal. MATLAB code has been developed for the simulation. One-dimensional (1-D) linear shallow water model has been considered here. Three different bathymetry profile are chosen for the investigation: (i) flat bathymetry (with or without seamount/basin); (ii) piecewise linear approximation to the real bathymetry of the off-coast of Digha; (iii) real bathymetry of the off-coast of Digha. Real bathymetry data has been obtained from the gridded data set of GEBCO (General Bathymetric Chart of the Oceans) published in 2022. Validation of the present numerical scheme has been performed by considering test problems with results existing in the literature. Comparative study of the effect of the bathymetries, on the propagation of shallow water waves, has been performed and demonstrated with figures and tables. Results of wave-bathymetry interaction obtained here are believed to be new with respect to the present study area and will definitely give some insight into the phenomena in this region.

Orbital stability of smooth solitons for the modified Camassa-Holm equation

The modified Camassa-Holm equation with cubic nonlinearity is completely integrable and is considered a model for the unidirectional propagation of shallow-water waves. The localized smooth-wave solution exists uniquely, up to translation, within a certain range of the linear dispersive parameter. By constructing conserved H1 and L1 quantities in terms of the momentum variable m, this study demonstrates that the smooth soliton, when regarded as a solution of the initial-value problem for the modified Camassa-Holm equation, is orbitally stable to perturbations in the Sobolev space H3. Furthermore, the global well-posedness of the solution is established for certain initial data in Hs with s≥3.

Dynamics of geometric shape solutions for space-time fractional modified equal width equation with beta derivative

The modified equal width equation describes the propagation of shallow water waves in which nonlinear and dispersive effects are significant, including phenomena such as wave breaking, soliton interactions, ion-acoustic waves, energy transfer in plasma, and nonlinear stress waves. The aim of this article is to establish some novel and generic solutions to the space-time fractional modified equal width (MEW) equation using the two variable (θ′/θ, 1/θ)-expansion technique, a modification of the (θ′/θ)-expansion method. A wide range of geometric shapes and inclusive soliton solutions have been constructed, comprising rational, trigonometric, and hyperbolic solutions, along with their integration to the equation under consideration. The two- and three-dimensional graphs of the solitons, including periodic, V-shaped, bell-shaped, singular periodic, flat kink-shaped, plane-shaped, peakon, and parabolic, illustrate the physical aspects of the solitary wave solution and the effect of the fractional parameter. The results demonstrate the efficiency, appropriateness, and reliability of the adopted technique for investigating fractional-order nonlinear evolution equations in science, technology, and engineering.

Exploring the dynamics of shallow water waves and nonlinear wave propagation in hyperelastic rods: Analytical insights into the Camassa–Holm equation

The objective of this paper is to examine the analytical properties of the nonlinear [Formula: see text]-dimensional Camassa–Holm equation ([Formula: see text]), a fundamental model within the domain of nonlinear evolution equations. The aforementioned equation serves as a valuable tool in elucidating the unidirectional propagation of shallow water waves over a level terrain, as well as in representing certain nonlinear wave phenomena seen in cylindrical hyperelastic rods. We use the Khater III ([Formula: see text]hat.III) and improved Kudryashov ([Formula: see text]ud) technique to provide accurate solutions, drawing inspiration from the intricate mathematical framework of the [Formula: see text] issue. He’s variational iteration ([Formula: see text]) technique is used as a numerical methodology to assess the correctness of the generated answers. This strategy reveals a notable concurrence between the analytical and numerical outcomes. This alignment guarantees the suitability of the acquired solutions within the framework of the studied model. The importance of this investigation lies in its ability to improve our comprehension of the intricate dynamics regulated by the [Formula: see text] equation and its connections with other nonlinear evolution equations that describe shallow water wave behaviors and nonlinear wave propagation in cylindrical hyperelastic rods. The results of the study demonstrate novel analytical approaches, expanding the range of potential solutions and offering valuable insights into the physical characteristics of the interconnected wave phenomena. This research offers valuable insights and methodologies for addressing intricate mathematical models in shallow water wave theory and studying nonlinear waves in hyperelastic materials, therefore making substantial advances to the subject of nonlinear partial differential equations.

Numerical study of nonlinear time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera equation arising in propagation of waves

In this manuscript, the numerical approximation of the solution for time-fractional Caudrey–Dodd–Gibbon–Sawada–Kotera (CDGSK) nonlinear partial differential equation is described involving the Caputo fractional derivative with the help of two novel techniques namely, Sumudu Residual Power Series (SRPS) method and Homotopy Perturbation Sumudu Transform (HPST) method. CDGSK equation describe the propagation of capillary waves and shallow water waves. The results obtained through numerical approximation shows that both SRPS and HPST under consideration are reliable and efficient. The present framework precisely reflects the behavior of the obtained result for various fractional orders. Additionally, we conduct a comparison between the exact solution and an approximate series solution to affirm the efficacy of the proposed techniques. This analysis highlights the efficiency and precision of both techniques in solving nonlinear fractional differential equations. All the numerical simulations are demonstrated graphically using Maple and Matlab.

The dynamical perspective of soliton solutions, bifurcation, chaotic and sensitivity analysis to the (3+1)-dimensional Boussinesq model

In this study, we examine multiple perspectives on soliton solutions to the (3+1)-dimensional Boussinesq model by applying the unified Riccati equation expansion (UREE) approach. The Boussinesq model examines wave propagation in shallow water, which is derived from the fluid dynamics of a dynamical system. The UREE approach allows us to derive a range of distinct solutions, such as single, periodic, dark, and rational wave solutions. Furthermore, we present the bifurcation, chaotic, and sensitivity analysis of the proposed model. We use planar dynamical system theory to analyze the structure and characteristics of the system’s phase portraits. The current study depends on a dynamic structure that has novel and unexplored results for this model. In addition, we display the behaviors of associated physical models in 3-dimensional, density, and 2-dimensional graphical structures. Our findings demonstrate that the UREE technique is a valuable mathematical tool in engineering and applied mathematics for studying wave propagation in nonlinear evolution equations.

New Traveling Wave Solutions to the Simplified Modified Camassa–Holm Equation and the Landau-Ginsburg-Higgs Equation

Researchers are interested in the (1+1)-dimensional Camassa-Holm and Landau-Ginzburg-Higgs equations as they allow for the study of unidirectional wave propagation in shallow waters with a flat seabed, as well as nonlinear media exhibiting dispersion systems and superconductivity. This work has effectively developed exact wave solutions to the stated models, which may have significant consequences for characterising the nonlinear dynamical behaviour related to the phenomena. The extended -expansion technique is employed to procure a diverse array of progressive wave solutions characterized by hyperbolic, trigonometric, and rational functions. The solutions are shown as 3D profiles with a variety of shapes, including kink, singular kink, periodic, singular periodic, etc. The physical significance of the solutions is discussed by these plots, and the approach used in this study is considered efficient and capable of finding analytical solutions for the nonlinear models. Dhaka Univ. J. Sci. 72(1): 7-13, 2024 (January)

Dynamical behavior of water wave phenomena for the 3D fractional WBBM equations using rational sine-Gordon expansion method

To examine the dynamical behavior of travelling wave solutions of the water wave phenomenon for the family of 3D fractional Wazwaz-Benjamin-Bona-Mahony (WBBM) equations, this work employs the rational Sine-Gordon expansion (RSGE) approach based on the conformable fractional derivative. The method generalizes the well-known sine-Gordon expansion using the sine-Gordon equation as an auxiliary equation. In contrast to the conventional sine-Gordon expansion method, it takes a more general approach, a rational function rather than a polynomial one of the solutions of the auxiliary equation. The method described above is used to generate various solutions of the WBBM equations for hyperbolic functions, including soliton, singular soliton, multiple-soliton, kink, cusp, lump-kink, kink double-soliton, etc. The RSGE method contributes to our understanding of nonlinear phenomena, provides exact solutions to nonlinear equations, aids in studying solitons, advances mathematical techniques, and finds applications in various scientific and engineering disciplines. The answers are graphically shown in three-dimensional (3D) surface plots and contour plots using the MATLAB program. The resolutions of the equation, which have appropriate parameters, exhibit the absolute wave configurations in all screens. Furthermore, it can be inferred that the physical characteristics of the discovered solutions and their features may aid in our understanding of the propagation of shallow water waves in nonlinear dynamics.

Lie symmetries, solitary wave solutions, and conservation laws of coupled KdV‐mKdV equation for internal gravity waves

The propagation of nonlinear solitary and shallow water waves can be examined by nonlinear evolution equations. The dynamics of nonlinear behavior are described by exact solutions having parametric functions. The integrability of nonlinear equation discloses various characteristics of real phenomenon with continuous and fluctuating background. This work aims to generate symmetry reductions and exact solutions of KdV‐mKdV equation, which is a completely integrable system of nonlinear equations and describes combined solitary wave effects in inhomogeneous medium. The infinitesimal generators and their commutative relations are deduced under invariance property of Lie groups. The invariance property provides a symmetry reduction of governing system. Therefore, a continuous process of reductions transmutes the KdV‐mKdV equation to an equivalent system of ODEs. This system of ODEs leads to the exact solutions involving several arbitrary constants and functions. Such obtained results are generalized than earlier works and never reported in literature. Furthermore, adjoint equation and conserved quantities associated with Lagrangian formulation are constructed under Lie symmetries. To reveal the significance of governing phenomenon, the exact solutions are traced via numerical simulation and thus the graphical structures demonstrate line multisoliton, single soliton, lumps, parabolic and stationary wave profiles.

Experimental Investigation of Pore Pressure on Sandy Seabed around Submarine Pipeline under Irregular Wave Loading.

The propagation of shallow-water waves may cause liquefaction of the seabed, thereby reducing its support capacity for pipelines and potentially leading to pipeline settlement or deformation. To ensure the stability of buried pipelines, it is crucial to consider the excess pore pressure induced by irregular waves thoroughly. This paper presents the findings of an experimental study on excess pore pressure caused by irregular waves on a sandy seabed. A series of two-dimensional wave flume experiments investigated the excess pore pressure generated by irregular waves. Based on the experimental results, this study examined the influences of irregular wave characteristics and pipeline proximity on excess pore pressure. Using test data, the signal analysis method was employed to categorize different modes of excess pore-water pressure growth into two types and explore the mechanism underlying pore pressure development under the influence of irregular waves.

Some Novel Fusion and Fission Phenomena for an Extended (2+1)-Dimensional Shallow Water Wave Equation

An extended (2+1)-dimensional shallow water wave (SWW) model which can describe the evolution of nonlinear shallow water wave propagation in two spatial and temporal coordinates, is systematically studied. The multi-linear variable separation approach is addressed to the extended (2+1)-dimensional SWW equation. The variable separation solution consisting of two arbitrary functions is obtained, by assumption, from a specific ansatz. By selecting these two arbitrary functions as the exponential and trigonometric forms, resonant dromion, lump, and solitoff solutions are derived. Meanwhile, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions, and periodic waves are studied with graphical and analytical methods. The results can be used to enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

Effect of initial conditions on a one-dimensional model of small-amplitude wave propagation in shallow water. II: Blowup for nonsmooth conditions

PurposeThe purpose of this paper is to analyze numerically the blowup in finite time of the solutions to a one-dimensional, bidirectional, nonlinear wave model equation for the propagation of small-amplitude waves in shallow water, as a function of the relaxation time, linear and nonlinear drift, power of the nonlinear advection flux, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of three types of initial conditions.Design/methodology/approachAn implicit, first-order accurate in time, finite difference method valid for semipositive relaxation times has been used to solve the equation in a truncated domain for three different initial conditions, a first-order time derivative initially equal to zero and several constant wave speeds.FindingsThe numerical experiments show a very rapid transient from the initial conditions to the formation of a leading propagating wave, whose duration depends strongly on the shape, amplitude and width of the initial data as well as on the coefficients of the bidirectional equation. The blowup times for the triangular conditions have been found to be larger than those for the Gaussian ones, and the latter are larger than those for rectangular conditions, thus indicating that the blowup time decreases as the smoothness of the initial conditions decreases. The blowup time has also been found to decrease as the relaxation time, degree of nonlinearity, linear drift coefficient and amplitude of the initial conditions are increased, and as the width of the initial condition is decreased, but it increases as the viscosity coefficient is increased. No blowup has been observed for relaxation times smaller than one-hundredth, viscosity coefficients larger than ten-thousandths, quadratic and cubic nonlinearities, and initial Gaussian, triangular and rectangular conditions of unity amplitude.Originality/valueThe blowup of a one-dimensional, bidirectional equation that is a model for the propagation of waves in shallow water, longitudinal displacement in homogeneous viscoelastic bars, nerve conduction, nonlinear acoustics and heat transfer in very small devices and/or at very high transfer rates has been determined numerically as a function of the linear and nonlinear drift coefficients, power of the nonlinear drift, viscosity coefficient, viscous attenuation, and amplitude, smoothness and width of the initial conditions for nonzero relaxation times.

Bosonization, symmetry reductions, mapping and deformation method for B-extension of Sawada–Kotera equation

An extended (2+1)-dimensional shallow water wave (SWW) model governs the evolution of nonlinear shallow water wave propagation in two spatial and a temporal coordinate. The multi-linear variable separation approach is applied to the SWW equation. The variable separation solution consisting of two arbitrary functions is given. By choosing the arbitrary functions as the exponential and trigonometric forms, some novel fission and fusion phenomena including the semifoldons, peakons, lump, dromions and periodic waves are graphically and analytically studied. The results enhance the variety of the dynamics of the nonlinear wave fields related by engineering and mathematical physics.

Analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation in shallow water wave propagation

In this study, the analytical soliton solutions of the beta time-fractional simplified modified Camassa-Holm equation, a mathematical model used to describe the propagation of shallow water waves characterized by weak dispersion and nonlinearities, is determined. The (G′/G,1/G)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$({G}^{\\prime}/G,1/G)$$\\end{document}-expansion method, a powerful and reliable technique, is exploited to formulate the soliton solutions for the equation. The process yields a wide range of solutions, including trigonometric, rational, and hyperbolic functions with free parameters. Various original soliton solutions are generated for different parameter values, including bell-shaped, anti-bell-shaped, periodic, compacton, singular bell-shaped, singular periodic and flat kink solitons. To visually comprehend the physical characteristics of the obtained solutions, two- and three-dimensional graphs, as well as contour plots, are plotted. Thorough comparisons with previous results are conducted to ensure the originality of the derived solutions. The insights gained from understanding soliton behavior in shallow water can be helpful in improving tsunami warnings, coastal protection systems, underwater data transmission, submarine cables, and marine sensing networks.

Line-soliton, lump and interaction solutions to the (2+1)-dimensional Hirota-Satsuma-Ito equation with time-dependent via Hirota bilinear forms

In this paper, we consider the (2+1)-dimensional generalized Hirota-Satsuma-Ito equation with time-dependent, which has applications in describing the propagation of shallow water waves. Based on the bilinear formalism and with the aid of symbolic computation, we obtain line-soliton, lump, one-lump-one-stripe and one-lump-one-soliton using various ansatze’s function. To realize dynamics, we use diverse plots to analyze these solutions. Obtained solutions are reliable in the mathematical physics and engineering.

2D Modeling of Wave Propagation in Shallow Water by the Method of Characteristics

2D Modeling of Wave Propagation in Shallow Water by the Method of Characteristics

A new paradigm of reservoir computing exploiting hydrodynamics

Nonlinear waves have played a significant role in the development of the science of complexity throughout history. More recently, they have also contributed to the emergence of a novel paradigm in reservoir computing called neuromorphic computing by waves. In this paradigm, information is transmitted through wave dynamics, which process the data to perform complex tasks with minimal energy consumption. Leveraging nonlinear hydrodynamic waves for computational purposes, we have created the Aqua-Photonic-Advantaged Computing Machine by Artificial Neural Networks (Aqua-PACMANNs). This system utilizes wave propagation in shallow water as the primary physical phenomenon, with minimal electronic components required, limited to a camera for detection purposes. As a proof of concept, we have successfully implemented an exclusive-not-or logic gate using the Aqua-PACMANN architecture. This accomplishment demonstrates the potential of fluid dynamic neuromorphic computing and opens up possibilities for a new class of computing systems.