Scattering of gravity waves by multiple surface-piercing floating membrane

Interaction of surface gravity waves with multiple vertically moored surface-piercing membrane breakwaters in finite water depth is analyzed based on the linearized theory of water waves. The study is carried out using least square approximation method to understand the effect of the vertical membrane as effective breakwater. Initially the problem is studied for a single membrane wave barrier but for the case of multiple membrane breakwaters the study is carried out using the method of wide-spacing approximation. In the present study, it is observed that the deflection of the membrane is reduced with the increase in the stiffness parameter of the mooring lines attached to the membrane. In the case of single surface-piercing membrane with moored and fixed edge conditions, the reflection and transmission coefficients are compared and analyzed in detail. The resonating pattern in the reflection coefficients are also observed for multiple floating membrane which can also be referred as Bragg's resonance. In the presence of the porosity constant the wave reflection is also observed to be decreasing and the change in the distance between the vertical floating breakwaters also helps in the attenuation of wave height. It is observed that the presence of multiple floating breakwater helps in the reduction of wave height in the transmitted region.     

Surface gravity wave interaction with elastic bottom

In the present paper, a hydroelastic model is developed to deal with surface gravity wave interaction with an elastic bed based on the small amplitude water wave theory and plate deflection in finite water depth. The elastic bottom bed is modelled as a thin elastic plate and is based on the Euler-Bernoulli beam equation. The wave characteristics in the presence of the elastic bed is analyzed in both the cases of deep and shallow water waves. Further, the linearized long wave equation is generalized to include bottom flexibility. A generalized expansion formula for the velocity potential is derived to deal with the boundary value problems associated with surface gravity waves having an elastic bed. The utility of the expansion formula is illustrated by demonstrating specific physical problems which will play significant role in the analysis of wave structure interaction problems. Behavior of the wave spectra are discussed in the case of closed basin having a free surface and an elastic bottom topography.     

Water wave scattering by bottom undulations in an ice-covered two-layer fluid

The scattering of plane surface waves by bottom undulations in an ice-covered ocean modelled as a two-layer fluid consisting of a layer of fresh water of lesser density above a deep layer of salt water, is investigated here by using a simplified perturbation analysis. In such a two-layer fluid there exist waves of two different modes, one with higher mode propagates along the interface and the other with lower mode propagates along the ice-cover. An incident wave of a particular mode gets reflected and transmitted by the bottom undulations into waves of both the modes so that transfer of wave energy from one mode to another takes place. The first-order reflection and transmission coefficients of two different modes are obtained due to incident waves of again two different modes by employing Fourier transform technique in the mathematical analysis. For sinusoidal bottom topography these coefficients are depicted graphically against the wavenumber. These figures show how the transfer of energy from one mode to another takes place.     

Interaction of ocean waves with floating and submerged horizontal flexible structures in three-dimensions

A three-dimensional general mathematical hydroelastic model dealing with the problem of wave interaction with a floating and a submerged flexible structure is developed based on small amplitude wave theory and linear structural response. The horizontal floating and submerged flexible structures are modelled with a thin plate theory. The linearized long wave equations based on shallow water approximations are derived and results are compared. Three-dimensional Green’s functions are derived using fundamental source potentials in water of finite and infinite depths. The expansion formulae associated with orthogonal mode-coupling relations are derived based on the application of Fourier transform in finite and infinite depths in case of finite width in three-dimensions. The usefulness of the expansion formula is demonstrated by analysing a physical problem of surface gravity wave interaction with a moored finite floating elastic plate in the presence of a finite submerged flexible membrane in three-dimensions. The numerical accuracy of the method is demonstrated by computing the complex values of reflected wave amplitudes for different modes of oscillation and mooring stiffness. Further, the effect of compressive force and modes of oscillations on a free oscillation hydroelastic waves in a closed channel of finite width and length for floating and submerged elastic plate system is analysed.     

Diffraction of Water Waves by A Vertically Floating Cylinder in A Two-Layer Fluid

In this paper, the diffraction of water waves by a vertically floating cylinder in a two-layer fluid of a finite depth is studied. Analytical expressions for the hydrodynamic loads on the vertically floating cylinder are obtained by use of the method of eigenfunction expansions. The hydrodynamic loads on the vertically floating cylinder in a two-layer fluid inelude not only the surge, heave and pitch exciting forces due to the incident wave of the surface-wave mode, but also those due to the incident wave of the internal-wave mode. This is different from the case of a homogenous fluid. Some given examples show that, for a two-layer fluid system with a small density difference, the hydrodynamic loads for the surface-wave mode do not differ significantly from those due to surface waves in a single-layer fluid, but the hydrodynamic loads for the internal-wave mode are important over a wide range of frequencies. Moreover, also considered are the free surface and interface elevations generated by the diffraction wave due to the incident wave of the surface-wave and interhal-wave modes, and transfer of energy between modes.     

Oblique wave interaction with a floating structure near a wall with stepped bottom

Diffraction of obliquely incident waves by a floating structure near a wall with step-type bottom topography is investigated under the three-dimensional small amplitude wave theory. Full solution of the problem under the potential flow approach is obtained by the matched eigenfunction expansion method. The wave-induced forces on the structure and on the wall, the reflection and transmission characteristics and the wave elevations in the free surface regions are studied for different incident wave angles, water depth ratios and dimension of the structure and the distance of the wall from the center of the structure. The problem is reformulated under shallow water approximations and results are compared with the finite depth results.     

Hydroelastic interaction between obliquely incident waves and a semi-infinite elastic plate on a two-layer fluid

The hydroelastic response of a semi-infinite thin elastic plate floating on a two-layer fluid of finite depth due to obliquely incident waves is investigated. The upper and lower fluids with different densities separated by a sharp and stable interface are assumed to be inviscid and incompressible and the motion to be irrotational. Simply time-harmonic incident waves of the surface and interfacial wave modes with a given angular frequency are considered within the framework of linear potential flow theory. With the aid of the methods of matched eigenfunction expansion and the inner product of the two-layer fluid, a closed system of simultaneous linear equations is derived for the reflection and transmission coefficients of the series solutions. Based on the dispersion relations for the gravity waves and the flexural–gravity waves in a two-layer fluid and Snell’s law for refraction, we obtain a critical angle for the incident waves of the surface wave mode and three critical angles for the incident waves of the interfacial wave mode, which are related to the existence of the propagating waves. Graphical representations of the series solutions show the interaction between the water waves and the plate. The effects of several physical parameters, including the density and depth ratios of the fluid and the thickness of the plate, on the wave scattering and the hydroelastic response of the plate are studied. It is found that the variation of the thickness of the plate may change the wave numbers and the critical angles. The density ratio is the main factor to influence the wave numbers of the interfacial wave modes. Finally, the stress state is considered.     

Scattering of Semidiurnal Internal Kelvin Wave at Step Bottom Topography

A three-dimensional, multi-level model was used to study the energy dissipation of semidiurnal internal Kelvin waves due to their interaction with bottom topography. A simplified topography consisting of a channel with an additional shallow bay was used to clarify the wave’s scattering process. When the first mode semidiurnal internal wave given at an open boundary arrives at the bay mouth, higher-mode internal waves are generated at a step bottom of the bay mouth. As a result, the energy of the first mode internal Kelvin wave is effectively decayed. The decay rate of the internal Kelvin wave depends on both the width and length of the additional bay. The maximum decay rate was found when a resonance condition occurs the bay, that is, the bay length is equal to a quarter of wave length of the first mode internal wave on the shallow region. The decay rate in the wide bay cases is higher than that in a narrow case, due to a contribution from the scattering due to the Poincare wave that emanates from the corners of the bay head. The decay rate with the additional bay is 1.1–1.8 times that of the case without the additional bay. The decay rate due to the scattering process is found to be of the same order as that of the internal and bottom friction.     

地形对陆架拦截波性质及机制的影响

陆架波的性质如频散关系、形成机制等受地形影响。研究地形对陆架波的性质影响具有重要意义。基于陆架拦截波理论,数值计算了分段线性地形下不同宽度陆架上陆架拦截波的频散关系、长波假设下波动的相速度、阻尼情况下的波动耗散率以及强迫波的外力影响因子。分析了陆架宽度及坡度对自由及强迫陆架拦截波性质的影响。陆架宽度影响陆架拦截波的频散关系。陆架变宽,使得长波频散曲线的斜率增大。陆架宽度的增加使第一模态陆架拦截波有明显的性质变化:相速度增大,波动受辐散影响的程度变大,摩擦衰减距离增大,且风应力旋度在波动的生成机制中起到的作用渐强。在宽的陆架上,研究陆架拦截波的生成及强迫波的振幅时,应充分考虑风应力旋度的作用。第二、三模态波动的相速度受陆架坡度的影响较大,但摩擦衰减距离基本都在200km左右,几乎不随陆架宽度改变,属于局地波。     

Interaction of current and flexural gravity waves

The interaction between current and flexural gravity waves generated due to a floating elastic plate is analyzed in two dimensions under the assumptions of linearized theory. For plane flexural gravity waves, explicit expressions for the water particle dynamics and trajectory are derived. The effect of current on the wavelength, phase velocity and group velocity of the flexural gravity waves is analyzed. Variations in wavelength and wave height due to the changes in current speed and direction are analyzed. Effects of structural rigidity and water depth on wavelength are discussed in brief. Simple numerical computations are performed and presented graphically to explain most of the theoretical findings in a lucid manner.     

Shoaling Surface Gravity Waves Cause a Force and a Torque on the Bottom

Freely propagating surface gravity waves are observed to slow down and to stop at a beach when the bottom has a relatively gentle upward slope toward the shore and the frequency range of the waves covers the most energetic wind waves (sea and swell). Essentially no wave reflection can be seen and the measured reflected energy is very small compared to that transmitted shoreward. One consequence of this is that the flux of the wave’s linear momentum decreases in the direction of wave propagation, which is equivalent to a time rate of change of the momentum. It takes a force to cause the time rate of change of the momentum. Therefore, the bottom exerts a force on the waves in order to decrease the momentum flux. By Newton’s third law (action equals reaction) the waves then impart an equal but opposite force to the bottom. In shallow (but finite) water depths the wave force per unit bottom area is calculated, for normal angle of incidence to the beach, to be directly proportional to the square of the wave amplitude and to the bottom slope and inversely proportional to the mean depth; it is independent of the wave frequency. Constants of proportionality are: 1/4, the fluid density and the acceleration of gravity. Swell attenuation near coasts and some characteristics of sand movement in the near-shore region are not inconsistent with the algebraic structure of the wave force formula. Since the force has a depth variation which is significantly faster than that of the dimensions of the particle orbits in the vertical direction, the bottom induces a torque on the fluid particles that decreases the angular momentum flux of the waves. By an extension of Newton’s third law, the waves also exert an equal but opposite torque on the bottom. And because the bottom force on the waves exists over a horizontal distance, it does work on the waves and decreases their energy flux. Thus, theoretically, the fluxes of energy, angular and linear momentum are not conserved for shoaling surface gravity waves. Mass flux, associated with the Stokes drift, is assumed to be conserved, and the wave frequency is constant for a steady medium.     

The prediction of the dynamic and structural motions of a floating-pier system in waves

In this study, a two-dimensional floating pier consists of single rectangular impermeable pontoon with side supporting pile-columns is studied. The purpose of this study is to present a theoretical solution for the linearized problem of incident waves exerting on a floating pier with pile-restrained. All boundary conditions are linearized in the problem, which is incorporated into a scattering problem and radiation problem with unit displacement. The method of separation of variables is used to solve for velocity potentials. For the radiation problem with unit heave and pitch amplitude, the boundary value problem with non-homogeneous boundary condition beneath the structure is solved by using a solution scheme. By calculating the wave force from velocity potential and solving the equation of motion of the floating structure simultaneously a close form theoretical solution for the problem is developed. The finite element method was also applied to calculate the dynamic responses on the supporting piles subjected to the pontoon motions and incident waves.     

Momentum balance in shoaling gravity waves: Comment on ’shoaling surface gravity waves cause a force and a torque on the bottom’ by K. E. Kenyon

In a recent paper, Kenyon (2004) proposed that the wave-induced energy flux is generally not conserved, and that shoaling waves cause a mean force and torque on the bottom. That force was equated to the divergence of the wave momentum flux estimated from the assumption that the wave-induced mass flux is conserved. This assumption and conclusions are contrary to a wide body of observations and theory. Most importantly, waves propagate in water, so that the momentum balance generally involves the mean water flow. Although the expression for the non-hydrostatic bottom force given by Kenyon is not supported by observations, a consistent review of existing theory shows that a smaller mean wave-induced force must be present in cases with bottom friction or wave reflection. That force exactly balances the change in wave momentum flux due to bottom friction and the exchange of wave momentum between incident and reflected wave components. The remainder of the wave momentum flux divergence, due to shoaling or wave breaking, is compensated by the mean flow, with a balance involving hydrostatic pressure forces that arise from a change in mean surface elevation that is very well verified by observations.     

A coupled-mode model for water wave scattering by horizontal, non-homogeneous current in general bottom topography

A coupled-mode model is developed for treating the wave–current–seabed interaction problem, with application to wave scattering by non-homogeneous, steady current over general bottom topography. The vertical distribution of the scattered wave potential is represented by a series of local vertical modes containing the propagating mode and all evanescent modes, plus additional terms accounting for the satisfaction of the free-surface and bottom boundary conditions. Using the above representation, in conjunction with unconstrained variational principle, an improved coupled system of differential equations on the horizontal plane, with respect to the modal amplitudes, is derived. In the case of small-amplitude waves, a linearised version of the above coupled-mode system is obtained, generalizing previous results by Athanassoulis and Belibassakis [J Fluid Mech 1999;389:275–301] for the propagation of small-amplitude water waves over variable bathymetry regions. Keeping only the propagating mode in the vertical expansion of the wave potential, the present system reduces to an one-equation model, that is shown to be compatible with mild-slope model concerning wave–current interaction over slowly varying topography, and in the case of no current it exactly reduces to the modified mild-slope equation. The present coupled-mode system is discretized on the horizontal plane by using second-order finite differences and numerically solved by iterations. Results are presented for various representative test cases demonstrating the usefulness of the model, as well as the importance of the first evanescent modes and the additional sloping-bottom mode when the bottom slope is not negligible. The analytical structure of the present model facilitates its extension to fully non-linear waves, and to wave scattering by currents with more general structure.     

Interaction of flexural gravity waves with shear current in shallow water

In the present study, the effect of shear current on the propagation of flexural gravity waves is analyzed under the assumptions of linearized shallow-water theory. Explicit expressions for the reflection and transmission coefficients associated with flexural gravity wave scattering by a step discontinuity in both water depth and current speed are derived. Further, trapping and scattering of flexural gravity waves by a jet-like shear current with a top-hat profile are examined and certain limiting conditions for the waves to exist are derived. The effects of change in water depth, current speed, incident wavelength and the angle of incidence on the group and phase velocities as well as on the reflection and transmission characteristics are analyzed through different numerical results.     

A note on the derivation of potential energy for two-dimensional water waves

The potential energy available in a two-dimensional progressive water wave can be calculated in numerous ways. One derivation of this energy based on the first law of thermodynamics and on the linearized velocity potential for waves, is presented in this paper. The energy densities and total energy expressions are given for deep and finite depth water waves. It is also shown that the travelling component of the energy for deep water waves is the potential energy component.     

Effect of higher-order bottom variation terms on the refraction of water waves in the extended mild-slope equations

This study investigates how the refraction of water waves is affected by the higher-order bottom effect terms proportional to the square of bottom slope and to the bottom curvature in the extended mild-slope equations. Numerical analyses are performed on two cases of waves propagating over a circular shoal and over a circular hollow. Numerical results are analyzed using the eikonal equation derived from the wave equations and the wave ray tracing technique. It is found that the higher-order bottom effect terms change the wavelength and, in turn, change the refraction of waves over a variable depth. In the case of waves over a circular shoal, the higher-order bottom effects increase the wavelength along the rim of shoal more than near the center of shoal, and intensify the degree of wave refraction. However, the discontinuity of higher-order bottom effects along the rim of shoal disperses the foci of wave rays. As a result, the amplification of wave energy behind the shoal is reduced. Conversely, in the case of waves over a circular hollow, the higher-order bottom effects decrease the wavelength near the center of the hollow in comparison with the case of neglecting higher-order bottom effects. Consequently, the degree of wave refraction is decreased, and the spreading of wave energy behind the hollow is reduced.     

Waves,long waves and nearshore morphology

Recent field measurements on beaches of different slopes have established that wave motion at periods substantially longer than the incident waves dominates the velocity field close to the shore. Analysis of a number of extensive data sets shows that much of this long wave motion is in the form of progessive edge waves, though forced wave motion, standing edge waves and free waves propagating away from the shore may also contribute to the energy.Theoretically, the drift velocities in bottom boundary layers due to edge waves show spatial patterns of convergence and divergence which may move sediment to form either regular crescentic or cuspate features when only one edge wave mode dominates, or a bewildering array of bars, bumps and holes when several phase-locked modes exist together.Convincing field demonstration of the link between nearshore topography and edge waves only exists for the special case of small-scale beach cusps on steep beaches, formed by edge waves at the subharmonic (twice the period) of the incident waves. At longer periods the link is proving more difficult to establish, due to the longer time-scales of topographic changes, the interaction between pre-existing topography and the water motion, and the observation of broad-banded edge wave motion which is not readily linked to topography with a well-defined scale.These ideas are, however, central to the study of nearshore processes, as most of the plausible alternate hypotheses do not seem to lead to quantitative predictions. Clearly, further theoretical and observational work is essential.     

Scattering of Oblique Water Waves by Two Unequal Surface-Piercing Vertical Thin Plates with Stepped Bottom Topography

Based on linear water-wave theory, this study investigated the scattering of oblique incident water waves by two unequal surface-piercing thin vertical rigid plates with stepped bottom topography. By using the matched eigenfunction expansion method and a least square approach, the analytical solutions are sought for the established boundary value problem. The effects of the incidence angle, location of step, depth ratio of deep to shallow waters, and column width between two plates, on the reflection coefficients, the horizontal wave forces acting on the two plates, and the mean surface elevation between the two plates, are numerically examined under a variety of wave conditions. The results show that the existence of the stepped bottom between two plates considerably impacts the hydrodynamic performances of the present system. It is found that the effect of stepped bottom on the reflection coefficient of the present two-plate structure is evident only with waves of the low dimensionless frequency. Moreover, the influence of the step location on the hydrodynamic performance of the present two-plate structure is slight if the step is placed in between the two plates.     

Numerical modeling of the long surface waves scattering for the 2011 Japan tsunami: Case study

The March 11, 2011, megaquake caused a catastrophic tsunami recorded throughout the Pacific. This paper presents an analysis of the sea-level records obtained from deep-water tsunami meters (DART and NEPTUNE). To evaluate the effect of the sea-level oscillations’ decay, a statistical analysis of observations and numerical modeling of tsunami generation and propagation have been conducted. The main goal is to uncover physical mechanisms of the tsunami wave field formation and evolution at scales up to tens of thousands of kilometers in space and a few days in time. It is shown that the tsunami lifetime is related to the wave-energy diffusion and dissipation processes. The decay time of the variance of the tsunami-generated level oscillations is about 1 day. Multiple reflections and scattering by irregularities of the bottom topography make the field of the secondary tsunami waves stochastic and incoherent: the distribution of the wave energy in the ocean reaches a statistical equilibrium in accordance with the Rayleigh-Jeans law of equipartition of the wave energy per degree of freedom. After the tsunami front has passed, the secondary-wave energy density turns out to be inversely proportional to the water depth.