Linear Water Wave Theory Research Articles

Wave radiation by a horizontal circular cylinder in a three-layer fluid

To construct cylindrical structures like circular pipe bridges or tunnels submerged in the almost still density-stratified ocean or seawater, the study of waves radiated by the cylinder is essential. This research solves the wave radiation problem by calculating non-dimensionalized added mass and damping coefficients to the mass of the fluid displaced by the submerged horizontal cylinder in either layer of a three-layer fluid, which is still otherwise. Under the linear theory of water waves, we investigate the circular cylinder's hydrodynamic forces by its swaying and heaving motion. The time-harmonic wave propagates with three distinct wavenumbers for a given frequency. The method of multipoles has been employed due to its rapid converging solutions by increasing the truncation limit. Potential functions are expressed into systems of linear algebraic equations, which are solved numerically for two sets of unknowns in each case by truncation. Then, the added mass and damping coefficients are obtained from the non-dimensionalized hydrodynamic forces when the horizontal circular cylinder is submerged in the lower, middle and upper layers, respectively. The obtained results are depicted graphically against wavenumber in numerous figures and analysed.

Trapped waves within the blocking frequency under compressed sea ice and two-dimensional current

The existence of trapped modes due to a horizontal submerged rigid cylinder placed below a compressed ice-covered surface is shown to exist within the frequency band of wave blocking by employing linearized water wave theory and with the inclusion of the effects of a two-dimensional current and the obliqueness of the propagating wave. A new modified multipole expansion method catering to the multiple propagating modes is proposed. The impact of different choices of propagating modes at a fixed frequency within the above-mentioned band on the trapped modes is detailed. Multiple trapped modes whose number depends on the direction and magnitude of the current exist within the frequency band. Out of the three choices of current types considered, while two of them result in the generation of trapped modes for all three possible choices of the propagating wave, the remaining one shows that only one propagating mode can generate a trapped wave. A break in the continuous spectrum of the trapped mode frequency due to the wave blocking and at the point of inflexion is presented. There exists a region of angle of propagation within which an abundance of the trapped mode is shown. The surface profiles for different angles of propagation and choices of propagating modes are illustrated graphically. The trapped modes associated with a higher angle of propagation decay at a faster rate.

Numerical investigation of diffraction and radiation of surface gravity waves by a rectangular structure over a stepped bottom with and without a vertical wall

A numerical investigation based on the boundary element method is conducted to analyze the effect of step bottom on the radiation of surface gravity wave by the rectangular marine structure under the assumption of a two-dimensional linearized water wave theory. The investigation is supplemented by analyzing the presence of a back wall located near the floating structure. The accuracy of the method is validated using the known special case of published results. Hydrodynamic coefficients such as added mass and damping coefficients and wave excitation force coefficients are evaluated in both cases. The influence of depth ratio and the distance of step from the center of the structure is calculated to understand the role of the stepped bottom on the hydrodynamic coefficients. On many occasions, these results are compared with the flat bottom case. The study results are helpful to understand the influence of the step-type bottom profile on the design of floating rectangular structures in various marine structure applications and wave energy converter devices.

Wave-triggered breakup in the marginal ice zone generates lognormal floe size distributions: a simulation study

Abstract. Fragmentation of the sea ice cover by ocean waves is an important mechanism impacting ice evolution. Fractured ice is more sensitive to melt, leading to a local reduction in ice concentration, facilitating wave propagation. A positive feedback loop, accelerating sea ice retreat, is then introduced. Despite recent efforts to incorporate this process and the resulting floe size distribution (FSD) into the sea ice components of global climate models (GCMs), the physics governing ice breakup under wave action remains poorly understood and its parametrisation highly simplified. We propose a two-dimensional numerical model of wave-induced sea ice breakup to estimate the FSD resulting from repeated fracture events. This model, based on linear water wave theory and visco-elastic sea ice rheology, solves for the scattering of an incoming time-harmonic wave by the ice cover and derives the corresponding strain field. Fracture occurs when the strain exceeds an empirical threshold. The geometry is then updated for the next iteration of the breakup procedure. The resulting FSD is analysed for both monochromatic and polychromatic forcings. For the latter results, FSDs obtained for discrete frequencies are combined following a prescribed wave spectrum. We find that under realistic wave forcing, lognormal FSDs emerge consistently in a large variety of model configurations. Care is taken to evaluate the statistical significance of this finding. This result contrasts with the power law FSD behaviour often assumed by modellers. We discuss the properties of these modelled distributions with respect to the ice rheological properties and the forcing waves. The projected output can be used to improve empirical parametrisations used to couple sea ice and ocean wave GCM components.

On analytical solutions for transient water waves radiated by cylinders

Analytical solutions for transient water waves radiated by a circular cylinder in deep water are studied in this paper. Within the framework of the potential flow and linear water wave theory, time-domain solutions are analytically derived based on decomposition of the velocity potential consisting of an instantaneous term and a memory term. These two terms are solved by satisfying the governing equations and initial boundary conditions. Transient concentric water waves are further studied and evaluated for a pulsating circular cylinder which expands and contracts radially. Results are compared with those from other numerical models with a good agreement in terms of wave elevation on the free surface. Analytical solutions obtained in this work can be considered as a benchmark in time-domain analysis for linear hydrodynamic problems, and are of help to the theory and application of wavemakers and transient water waves.

Numerical and Experimental Study on the Bragg Reflection of Water Waves by Multiple Vertical Thin Plates

The Bragg reflection of water waves by multiple vertical thin plates is investigated numerically and experimentally. The problem of surface gravity wave interaction with multiple vertical thin plates is formulated mathematically under the assumption of linear water wave theory. Numerical results of the reflection and transmission coefficients are obtained based on the dual boundary element method and compared with the results in the literature. The characteristics of Bragg reflection are represented by the occurring condition, the primary reflection coefficient and the effective bandwidth. The effects of the number, the immerged depth and the spacing of the plates on the characteristics of Bragg reflection are further analyzed systematically. Model experiments are conducted and the experimental results are used to compare with the numerical results of Bragg reflection. This study can provide guidance on the design of multiple vertical thin plates as effective breakwaters by taking advantage of Bragg reflection.

Gravity wave scattering by retrofitted circular breakwaters using dual boundary integral formulation

In the present study, the gravity wave scattering by circular edged breakwaters in the presence of retrofits is analysed within the framework of linearized water wave theory. Various types of retrofits (say, porous shield) such as quarter-circular retrofit, semi-rectangular retrofit, semi-circular retrofit, and rectangular retrofit are investigated to protect the newly developed or existing circular breakwaters from the incident wave stroke. A dual boundary integral formulation is used to solve the boundary value problem considering the quadratic pressure drop boundary condition for the waves past the thin porous barriers. A generalized computer code is developed to accommodate the domain changes and is validated with the known results available in the literature. Various results such as the scattering coefficients (wave reflection, transmission and wave energy loss) and force coefficients (vertical and horizontal force on the breakwater and retrofit) are computed as a function of non-dimensional wave number and discussed. The sharp-edged retrofits (rectangular and semi-rectangular) display a considerable reduction of wave transmission as compared with smooth-edged retrofits, and the semi-circular breakwater with rectangular retrofit is suggested for further development of the porous structures. The effect of porous shield alone, breakwater alone, both breakwater and shield on incident wave energy distribution is also reported. The minimal wave transmission and higher wave damping coefficients attended occurred by the proposed breakwaters are identified when the shield porosity varies within 0.1≤μ≤0.2.

Numerical Approach on Oblique Wave Scattering by a Wide Rectangular Impediment With a Vent Placed Under a Finite Depth Water Body With Ice-Covered Surface

Abstract In this paper, we investigate the scattering of obliquely incident surface water waves on a hurdle as a form of a thick symmetric wall with a gap immersed in a finite depth water body having a cover of a thin ice sheet. In the context of the linear theory of water waves, this two-dimensional problem is formulated as a first-kind integral equation by splitting the velocity potential into symmetric and antisymmetric parts. The integral equation is tackled by using two numerical methods. The first method is the boundary element method where the range of integration is divided into a finite number of small line elements, and choosing the unknown function of the integral equation as constant in each line interval, we reduce the integral equation into a linear system of an algebraic equation. The second method is the multi-term Galerkin Approximation method where the basis functions are chosen as ultraspherical Gegenbauer polynomials to reduce the integral equation to a system of an algebraic equation. These systems of equations are then solved to obtain the unknown function of the integral equation in both methods. Here, very accurate numerical estimates are obtained for the reflection coefficient by both methods which are depicted graphically against the wave number for different parameters involving this problem. Also, there is a good agreement between the results of the reflection coefficient by the two methods.

Scattering of linear gravity-capillary waves by a completely submerged vertical porous elastic plate

The problem of water wave scattering by a vertical porous elastic plate completely submerged in water of infinite depth in the presence of surface tension at the free surface is investigated. Using the concepts of linear water wave theory and Green's Integral theorem, a second kind Fredholm integral equation is obtained which is hypersingular in nature. The obtained hypersingular integral equation is solved numerically. A new mathematical form of the energy identity has been derived which involves an energy loss term that occurs due to the combined effects of surface tension and porous nature of the plate. Numerical estimates for physical quantities like reflection coefficient, energy loss coefficient, hydrodynamic force, and plate deflection are evaluated and then depicted graphically for a certain set of parametric values. The present methodology is validated using the energy identity and the results present in the existing literature. The effect of surface tension on the propagation of gravity waves is analyzed with the help of the various hydrodynamic quantities, which portrays that it is not always insignificant.

Performance of a submerged piezoelectric wave energy converter device floating over an undulated seabed

In this study, the power absorption of a submerged piezoelectric wave energy converter (PWEC) device floating over a rippled seabed is investigated under the linear water wave theory. The boundary element method (BEM) is used to solve the physical problem analogs to the hydrodynamics of the PWEC device. The effect of different structural parameters such as the length of the piezoelectric plate, submergence depth, amplitude, and the number of ripples of the sinusoidal seabed on the power generation of the PWEC device is examined. The study demonstrates that the PWEC plate having moderate length can produce the highest power in the shortwave regime. Further, positioning the PWEC near the free surface can generate more power from ocean wave energy.

Sloshing reduction in three-dimensional rectangular container using rigid baffles

In this paper, liquid sloshing in the three-dimensional rigid rectangular container has been investigated. The effects of a rigid baffle placed on the free surface of the liquid have been analyzed on sloshing frequencies. The tank is partially filled with an ideal and incompressible fluid and the motion of the fluid is irrotational. Linear water wave theory is used to formulate the boundary value problem (BVP). The analytical solution of BVP is obtained in terms of velocity potential with unknown frequency. Mode shapes of the liquid domain and rigid baffle are reported using ANSYS software.

A virtual laboratory for conducting “hands-on” experiments on water wave mechanics

Water wave mechanics are an important topic for scientist and practitioners of civil engineering due to the key role in coastal structure design and the understanding of beach erosion and coastal flooding. Different water wave theories have been proposed over the past two centuries to describe wave transformation, being the so-called linear water wave theory; the most fundamental but still widely used despite its limitations. The use of laboratory facilities in water wave courses, at both undergraduate and graduate levels, is fundamental to provide hands on experience and training on the experimental design and data analysis. Furthermore, it allows to evaluate the validity and limitations of linear wave theory. Unfortunately, such facilities are scarce and their use was further complicated owing to social distancing measures during the COVID-19 outbreak. On the other hand, the advent of computer technology and Computation Fluid Dynamics codes have allowed the development of virtual laboratories for science, technology, and engineering. Here, a widely validated numerical model, which solves the Reynolds Averaged Navier-Stokes (RANS) equations, is employed to develop a virtual wave flume for educational purposes. The virtual laboratory consists of three components: 1) An educational component consisting of lab assignments including background theory, mainly based on the water wave mechanics book by Dean and Dalrymple (1991), and the description of the experiment to be conducted in the virtual wave flume; 2) Synthetic data, corresponding to different RANS simulations, including the spatial-temporal variability of different hydrodynamic variables corresponding to realistic laboratory experiments; and 3) The Graphical User Interface (GUI) allowing the user to select the experiment and download the lab assignment instructions. Within each experiment the user may select different wave conditions and deploy the virtual “sensors” interactively. The experiment and time series corresponding to the virtual sensors can be visualized in real-time. The data acquired with the virtual sensors can be exported as a binary file for further analysis and comparison with theory. The virtual laboratory has been employed as a supporting tool in the Water Wave Mechanics and Nearshore Hydrodynamics graduate courses taught at the National Autonomous University of Mexico. A preliminary assessment of learning outcomes revealed that students agree on the usefulness of the virtual wave flume to the understanding of wave transformation processes and the enjoyment of the courses. The documentation, pre-run numerical simulations, and GUI are integrated in a self-application program developed in Matlab® available for download from a public repository.

Radiation of water waves by a heaving submerged disc in a three-layer fluid

A thin rigid disc is submerged in the infinitely deep lower layer of a three-layer fluid. Using linear water wave theory, the three-dimensional problem of wave radiation due to heaving motion of the disc is considered. The problem is first reduced to solving a hypersingular boundary integral equation. This integral equation is then reduced to a system of one-dimensional Fredholm integral equations of second kind. The solution of each integral equation is associated with a radial component of the difference of potential function across the plate. The solution of these integral equations has been used to compute added mass and damping coefficient numerically. Numerical results for a heaving disc submerged in a single layer, a two-layer, and a three-layer fluid have been depicted. Also, the effects of different parameters like depth of upper layer, middle layer, distance of the disc from the interface, and the density ratios of the fluid layers on these physical quantities have been demonstrated.

Scattering of surface waves by a vertical truncated structured cylinder.

This paper describes the solution to the problem of scattering of plane incident waves on water of constant depth by a bottom mounted circular cylinder, extending partially through the depth, which has an internal structure comprised of closely spaced thin vertical barriers between which fluid is allowed to flow. The problem is solved under full depth-dependent linearized water wave theory using an effective medium equation to describe the fluid motion in cylinder and effective boundary conditions to match that flow to the fluid region outside the cylinder. The interest in this problem lies in the development of novel solution methods for fully three-dimensional water wave interaction with bathymetric plate arrays. Results computed using this theory are compared with a shallow water approximation based on the recent work of Marangos & Porter (2021 Shallow water theory for structured bathymetry. Proc. R. Soc. A 477, 20210421.) and with accurate computations of an exact representation of the geometry using a discrete set of plates. Other results highlight the resonant directional lensing effects of this type of cylindrical plate array device.

Submerged breakwater of a flexible porous membrane with a vertical flexible porous wall over variable bottom topography

An analytical model of oblique wave interaction with a breakwater consisting of a submerged horizontal flexible porous membrane near a vertical porous barrier over variable bottom topography is developed based on linear water wave theory. Using Green's second identity, a mild-slope equation for oblique waves propagating over a submerged porous membrane and a finite variable bottom topography is derived. The analytical solution of the referred problem is obtained by using the eigenfunction expansion method along with the mild-slope equation and Darcy's law is used for wave past horizontal and vertical porous structure. The present results are compared with existing published analytical results and experimental data and observed that they have a good level of agreement. The effect of different design parameters on the wave quantities and wave forces on the porous wall in variable bottom depth is analyzed. It is found that full-wave reflection is drastically reduced and the wave energy is dissipated by the submerged horizontal flexible porous membrane and porous barrier. The analysis of the reflection and dissipation coefficients as well as wave forces on the vertical wall in terms of wave energy dissipation and wave loads by the proposed submerged flexible porous breakwater will be helpful for better understanding in the design of various types of protecting infrastructures for marine operations.

The influence of oblique waves on the hydrodynamic efficiency of an onshore OWC wave energy converter

In this work, the maximum theoretical hydrodynamic efficiency of an onshore Oscillating Water Column (OWC) device is investigated in relation to the incidence angle of the wave and the configuration of the chamber. Linearized water wave theory is used, and the full solution of the associated boundary value problem (BVP) is achieved by using the simple matched eigenfunction expansion method (EEM) and the boundary element method (BEM) with second-order elements. This study is novel in using these techniques to address the influence of oblique water waves on the efficiency of an OWC with a thick front barrier. The hydrodynamic efficiency is analyzed for different directions of wave incidence and OWC chamber geometrical configurations. The semi-analytical and numerical approaches were found to be in good agreement with cases published in the specialized literature. The results show that both a thick front wall and a large incidence angle of the wave can significantly narrow the hydrodynamic efficiency band and modify the resonant frequency.

Efficient asymptotics of solutions to the Cauchy problem with localized initial data for linear systems of differential and pseudodifferential equations

We say that the initial data in the Cauchy problem are localized if they are given by functions concentrated in a neighbourhood of a submanifold of positive codimension, and the size of this neighbourhood depends on a small parameter and tends to zero together with the parameter. Although the solutions of linear differential and pseudodifferential equations with localized initial data constitute a relatively narrow subclass of the set of all solutions, they are very important from the point of view of physical applications. Such solutions, which arise in many branches of mathematical physics, describe the propagation of perturbations of various natural phenomena (tsunami waves caused by an underwater earthquake, electromagnetic waves emitted by antennas, etc.), and there is extensive literature devoted to such solutions (including the study of their asymptotic behaviour). It is natural to say that an asymptotics is efficient when it makes it possible to examine the problem quickly enough with relatively few computations. The notion of efficiency depends on the available computational tools and has changed significantly with the advent of Wolfram Mathematica, Matlab, and similar computing systems, which provide fundamentally new possibilities for the operational implementation and visualization of mathematical constructions, but which also impose new requirements on the construction of the asymptotics. We give an overview of modern methods for constructing efficient asymptotics in problems with localized initial data. The class of equations and systems under consideration includes the Schrödinger and Dirac equations, the Maxwell equations, the linearized gasdynamic and hydrodynamic equations, the equations of the linear theory of surface water waves, the equations of the theory of elasticity, the acoustic equations, and so on. Bibliography: 109 titles.

Surface Gravity Wave Interaction With a Submerged Composite Wavy Porous Plate Attached to a Vertical Wall

Abstract The hydrodynamic performance of composite wavy submerged porous plate attached to a wall is investigated using numerical method multi-domain boundary element method in the linearized water wave theory in which wave past the porous barriers are modeled using Darcy’s law. Effect of the presence of wall on the hydrodynamic parameters like reflection coefficient, dimensionless wave height, wave force and moment acting on the composite wavy porous structure, and horizontal force on the vertical rigid wall is analyzed for various physical parameters like the number of ripple wavelength, structural porosity, submergence depth, and relative amplitude of composite wavy porous plate. The study demonstrates that the efficiency of hydrodynamic characteristics of the composite wavy porous plate attached to the wall is better compared to a horizontal porous plate attached to the wall of the same applicability conditions (around 27% reduction in wave reflection). Moreover, optimum performance of this kind of breakwater system is increased by considering the lower submergence, higher relative ripple amplitude, appropriate relative amplitude, and suitable moderate porosity of the structure in the range of wavenumber 1 < k0h < 5. Dimensionless wave height, horizontal load on the impermeable wall are reduced to zero, while substantial minimization of vertical load on the structure with suitable consideration of the other influencing parameters of porosity Gp = 0.3, relative amplitude Ds/h = 0.1, relative ripple wavelength m > 3, and submergence depth h1/h = 0.3. The present structural arrangement will be useful for attenuating wave effects on the sea wall.

Wave attenuation properties of rubble mound breakwater in tandem with a floating dock against oblique regular waves

This work examines the impact of oblique wave action on a combination of a rubble mound breakwater and a floating dock. The influence of the breakwater on safety of the floating dock is examined in various configurations. The breakwater is approximated by a submerged trapezoidal structure and the floating dock by a surface piercing rigid rectangular structure. A numerical solution is developed combining the eigenfunction expansion method and the multi-domain boundary element method while assuming linear water wave theory. Reflection and transmission coefficients along with wave forces on the dock are evaluated and analyzed over a wide parameter range. The study reveals that the sloping angle of the trapezoidal breakwater plays a vital role in dissipating the wave load on the floating dock and the case of a trapezoidal breakwater with a vertical lee side faces is also examined. The reflection and transmission of waves are found to be dominated by the floating dock parameters in the system, while the wave forces on the dock are reduced by increasing the length and breadth of the breakwater and frictional coefficient of the porous structure, respectively. The combination of two structures in different tandem configurations is proposed as an effective coastal protection system.

Bragg Scattering of Surface Gravity Waves Due to Multiple Bottom Undulations and a Semi-Infinite Floating Flexible Structure

Surface gravity wave interaction with a semi-infinite floating elastic plate in the presence of multiple undulations has been studied under the assumption of linearized water wave theory and small amplitude structural response. The elastic plate is modeled using the Euler-Bernoulli beam equation, whilst the multiple undulations are categorized as an array of submerged trenches or breakwaters. The numerical solution obtained in finite water depth using the boundary element method is validated with the semi-analytic solution obtained under shallow water approximation. Bragg resonance occurs due to the scattering of surface waves by an array of trenches or breakwaters irrespective of the presence of the floating semi-infinite plate. The zero-minima in wave reflection occur when the width of the trench and breakwater is an integer multiple of 0.6 and 0.35 times wavelength, respectively, as the number of trenches or breakwaters increases. In contrast to trenches and breakwaters in isolation, non-zero minima in wave reflection occur in the presence of a semi-infinite plate. Moreover, the number of complete cycles in trenches is less than the number of complete cycles in breakwaters, irrespective of the presence of the floating structure. The frequency of occurrence of zero minimum in wave reflection is reduced in the presence of the semi-infinite plate, and wave reflection increases with an increase in rigidity of the floating plate. Time-dependent simulation of free surface displacement and plate deflection due to multiple undulations of seabed in the presence of the semi-infinite floating plate is demonstrated in different cases.