Surface Wave Problem Research Articles

Problem of Surface Waves on Water in Higher School Laboratory Workshop

Traditionally, in lecture courses on General Physics, the example of waves propagating on the water surface is used for the qualitative description of wave phenomena. At the same time, in corresponding laboratory courses, there are almost no quantitative tasks on the same topic, the purpose of which would be to measure the surface wave characteristics. The reason for this difference must be associated with the complexity of the physical mechanism of generation and propagation of waves on the water surface. In this work, a laboratory problem has been developed to measure the group velocity of circular waves traveling on the water surface. It uses a standard laboratory water tank equipped with a surface wave generator. The proposed experiment offers an opportunity for quantitative measurement of surface wave characteristics, a dispute of only qualitative observations of wave phenomena commonly found in current curricula, due to the introduction of two improvements. First, to produce the water waves’ rich in-contrast representations on a screen, the optimal generator amplitude is specially set for each experiment; and second, to measure the group wavelength, the device magnification is preliminary calculated based on the laws of geometric optics. The tests carried out have shown that the relative accuracy of these measurements is quite acceptable: within a few percent. The introduction of the problem of surface waves on water into the General Physics laboratory workshop for engineering specialties of technical universities will contribute to improving the quality of mastering the subject by students.

On the decay of viscous surface waves in 3D

We consider the incompressible viscous surface wave problem in the setting that the fluid domain is a horizontal infinite layer in 3D. The fluid dynamics is governed by the gravity-driven incompressible Navier–Stokes equations, and the effect of surface tension is ignored on the upper free surface. We prove the optimal time-decay rate of the low-order energy of the solution with minimal derivative count 3, which implies that the Lipschitz norm of the velocity decays at the rate (1+t)−1. This together with a time-weighted estimate for the highest order spatial derivatives of the free surface function leads to the boundedness of the high-order energy, which improves the result of Wang [9].

Electromagnetic signal propagation through lossy media via surface electromagnetic waves.

A theory of surface electromagnetic waves in gradient media exhibiting arbitrary surface gradients of dielectric permittivity and magnetic permeability has been developed. Novel low-loss propagating surface wave solutions have been found in the gradient media in which both dielectric permittivity and magnetic permeability are dominated by their imaginary parts. Several examples of gradient geometries in which the surface wave problem may be solved analytically have been found. Examples of practically useful surface wave geometries spanning from radio communication underwater to UV nanophotonics have been demonstrated.

Asymptotic formula of Rayleigh wave speed in an orthotropic half-space coated by a thin piezoelectric film

ABSTRACT Piezoelectric thin films have been widely used in electro-acoustic devices including ones measuring Rayleigh wave speed. The main objective of this work is to find an explicit formula of Rayleigh speed taking into account the effect of the thin piezoelectric film. First the problem of Rayleigh surface waves propagating along an orthotropic substrate coated by a thin piezoelectric film is considered using Stroh’s formalism approach. Then, the exact secular equation of Rayleigh wave is obtained in implicit form. In case when the thickness of the thin layer is much smaller than the considering wavelength of Rayleigh waves, an asymptotic formula of Rayleigh speed is derived from the asymptotic form of the exact secular equation. The obtained asymptotic formula is simple enough and could be used in practice when Rayleigh speed is measured by using piezoelectric device. Numerical illustrations are carried out to show the validity of the obtained approximate formulas and its potential application in inverse problem.

(2+1)-dimensional KdV, fifth-order KdV, and Gardner equations derived from the ideal fluid model. Soliton, cnoidal and superposition solutions

We study the problem of gravity surface waves for an ideal fluid model in the (2+1)-dimensional case. We apply a systematic procedure to derive the Boussinesq equations for a given relation between the orders of four expansion parameters, the amplitude parameter α, the long-wavelength parameter β, the transverse wavelength parameter γ, and the bottom variation parameter δ. Such an approach, known in (1+1)-dimensional theory, is extended for the first time for (2+1)-dimensional shallow water waves. Using this procedure, we derived the only possible (2+1)-dimensional extensions of the Korteweg–de Vries equation, the fifth-order KdV equation and the Gardner equation in three cases of the relationship between these parameters. All these equations are non-local. When the bottom is flat, the (2+1)-dimensional KdV equation can be transformed to the Kadomtsev–Petviashvili equation in a fixed reference frame and next to the classical KP equation in a moving frame. Thus, the Kadomtsev–Petviashvili equation gained a derivation from the fundamental laws of hydrodynamics. We have found soliton, cnoidal, and superposition solutions (essentially one-dimensional) to the (2+1)-dimensional Korteweg–de Vries equation and the Kadomtsev–Petviashvili equation.

On the formulation of energy conservation in the eeKdV equation

The Korteweg-de Vries (KdV) equation is a well-known model equation for unidirectional shallow-water (long) surface waves. The equation includes dispersion and weak non-linearity. The derivation of the equation originates in assuming that the velocity potential takes the form of an asymptotic expansion, and applying this in the classical surface wave problem. While a typical assumption on the relative size of non-dimensional key parameters introduced in the derivation will give the KdV equation as a final result, one can change the assumption on the relative size of parameters, and end up with an equation including terms in higher orders in desired parameters. The present article presents the derivation of an extended form referred to as the eeKdV equation.Information regarding various properties of the flow can be found by studying the derivation of the eeKdV equation itself, and some of the relations found can be used for studying the energy balance of a system modeled by the equation. In line with previous work for the KdV equation, we present here corresponding formulations of energy balance laws for an inertial reference frame, in context of the eeKdV equation. We also present a partial verification of the KdV and eeKdV energy flux expressions by looking at a far-field, uniform flow situation, as well as performing a numerical study to confirm the assumed behaviour of the error in the eeKdV energy equation in the context of a undular bore flow setup. Further, conserved integrals for the eeKdV equation are presented and numerically checked.

Analysis of a simplification strategy in a nonhydrostatic model for surface and internal wave problems

Analysis of a simplification strategy in a nonhydrostatic model for surface and internal wave problems

Well-posedness of the surface wave problem for two dimensional micropolar fluids

We consider a viscous fluid in a finite-depth domain of two dimensions, with a free moving boundary and a fixed solid boundary. The fluid dynamics are governed by gravity-driven incompressible micropolar equations, and the surface tension is neglected on the free surface. The main result of this paper is to prove the global well-posedness of the surface wave problem when the fluid domain is horizontally periodic. And the solutions decay to equilibrium at an almost exponential rate. This is achieved by establishing delicate estimates when dealing with the strong coupling of fluid velocity and micro-rotation velocity. Our proof relies on two-tier nonlinear energy method developed by Guo and Tice [17–19].

Hamiltonian formulation of the stochastic surface wave problem

We devise a stochastic Hamiltonian formulation of the water wave problem. This stochastic representation is built within the framework of the modelling under location uncertainty. Starting from restriction to the free surface of the general stochastic fluid motion equations, we show how one can naturally deduce Hamiltonian structure under a small noise assumption. Moreover, as in the classical water wave theory, the non-local Dirichlet–Neumann operator appears explicitly in the energy functional. This, in particular, allows us, in the same way as in deterministic setting, to conduct systematic approximations of the Dirichlet–Neumann operator and to infer different simplified wave models including noise in a natural way.

A Time-Domain Analytic Solution of Flow-Induced Undular Bores

In this study, the problem of surface waves induced by water flow in a flow channel was investigated. The mathematical model based on the potential wave theory was established, and a new analytic solution to the corresponding initial and boundary value problem was proposed. To confirm our analytic solution, the mathematical model was applied to simulate experiments conducted in a flow channel in the laboratory. Using our analytic solution, water surface elevations and flow velocities at certain locations in the channel were compared with experimental results. Comparisons between our analytic solution and experimental results confirmed our theory that amplitudes and propagating phases are in very close agreement. Our analytic solution can be used to calculate variations in pressure and velocity along the water depth, which are expensive to calibrate and obtain in experiments. Although our analytic solution was established based on linear theory, it is very practical for applications studying the basic properties of surface elevation, velocity, and pressure of the flow field induced by water current both in space and time.

Boussinesq’s equations for (2+1)-dimensional surface gravity waves in an ideal fluid model

We study the problem of gravity surface waves for the ideal fluid model in (2+1)-dimensional case. We apply a systematic procedure for deriving the Boussinesq equations for a prescribed relationship between the orders of four expansion parameters, the amplitude parameter $\alpha$, the long-wavelength parameter $\beta$, the transverse wavelength parameter $\gamma$, and the bottom variation parameter $\delta$. We also take into account surface tension effects when relevant. For all considered cases, the (2+1)-dimensional Boussinesq equations can not be reduced to a single nonlinear wave equation for surface elevation function. On the other hand, they can be reduced to a single, highly nonlinear partial differential equation for an auxiliary function $f(x,y,t)$ which determines the velocity potential but is not directly observed quantity. The solution $f$ of this equation, if known, determines the surface elevation function. We also show that limiting the obtained the Boussinesq equations to (1+1)-dimensions one recovers well-known cases of the KdV, extended KdV, fifth-order KdV, and Gardner equations.

Finite element modeling of extreme wave analysis with dissipation in arbitrary shaped domain

A finite element numerical model is presented considering the effect of refraction, diffraction, and energy dissipation by breakwater. As wave move from deep to shallow water coastal regions, wave parameter changes as a result of depth variations, presence of irregular shore boundaries, and wave reflections from the coastline. Therefore, to understand the wave transformation phenomenon near the coastal region, mild slope equation is solved using the finite element method. The numerical model is validated against the experimental and numerical results available for the rectangular domain and is applied to a real coastal Paradip port, situated in Odissa (India). The wave height is computed at the selected key stations in the port. From the simulated results, it is observed that the present numerical model with dissipation parameter is an efficient and useful tool for the analysis of water surface wave problem in realistic domains.

A robust method for surface wave dispersion in anisotropic semi-infinite periodically layered structures with coating layers

This study presents a robust method towards the computation of dispersion curves of surface waves in an anisotropic semi-infinite periodically layered structure (PLS) coated by a stack of layers. Using the properties of the symplectic transfer matrix to analyze the relationship between the eigenequations of surface waves in a semi-infinite PLS and of its corresponding finite PLS, the eigenfrequencies within the stopbands for surface waves in a semi-infinite PLS are solved by a finite PLS consisting of a sufficient number of unit cells. The eigenvalues of a finite PLS are calculated by combining the precise integration method with the Wittrick–Williams (W-W) algorithm, which makes the present method accurate, efficient and numerically stable. On this basis, using the concept of the eigenvalue count in the W-W algorithm, the eigenfrequencies of surface waves in a semi-infinite PLS coated by a stack of layers are accurately computed for any given wavenumber. Unlike most existing approaches that determine dispersion curves by directly seeking roots from the transcendental eigenequation, the present method takes full advantages of the eigenvalue count to find all eigenfrequencies without the possibility of any being missed. The method can be applicable for surface wave problems in an arbitrarily anisotropic semi-infinite PLS and does not restrict layer number, layer thickness and material properties.

Thermoelastic waves in double porosity materials

The focus of this paper is on the propagation of thermoelastic waves with assigned wavelength within the context of theory of thermoelasticity for materials with double porosity. It is shown that there exist two shear waves that are undamped in time, non-dispersive and that are unaltered by the presence of pore system or by thermal effects. Furthermore, there also exist other four longitudinal wave solutions that are dispersive and damped in time: one longitudinal quasi-elastic wave and two quasi-pore modes and one quasi-thermal mode. The dispersion relation is explicitly established like a fifth degree algebraic equation with real coefficients and hence it always allows some real roots, thus predicting existence of some standing waves. A numerical analysis for Berea sandstone shows that the speed of propagation of the longitudinal quasi-elastic wave is larger than the wave speed for its counterpart from the classical elastic model or that in the double porosity elastic model. The propagation of the Rayleigh surface waves is addressed and the corresponding secular equation is explicitly established. As an illustrative example it is used Berea sandstone when the secular equation is solved by means of the graphical method to prove that the Rayleigh surface wave problem admits a solution.

An ISPH with k–ε closure for simulating turbulence under solitary waves

In this paper an Incompressible Smoothed Particle Hydrodynamics (ISPH) method solving the 2D RANS (Reynolds Averaged Navier-Stokes) equations with the k–ε turbulence closure is constructed. In the present model, the concept of “massless ISPH” utilizing the definition of “particle density” (number of computational particles within unit volume) is stressed. The skills of this numerical model are tested by applying to two laboratory experiments: (1) A non-breaking solitary wave propagating over a bottom-mounted barrier and (2) a solitary wave breaking on a 1 on 50 slope. In the former case flow separation occurs behind the barrier as the wave crest passes by and a vortex is generated, which later interacts with free surface causing breaking. In the latter case wave breaking and bottom friction both generate significant turbulence. For both cases, the effects of initial seeding of turbulent kinetic energy, required in the k–ε model, are studied and it is concluded that initial values of O (10−10) to O (10−8) m2/s2 should be used. An adaptive wall boundary condition for k–ε turbulence model is employed to avoid the unrealistic production of turbulence near the wall boundary. The numerical results, in terms of free surface profile, mean velocity field, vorticity field, turbulent kinetic energy and turbulent shear stress, are compared with experimental data. Very reasonable agreement is observed. This paper presents the first comprehensively validated 2D ISPH model with the k–ε turbulence closure, which can be applied to transient free surface wave problems.

О численном моделировании поверхностных волн в бассейне с подвижными непроницаемыми границами

Представлены результаты, связанные с разработкой вычислительного алгоритма для изучения поверхностных волн в областях с подвижными непроницаемыми границами (подвижные боковые стенки бассейна, подвижные фрагменты дна). Алгоритм основан на конечноразностной аппроксимации на подвижных сетках двумерной модели потенциальных течений несжимаемой жидкости со свободной границей. Предложены новые начальные данные для задачи о движении уединенной волны, неотражающие условия, способ задания нулевого итерационного приближения для расчета потенциала скорости и результаты исследования устойчивости численного метода. Показано, что поверхностные волны, вызванные подводным оползнем, существенно влияют на процесс наката на берег одиночной волны и могут увеличить ее максимальный заплеск. In numerical simulation of fluid flows with a free surface, the most difficult problems are those in which not only the free boundary is mobile, but also some other parts of the boundary of the region occupied by the liquid. For example, these are the problems of surface waves caused by underwater and coastal landslides, the generation of waves by wavemakers in laboratory flumes and tanks, the problem of waves runup on the shore, the interaction of waves with moving wave protection walls and problems addressed moving semisubmerged or fully immersed objects. The purpose of this paper is to analyze the properties and evaluate the capabilities of the computational algorithm based on the finitedifference approximation of the equations of potential fluid flows with a free boundary and designed to study surface waves in a confined basin, part of the impermeable boundary of which can be mobile. The algorithm relies on the use of curvilinear grids that adapt to all parts of the boundary, both moving and stationary. New initial data are proposed for the problem of motion of a solitary or a single wave, consistent with the initial data for the shallow water equations of the second longwave approximation. New nonreflecting conditions have been developed that allow waves to be emitted across the boundary of a flow region with very little reflection. A new initial approximation is proposed for the iterative process of calculating the potential of the velocity vector. By using this approach we significantly reduce the number of iterations at each time step. The original stability condition of the linearized difference scheme is derived. The reasons for the appearance of two peaks in the chronograms of pressure when the long waves of large amplitude roll onto a vertical wall are discussed. The capabilities of the numerical algorithm are demonstrated on the problem of generating waves by a moving wall travelling in the initial part of the flume. The results of the calculations well reproduce the experimental data, in particular, the decrease in the length and increase in the amplitude of the wave when it moves towards the shallow part of the flume, as well as the formation of a dispersion tail as the waves reverse motion after reflection from the vertical wall installed at the end of the flume. The developed algorithm was used to study the process of generation for surface waves by an underwater landslide moving along an uneven bottom, and the interaction of these landslide waves with a single surface wave moving towards the shore. It is shown that the surface waves caused by an underwater landslide significantly affect the process of rolling of a single wave on the shore and it could increase its maximum splash.

The Viscous Surface Wave Problem with Generalized Surface Energies

We study a three-dimensional incompressible viscous fluid in a horizontally periodic domain with finite depth whose free boundary is the graph of a function. The fluid is subject to gravity and gen...

Rayleigh waves in isotropic strongly elliptic thermoelastic materials with microtemperatures

This paper is concerned with the linear theory of thermoelasticity with microtemperatures, based on the entropy balance proposed by Green and Naghdi, which permits the transmission of heat as thermal waves of finite speed. We analyze the behavior of Rayleigh waves in an unbounded isotropic homogeneous strongly elliptic thermoelastic material with microtemperatures. The related solution of the Rayleigh surface wave problem is expressed as a linear combination of the elements of the bases of the kernels of appropriate matrices. The secular equation is established and afterwards an explicit form is written when some coupling constitutive coefficients vanish. Then, we solve numerically the secular equation by mean of a grafical metod and by taking arbitrary data for strongly elliptic thermoelastic material.

3D hydroelastic analysis of very large floating bodies over variable bathymetry regions

The coupled-mode model developed by Belibassakis and Athanassoulis (J Fluid Mech 531:221–249, 2005) is extended and applied to the hydroelastic analysis of three-dimensional large floating bodies of shallow draft lying over variable bathymetry regions. The method is also applicable to the problem of wave interaction with ice sheets of small thickness. A general bathymetry is assumed, characterized by a continuous depth function, joining two regions of constant, but possibly different, depth. We consider the scattering problem of harmonic incident surface waves, under the combined effects of variable bathymetry and a floating elastic plate of orthogonal planform shape. Under the assumption of small-amplitude waves and plate deflections, the hydroelastic problem is formulated within the context of linearized water-wave and thin elastic-plate theory. To consistently treat the wave field beneath the elastic floating plate, down to the sloping bottom boundary, a complete, local, hydroelastic-mode series expansion of the wave field is used, enhanced by an appropriate sloping-bottom mode. The latter enables the consistent satisfaction of the Neumann bottom-boundary condition on a general topography. Numerical results concerning floating structures over flat and inhomogeneous seabed are presented, and the effects of wave direction, bottom slope and bottom corrugations on the hydroelastic responses are discussed.

Analytical models of stationary nonlinear gravitational waves

Euler’s equations with standard boundary conditions for the problem of potential surface waves of an arbitrary amplitude in a homogeneous liquid layer with a flat bottom are converted into the new system, including integral and differential equations for the of the potential and its time derivative near the surface. The basic formula of the theory of infinitesimal waves, paired Korteweg-de Vries (KdV) and Kadomtsev− Petviashvili (KP) equations, the envelope Zakharov−Shabat soliton follows from the system in limiting case. The resulting generalized equation, unlike traditional KdFand KP-equations is suitable for the description of waves on the surface of the initially quiescent fluid. A new exact solutions for gravity waves in a deep water, expressed in terms of complex Lambert’s functions are constructed.