Equation For Surface Waves Research Articles

Low-frequency surface waves on semi-bounded magnetized quantum plasma

The propagation of low-frequency electrostatic surface waves on the interface between a vacuum and an electron-ion quantum plasma is studied in the direction perpendicular to an external static magnetic field which is parallel to the interface. A new dispersion equation is derived by employing both the quantum magnetohydrodynamic and Poisson equations. It is shown that the dispersion equations for forward and backward-going surface waves are different from each other.

Extended mild-slope equation for surface waves interacting with a vertically sheared current

Propagation of water waves in coastal zones is mainly affected by the influence of currents and bathymetry variations. Models describing wave propagation in coastal zones are often based on the numerical solution of the Mild Slope equation (Kirby, 1984). In this work, an extension of this equation is derived, taking into account the linear variation of the current with depth, which results in a constant horizontal vorticity, slowly varying horizontally, within the background current field. The present approach is based on the asymptotic expansion of the depth-integrated lagrangian, assuming the linear variation of the background current with depth. With the aid of selected examples the role of this horizontal vorticity, associated with the assumed background current velocity profile, is then illustrated and emphasized, demonstrating its effect on the propagation of water waves in coastal areas.

Modified shallow water equations for significantly varying seabeds

In the present study, we propose a modified version of the Nonlinear Shallow Water Equations (Saint-Venant or NSWE) for irrotational surface waves in the case when the bottom undergoes some significant variations in space and time. The model is derived from a variational principle by choosing an appropriate shallow water ansatz and imposing some constraints. Our derivation procedure does not explicitly involve any small parameter and is straightforward. The novel system is a non-dispersive non-hydrostatic extension of the classical Saint-Venant equations. A key feature of the new model is that, like the classical NSWE, it is hyperbolic and thus similar numerical methods can be used. We also propose a finite volume discretisation of the obtained hyperbolic system. Several test-cases are presented to highlight the added value of the new model. Some implications to tsunami wave modeling are also discussed.

Propagation of Long-Wavelength Nonlinear Slow Sausage Waves in Stratified Magnetic Flux Tubes

The propagation of nonlinear, long-wavelength, slow sausage waves in an expanding magnetic flux tube, embedded in a non-magnetic stratified environment, is discussed. The governing equation for surface waves, which is akin to the Leibovich–Roberts equation, is derived using the method of multiple scales. The solitary wave solution of the equation is obtained numerically. The results obtained are illustrative of a solitary wave whose properties are highly dependent on the degree of stratification.

Local well-posedness and persistence properties for the variable depth KDV general equations in Besov space $B^{ \frac 32 }_{2,1}$

In this paper, we consider a nonlinear evolution equation for surface waves in shallow water over uneven bottom. First, the local well-posedness is obtained in Besov space $B^{ \frac 32 }_{2,1}$. Then, persistence properties on strong solutions are also investigated.

Macroeconomic Surface Waves

This paper develops parallels between economics and physics and describes macroeconomics as multi-agent systems on economic space alike to multi-particle systems. Economic agents play roles of simple units of macroeconomics alike to “economic particles” and risk ratings of economic agents define their coordinates on economic space. Economic agents fill macroeconomic area on economic space with coordinates between minimum and maximum risk ratings that define most risky and most secure economic agents respectively. Borders of such macroeconomic area may be disturbed by economic and financial shocks. Such perturbations of macroeconomic borders may generate macroeconomic surface waves alike to surface waves in fluids. These macroeconomic surface waves can propagate along macroeconomic borders and can induce perturbations of macroeconomic variables inside macroeconomic domain. We derive macroeconomic surface wave equations on economic space for simplest local model of interaction of economic agents on economic space. Varieties of waves in simplest macroeconomic models on economic space indicate diversity and importance of hidden wave processes for economic modeling and crises forecasting.

Singular solutions for a class of traveling wave equations arising in hydrodynamics

We give an exhaustive characterization of singular weak solutions for ordinary differential equations of the form üu+12u̇2+F′(u)=0, where F is an analytic function. Our motivation stems from the fact that in the context of hydrodynamics several prominent equations are reducible to an equation of this form upon passing to a moving frame. We construct peaked and cusped waves, fronts with finite-time decay and compact solitary waves. We prove that one cannot obtain peaked and compactly supported traveling waves for the same equation. In particular, a peaked traveling wave cannot have compact support and vice versa. To exemplify the approach we apply our results to the Camassa–Holm equation and the equation for surface waves of moderate amplitude, and show how the different types of singular solutions can be obtained varying the energy level of the corresponding planar Hamiltonian systems.

On Propagation of Rayleigh Type Surface Wave in a Micropolar Piezoelectric Medium

In the present paper, the governing equations of a linear transversely isotropic micropolar piezoelectric medium are specialized for x-z plane after using symmetry relations in constitutive coefficients. These equations are solved for the general surface wave solutions in the medium. Following radiation conditions in the half-space, the particular solutions are obtained, which satisfy the appropriate boundary conditions at the stress-free surface of the half-space. A secular equation for Rayleigh type surface wave is obtained. An iteration method is applied to compute the non-dimensional wave speed of the Rayleigh surface wave for specific material parameters. The effects of piezoelectricity, non-dimensional frequency and non-dimensional material constant, charge free surface and electrically shorted surface are shown graphically on the wave speed of Rayleigh wave.

Travelling wave solutions for a surface wave equation in fluid mechanics

This paper considers a non-linear wave equation arising in fluid mechanics. The exact traveling wave solutions of this equation are given by using G'/G-expansion method. This process can be reduced to solve a system of determining equations, which is large and difficult. To reduce this process, we used Wu elimination method. Example shows that this method is effective.

Coupled response of liquid in a rigid cylindrical container equipped with an elastic annular baffle

The coupled dynamic response of liquid partially filled in a rigid cylindrical container equipped with an elastic baffle under lateral excitation has been studied. Firstly, the liquid domain is divided into four simple sub-domains so that the non-convex liquid domain changes to four convex sub-domains. Therefore, the liquid velocity potential in each liquid sub-domain has continuous boundary conditions of class C1. Based on the superposition principle, the analytical solution of the liquid velocity potential corresponding to each liquid sub-domain is obtained by means of the method of separation of variables. Then, the wet mode of the elastic annular baffle is expressed in terms of the dry-modal functions. The free surface wave equation, the interface equations and the coupled liquid-baffle vibration equations are all expressed in terms of the Fourier series along the liquid height and the Bessel series in the radial direction, respectively. The orthogonality among the coupled modes of the system is demonstrated. Finally, the total velocity potential function under lateral excitation is taken as the sum of the container potential function and the liquid perturbed potential function. The coupled dynamic response equation of the liquid-baffle system is established by combining the free surface wave equation and the governing vibration equation of the baffle. The surface wave height, the hydrodynamic pressure distribution, the resultant hydrodynamic force and moment for a container subjected to lateral excitations are discussed in detail.

A New Development of Reduced Hamiltonian Equations for Ocean Surface Waves: an Extension From Small to Finite Amplitude

The reduced 3-wave and 4-wave Hamiltonian equations for ocean surface waves were widely used for the simplified structure with symmetric polynomial kernels and for the conservation of energy, etc. However, according to the related assumption for approximation in derivation, the range of applicability was limited to weakly nonlinear waves of small amplitude. Here the following issue was further studied: for nonlinear waves of finite amplitude within a certain range, was it also possible to obtain reduced Hamiltonian equations with symmetric polynomial kernels in a sense of sufficient accuracy? Because of complicated strongly nonlinear coupling, few development in this significant respect had been made as yet. A new approach was proposed based on the Chebyshev polynomials to best approximate the primitive water wave equations in the exact sense of strongly nonlinear coupling and derive new reduced Hamiltonian equations with symmetric polynomial kernels. The new results exhibit an extension from a weakly nonlinear case in which the product of the wave number and the wave steepness is small to a nonlinear case in which this product goes up to about 1.035. Moreover, within this range, the approximation errors are lower than 5%, and in particular, the new results prove exact whenever the said product lies close to 0.9.

On the diffusion-strain coupling and dispersion of surface waves in transversely isotropic laser-excited solids

The present paper is aimed at studying the boundary value problem in elasticity theory concerning the propagation behavior of harmonic waves and vibrations on the surface of the transversely isotropic laser-excited crystalline solids with atomic defect generation. Coupled dynamical diffusion-–deformation interaction model is employed to study this problem. The frequency equations of surface waves in closed form are derived and discussed. The three motions, namely, longitudinal, transverse, and diffusion of the medium are found to be dispersive and coupled with each other due to the defect concentration changes and anisotropic effects. The phase velocity and attenuation coefficient of the surface waves get modified due-to the defect–strain coupling and anisotropic effects, and are also influenced by the defect relaxation time. A softening of frequencies of surface acoustic waves (instability of frequencies) is obtained. Relevant results of previous investigations are deduced as special and limiting cases.

Quantitative methods for reconstructing tissue biomechanical properties in optical coherence elastography: a comparison study

We present a systematic analysis of the accuracy of five different methods for extracting the biomechanical properties of soft samples using optical coherence elastography (OCE). OCE is an emerging noninvasive technique, which allows assessment of biomechanical properties of tissues with micrometer spatial resolution. However, in order to accurately extract biomechanical properties from OCE measurements, application of a proper mechanical model is required. In this study, we utilize tissue-mimicking phantoms with controlled elastic properties and investigate the feasibilities of four available methods for reconstructing elasticity (Young’s modulus) based on OCE measurements of an air-pulse induced elastic wave. The approaches are based on the shear wave equation (SWE), the surface wave equation (SuWE), Rayleigh-Lamb frequency equation (RLFE), and finite element method (FEM), Elasticity values were compared with uniaxial mechanical testing. The results show that the RLFE and the FEM are more robust in quantitatively assessing elasticity than the other simplified models. This study provides a foundation and reference for reconstructing the biomechanical properties of tissues from OCE data, which is important for the further development of noninvasive elastography methods.

Rotation and Magnetic Field Effect on Surface Waves Propagation in an Elastic Layer Lying over a Generalized Thermoelastic Diffusive Half-Space with Imperfect Boundary

The aim of the present investigation is to study the effects of magnetic field, relaxation times, and rotation on the propagation of surface waves with imperfect boundary. The propagation between an isotropic elastic layer of finite thickness and a homogenous isotropic thermodiffusive elastic half-space with rotation in the context of Green-Lindsay (GL) model is studied. The secular equation for surface waves in compact form is derived after developing the mathematical model. The phase velocity and attenuation coefficient are obtained for stiffness, and then deduced for normal stiffness, tangential stiffness and welded contact. The amplitudes of displacements, temperature, and concentration are computed analytically at the free plane boundary. Some special cases are illustrated and compared with previous results obtained by other authors. The effects of rotation, magnetic field, and relaxation times on the speed, attenuation coefficient, and the amplitudes of displacements, temperature, and concentration are displayed graphically.

Surface Wave Propagation in a Microstretch Thermoelastic Diffusion Material under an Inviscid Liquid Layer

The present investigation deals with the propagation of Rayleigh type surface waves in an isotropic microstretch thermoelastic diffusion solid half space under a layer of inviscid liquid. The secular equation for surface waves in compact form is derived after developing the mathematical model. The dispersion curves giving the phase velocity and attenuation coefficients with wave number are plotted graphically to depict the effect of an imperfect boundary alongwith the relaxation times in a microstretch thermoelastic diffusion solid half space under a homogeneous inviscid liquid layer for thermally insulated, impermeable boundaries and isothermal, isoconcentrated boundaries, respectively. In addition, normal velocity component is also plotted in the liquid layer. Several cases of interest under different conditions are also deduced and discussed.

Traveling surface waves of moderate amplitude in shallow water

We study traveling wave solutions of an equation for surface waves of moderate amplitude arising as a shallow water approximation of the Euler equations for inviscid, incompressible and homogeneous fluids. We obtain solitary waves of elevation and depression, including a family of solitary waves with compact support, where the amplitude may increase or decrease with respect to the wave speed. Our approach is based on techniques from dynamical systems and relies on a reformulation of the evolution equation as an autonomous Hamiltonian system which facilitates an explicit expression for bounded orbits in the phase plane to establish existence of the corresponding periodic and solitary traveling wave solutions.

Homogenization in the problem of long water waves over a bottom site with fast oscillations

The system of equations of gravity surface waves is considered in the case where the basin’s bottom is given by a rapidly oscillating function against a background of slow variations of the bottom. Under the assumption that the lengths of the waves under study are greater than the characteristic length of the basin bottom’s oscillations but can be much less than the characteristic dimensions of the domain where these waves propagate, the adiabatic approximation is used to pass to a reduced homogenized equation of wave equation type or to the linearized Boussinesq equation with dispersion that is “anomalous” in the theory of surface waves (equations of wave equation type with added fourth derivatives). The rapidly varying solutions of the reduced equation can be found (and they were also found in the authors’ works) by asymptotic methods, for example, by the WKB method, and in the case of focal points, by the Maslov canonical operator and its generalizations.

Non-uniform continuity of the flow map for an evolution equation modeling shallow water waves of moderate amplitude

We prove that the flow map associated to a model equation for surface waves of moderate amplitude in shallow water is not uniformly continuous in the Sobolev space Hs with s>3/2. The main idea is to consider two suitable sequences of smooth initial data whose difference converges to zero in Hs, but such that neither of them is convergent. Our main theorem shows that the exact solutions corresponding to these sequences of data are uniformly bounded in Hs on a uniform existence interval, but the difference of the two solution sequences is bounded away from zero in Hs at any positive time in this interval. The result is obtained by approximating the solutions corresponding to these initial data by explicit formulae and by estimating the approximation error in suitable Sobolev norms.

An integrable evolution equation for surface waves in deep water

In order to describe the dynamics of monochromatic surface waves in deep water, we derive a nonlinear and dispersive system of equations for the free surface elevation and the free surface velocity from the Euler equations in infinite depth. From it, and using a multiscale perturbative method, an asymptotic model for small wave steepness ratio is derived. The model is shown to be completely integrable. The Lax pair, the first conserved quantities as well as the symmetries are exhibited. Theoretical and numerical studies reveal that it supports periodic progressive Stokes waves which peak and break in finite time. Comparison between the limiting wave solution of the asymptotic model and classical results is performed.

Local well-posedness and wave breaking results for periodic solutions of a shallow water equation for waves of moderate amplitude

We study the local well-posedness of a periodic nonlinear equation for surface waves of moderate amplitude in shallow water. We use an approach proposed by Kato which is based on semigroup theory for quasi-linear equations. We also show that singularities for the model equation can occur only in the form of wave breaking, in particular surging breakers.