Shallow Water Wave Theory Research Articles

Transient Shallow Water Wave Interactions with a Partially Fragmented Ice Shelf

This work investigates the interaction between water waves and multiple ice shelf fragments in front of a semi-infinite ice sheet. The hydrodynamics are modelled using shallow water wave theory and the ice shelf vibration is modelled using Euler–Bernoulli beam theory. The ensuing multiple scattering problem is solved in the frequency domain using the transfer matrix method. The appropriate conservation of energy identity is derived in order to validate our numerical calculations. The transient scattering problem for incident wave packets is constructed from the frequency domain solutions. By incorporating multiple scattering, this paper extends previous models that have only considered a continuous semi-infinite ice shelf. This paper serves as a fundamental step towards developing a comprehensive model to simulate the breakup of ice shelves.

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.

The Moving Element Method for the Dynamic Responses of Floating Double-Plate Structures on the Shallow Water

In this paper, the dynamic behavior of floating double-plate structures under a moving load is investigated in great detail with different factors utilizing the moving element method (MEM). The structures in this study are very large floating structures (VLFSs). The Kirchhoff–Love theory is used to model two floating plates connected by a Winkler-type elastic core, whilst the water surrounding the structure is simulated by the shallow linear water wave theory. In order to improve the computational efficiency of moving load problems, the MEM is adopted. Since the MEM uses fewer discrete elements than those of the traditional finite element method (FEM), the computational cost is reduced. The proposed two-plate model helps to reduce the influence of the surrounding water on the upper plate. Several numerical results are presented to investigate the influence of the ratio of thickness between layers as well as core stiffness on the displacements with different water depths, and load speeds.

Directing Shallow-Water Waves Using Fixed Varying Bathymetry Designed by Recurrent Neural Networks

Directing shallow-water waves and their energy is highly desired in many ocean engineering applications. Coastal infrastructures can be protected by reflecting shallow-water waves to deep water. Wave energy harvesting efficiency can be improved by focusing shallow-water waves on wave energy converters. Changing water depth can effectively affect wave celerity and therefore the propagation of shallow-water waves. However, determining spatially varying bathymetry that can direct shallow-water waves to a designed location is not trivial. In this paper, we propose a novel machine learning method to design and optimize spatially varying bathymetry for directing shallow-water waves, in which the bathymetry is assumed fixed in time without considering morphodynamics. Shallow-water wave theory was applied to establish the mapping between water wave mechanics and recurrent neural networks (RNNs). Two wave-equivalent RNNs were developed to model shallow-water waves over fixed varying bathymetry. The resulting RNNs were trained to optimize bathymetry for wave energy focusing. We demonstrate that the bathymetry optimized by the wave-equivalent RNNs can effectively reflect and refract wave energy to various designed locations. We also foresee the potential that new engineering tools can be similarly developed based on the mathematical equivalence between wave mechanics and recurrent neural networks.

A new (3+1) dimensional Hirota bilinear equation: Painlavé integrability, Lie symmetry analysis, and conservation laws

This study's subject is a (3 + 1) dimensional new Hirota bilinear (NHB) equation that appears in the theory of shallow water waves. We investigate how particular dispersive waves behave in an NHB equation. In this regard, we first test the Painlavé integrability using the WTC-Kruskal method and second perform Lie point symmetry analysis on NHB. The algorithm outputs the Lie algebra, the symmetry reductions, and group invariant solutions. Using Lie point symmetries, NHB transforms into an ordinary differential equation. The integration architectures to solve this equation are the Bernoulli sub-ODE, 1/ G, and modified Kudryashov methods. We plot their graphs and observe how the solutions behaved in understanding the physical phenomenon. Additionally, we discuss the model utilizing nonlinear self-adjointness and Ibragimov's approach to generate conservation laws for each Lie symmetry generator.

Similarity reductions of peakon equations: the $$b$$-family

The $$b$$ -family is a one-parameter family of Hamiltonian partial differential equations of nonevolutionary type, which arises in shallow water wave theory. It admits a variety of solutions, including the celebrated peakons, which are weak solutions in the form of peaked solitons with a discontinuous first derivative at the peaks, as well as other interesting solutions that have been obtained in exact form and/or numerically. In each of the special cases $$b=2$$ and $$b=3$$ (the Camassa–Holm and Degasperis–Procesi equations, respectively), the equation is completely integrable, in the sense that it admits a Lax pair and an infinite hierarchy of commuting local symmetries, but for other values of the parameter $$b$$ it is nonintegrable. After a discussion of traveling waves via the use of a reciprocal transformation, which reduces to a hodograph transformation at the level of the ordinary differential equation satisfied by these solutions, we apply the same technique to the scaling similarity solutions of the $$b$$ -family and show that when $$b=2$$ or $$b=3$$ , this similarity reduction is related by a hodograph transformation to particular cases of the Painlevé III equation, while for all other choices of $$b$$ the resulting ordinary differential equation is not of Painlevé type.

Interaction of solitary wave with a concentric structure with multiple porous outer walls

This paper gives an analytical solution for a plane solitary wave interaction with a concentric structure with coaxial multiple-layered porous walls. An analytical solution is obtained based on shallow water wave theory and eigenfunction expression matching. Furthermore, a series of simultaneous equations are used to determine potential coefficients. The accuracy of the present model is verified by comparing its output with published results. Meanwhile, the impact of various important parameters (i.e., the number of porous walls, annular spacing and porosity of walls) with respect to wave forces and relative wave height is examined. It is sufficient to safeguard the inner cylinder by using two-three porous walls. For multiple walls with different porous-effect parameters, the arrangement of the selected porous effect parameters has a marginal influence on the force of the impermeable cylinder. However, the minimum wave run-up on it can be observed when the coefficient of the porous walls increases from the outside to the inside. In addition, an impermeable cylinder with two concentric porous walls is investigated owing to its higher application prospects.

On nonclassical symmetries, Painlevé analysis and singular, periodic and solitary wave solutions of generalized Hirota-Satsuma coupled KdV system

In this work, we investigate the generalized Hirota-Satsuma coupled KdV system which is materialized in the theory of shallow water waves. Nonclassical symmetries of governing system are derived using Bluman and Cole method. The obtained symmetries are in the form of Jacobi elliptic functions, which are more generalized than earlier obtained symmetries for the governing system. Some new singular, periodic and solitary wave solutions of the governing system are also derived by assuming the Jacobi elliptic function type solutions. The obtained exact solutions for the considered system are verified by graphical representation. The Painlevé property of the considered system is also checked with the help of Kruksal’s method and symbolic computational tool Maple which shows the integrable behaviour of the considered system.

RBF-PS method for approximation and eventual periodicity of fractional and integer type KdV equations

In engineering and mathematical physics nonlinear evolutionary equations play an important role. KdV equation is one of the famous evolution equation appeared in the theory of shallow water waves with large wave length and small wave amplitude. Investigations have shown an important phenomena for some evolution equations, related to their large-time behavior also known as eventual time periodicity of initial–boundary-value problems (IBVPs). Here in this study we numerically investigate, the periodicity of the solutions for integer as well as fractional type KdV equations on a limited domain with periodic boundary conditions exploiting meshfree technique called as Radial Basis Function Pseudo Spectral (RBF-PS) method.

Unimodular matrix on shallow water wave theory. Unimodularity through matrix method

Unimodular matrix on shallow water wave theory. Unimodularity through matrix method

Solving Whitham-Broer-Kaup-Like Equations Numerically by using Hybrid Differential Transform Method and Finite Differences Method

This paper aims to propose a hybrid approach of two powerful methods, namely the differential transform and finite difference methods, to obtain the solution of the coupled Whitham-Broer-Kaup-Like equations which arises in shallow-water wave theory. The capability of the method to such problems is verified by taking different parameters and initial conditions. The numerical simulations are depicted in 2D and 3D graphs. It is shown that the used approach returns accurate solutions for this type of problems in comparison with the analytic ones.

Dynamical behaviour of shallow water waves and solitary wave solutions of the Dullin-Gottwald-Holm dynamical system

In this article, we recover a variety of new families of shallow water wave and solitary wave solutions to the (1+1)-dimensional Dullin “Gottwald” Holm (DGH) system by employing new extended direct algebraic method (NEDAM). The derived results are obtained in diverse hyperbolic and periodic function forms. The attained solutions are new addition in the study of solitary wave and shallow water wave theory. In addition, 3-dimensional, 2-dimensional, and contour graphs of secured results are plotted in order to observe their dynamics with the choices of involved parameters. On the basis of achieved outcomes, we may claim that the proposed computational method is direct, dynamics, well organized, and will be useful for solving the more complicated nonlinear problems in diverse areas together with symbolic computations.

Concentrating standing waves for Davey–Stewartson systems

In this paper, we study the existence and concentration behaviour of multi-peak standing waves for a singularly perturbed Davey–Stewartson system, which arises in the theory of shallow water waves. For this purpose, we first give a sharp threshold of the existence of ground-state solutions to the related limiting problem. Next, combining the penalization method and the regularity theory of elliptic equations, we construct a family of positive solutions concentrating around any prescribed finite set of local minima, possibly degenerate, of the potential. A feature of this analysis is that we do not need any uniqueness or non-degeneracy conditions for the limiting equation. To the best of our knowledge, this paper is the first study dealing with the study of concentrating solutions for Davey–Stewartson systems. We emphasize that with respect to the classical Schrödinger equation, the presence of a singular integral operator in the Davey–Stewartson system forces the implementation of new ideas to obtain the existence of multi-peak solutions.

선수갑판 침수의 영향을 고려한 선체운동과 파랑하중의 비선형 해석

In rough sea, ships including fishing vessel sometimes suffer a severe bow deck wetness which may cause structural damage on bow deck and equipment. In this paper, a fully coupled method for the nonlinear computation of ship motion and bow deck water flow is developed and validated through example calculations. The interaction effects of ship motion and deck water flow are taken into account in computation by an interactive iteration scheme at each time step. Heave pitch motion computation is performed by the nonlinear strip method in time domain and deck flow computation is performed by the time domain finite difference method based on the shallow water wave theory. The two computations are coupled to take into account the interaction effects between them. Heave, pitch, bow deck water height, deck pressure and wall pressure are calculated and compared with experimental measurements. Good agreements are shown in the comparisons and the present computation method has been validated.

Design of tuned liquid sloshing dampers using nonlinear constraint optimization for across-wind response control of benchmark tall building

The present study proposes an optimal design methodology of tuned liquid sloshing dampers (TLSDs) installed in wind-excited benchmark tall building. The TLSDs design parameters were determined using proposed optimal design methodology based on nonlinear constraint optimization technique. Top floor peak acceleration was considered as the objective function and maximum value of sloshing depth of liquid as a nonlinear constraint for the design purpose. The classical nonlinear shallow water wave theory was used for the simulation of liquid sloshing, and the basic equations of nonlinear liquid sloshing were solved using the Lax Finite Difference Scheme. The governing equations of motion of combined structure-TLSDs system were expressed as state-space variables and structural response was numerically simulated. The findings were presented and assessed on the basis of a comparison between power spectral density functions, time history analysis, root mean square, peak response, and performance criteria for benchmark building with and without TLSDs. The off-tuning effect of TLSDs due to ±15% uncertainty in the building stiffness was also investigated. Further, an attempt has been made to present the control effectiveness of TLSDs in time–frequency domain (wavelet scalogram) using continuous wavelet transformation technique. The optimal TLSDs design based on the proposed methodology was found to be quite effective in serviceability-based design of benchmark tall building under wind loads and can be very useful for the practical design of TLSDs.

Numerical and analytical study of undular bores governed by the full water wave equations and bidirectional Whitham–Boussinesq equations

Undular bores, also termed dispersive shock waves, generated by an initial discontinuity in height as governed by two forms of the Boussinesq system of weakly nonlinear shallow water wave theory, the standard formulation and a Hamiltonian formulation, two related Whitham–Boussinesq equations, and the full water wave equations for gravity surface waves are studied and compared. It is found that the Whitham–Boussinesq systems give solutions in excellent agreement with numerical solutions of the full water wave equations for the positions of the leading and trailing edges of the bore up until the onset on modulational instability. The Whitham–Boussinesq systems, which are far simpler than the full water wave equations, can then be used to accurately model surface water wave undular bores. Finally, comparisons with numerical solutions of the full water wave equations show that the Whitham–Boussinesq systems give a slightly lower threshold for the onset of modulational instability in terms of the height of the initial step generating the undular bore.

Complex Korteweg-de Vries equation: A deeper theory of shallow water waves.

Using Levi-Cività's theory of ideal fluids, we derive the complex Korteweg-de Vries (KdV) equation, describing the complex velocity of a shallow fluid up to first order. We use perturbation theory, and the long wave, slowly varying velocity approximations for shallow water. The complex KdV equation describes the nontrivial dynamics of all water particles from the surface to the bottom of the water layer. A crucial step made in our work is the proof that a natural consequence of the complex KdV theory is that the wave elevation is described by the real KdV equation. The complex KdV approach in the theory of shallow fluids is thus more fundamental than the one based on the real KdV equation. We demonstrate how it allows direct calculation of the particle trajectories at any point of the fluid, and that these results agree well with numerical simulations of other authors.

Path planning for autonomous underwater vehicle based on an enhanced water wave optimization algorithm

The water wave optimization (WWO) algorithm is inspired by shallow water wave theory and mainly simulates propagation, refraction and breaking to obtain the global optimal solution in the search space. Due to its premature convergence and low optimization efficiency, the basic WWO has a slow convergence speed and low calculation accuracy. To improve the overall optimization performance of the basic WWO, an enhanced WWO based on the elite opposition-based learning strategy and the simplex method (ESWWO) is proposed to solve the function optimization problem and path planning problem for an autonomous underwater vehicle (AUV). The elite opposition-based learning strategy increases the diversity of the population and enhances the global search ability to avoid falling into the local optimum. The simplex method has a fast search speed and strong local search ability to obtain a very accurate solution. The ESWWO algorithm can not only achieve complementary advantages to improve the optimization efficiency of the basic WWO but can also balance exploration and exploitation to obtain the global optimal solution. For the function optimization problem, the ESWWO has strong stability and robustness, and the fitness values of the ESWWO are better than those of other algorithms. For the AUV path planning problem, the ESWWO can avoid threat areas with a minimum fuel cost to obtain the optimal path. The experimental results show that the overall optimization performance of the ESWWO algorithm is superior to that of other algorithms, and thus, ESWWO is an effective and feasible method for solving the function optimization problem and AUV path planning problem.

Local and global pathwise solutions for a stochastically perturbed nonlinear dispersive PDE

In this paper, we consider the periodic Cauchy problem for a stochastically perturbed nonlinear dispersive partial differential equation with cubic nonlinearity, which involves the integrable Novikov equation arising from the shallow water wave theory as a special case. We first establish the existence and uniqueness of local pathwise solutions in Sobolev spaces Hs(T)(s>32) with nonlinear multiplicative noise, where the key ingredients are the stochastic compactness method, the Skorokhod representation theorem and the Gyöngy–Krylov characterization of convergence in probability. In the case of linear multiplicative noise, we investigate the conditions which lead to the blow-up phenomena and global existence of pathwise solution. Finally, we show that the linear multiplicative noise has a dissipative effect on the periodic peakon solutions to the associated deterministic Novikov equation.

Sloshing in Regular Polygonal Basins—Frequencies, Modes, and Tileability

Abstract The classical theory of small amplitude shallow water waves is applied to regular polygonal basins. The natural frequencies of the basins are related to the eigenvalues of the Helmholtz equation. Exact solutions are presented for triangular, square, and circular basins while pentagonal, hexagonal, and octagonal basins are solved, for the first time, by an efficient Ritz method. The first five eigenvalues of each basin are tabulated and the corresponding mode shapes are discussed. Tileability conditions are presented. Some modes (eigenmodes) can be tiled into larger domains.