Linear Water Wave Theory Research Articles

Wave reflection by a vertical wall with a horizontal submerged porous plate

By applying the linear water wave theory and the eigenfunction expansion method, the wave reflection by a vertical wall with a horizontal submerged porous plate is investigated in this paper. The numerical results, concerning the effects of the dimensionless plate length, the relative water depth, and the porous effect parameter of the plate on the wave loads on the plate and the wave height near the wall as well as the reflection coefficient, are discussed. It is found that the submerged plate increases the complexity of the phenomenon related to the wave reflection and refraction in the close region of the wall, and leads to the occurrence of the phenomenon of wave trapping. The results indicate that there may exist a process of focusing wave energy near the wall for small dimensionless porous effect parameters, whereas the increase of the dimensionless porous effect parameter decreases gradually the wave height until setdown occurs. The behavior of a larger plate with proper porosity is similar to that of a wave absorber which can significantly suppress not only the wave height above the plate but also the reflection waves. The ability of the porous plate to reduce the wave height on the wall surface is, in general, directly proportional to the dimensionless plate length and may be strongest for a proper value of the dimensionless porous effect parameter. It is also demonstrated that the wave loads on a porous plate are smaller than those on an impermeable plate.

Scattering of water wave by a submerged horizontal plate and a submerged permeable breakwater

Based on a two-dimensional linear water wave theory, this study develops the boundary element method (BEM) to examine normally incident wave scattering by a fixed, submerged, horizontal, impermeable plate and a submerged permeable breakwater in water of finite depth. Numerical results for the transmission coefficients are also presented. In addition, the numerical technique's accuracy is demonstrated by comparing the numerical results with previously published numerical and experimental ones. According to that comparison, the transmission coefficient relies not only on the submergence of the horizontal impermeable plate and the height of the permeable breakwater, but also on the distance between horizontal plate and permeable breakwater. Results presented herein confirm that the transmission coefficient is minimum for the distance approximately equal to four times the water depth.

A Basic Investigation on Deflection Wave Propagation and Strength of Very Large Floating Structures (The 3rd Report)

This paper deals with the dynamic response of Very Large Floating Structures (VLFS) in regular waves considering the deflection wave propagation (VLFS-wave) using a simple beam or plate modeling.Firstly, two kinds of dispersion relations of the VLFS-wave are derived based on the analogy of linear water wave theory, and it is shown that the eigen functions of the VLFS-wave are independent, but not orthogonal. Next, the function separation method-the method for obtaining the relationships among progressive waves-which are the incident wave, reflecting wave, VLFS-wave, and so on, is proposed. Finally, using this method, the stationary wave, reflection and refraction waves of the VLFS are considered. The law of reflection and refraction between the incident wave, reflecting wave and the VLFS-wave are derived, and the simplified dynamic response formulae are proposed for the stationary wave of a 2-dimensional beam, and for a semi-infinite large plate in which case the stationary wave is neglected.

The hydrodynamic coefficients for an oscillating rectangular structure on a free surface with sidewall

Based on a two dimensional linear water wave theory, the boundary element method (BEM) is developed and applied to study the heave and the sway problem of a floating rectangular structure in water to finite depth with one side of the boundary is a vertical sidewall and the other boundary is an open boundary. Numerical results for the added mass and radiation damping coefficients are presented. These coefficients are not only depend on the submergence and the width of the structure, but also depend on the clearance between structure and sidewall. Negative added mass and sharp peaks in the damping and added mass coefficients have been found when the clearance with a value close to integral times of half wave length of wave generated by oscillation structure. The important effect of the clearance on the added mass and radiation damping coefficients are discussed in detail. An analytical solution method is also presented. The BEM solution is compared with the analytical solution, and the comparison shows good agreement.

A Basic Investigation on Deflection Wave Propagation and Strength of Very Large Floating Structures (The 2nd Report)

This paper deals with the dynamic response of Very Large Floating Structures (VLFS) in regular waves considering the deflection wave propagation (VLFS-wave) using a simple beam modeling. Firstly, the dispersion relation of VLFS-wave is derived based on the analogy of linear water wave theory, and the relation between the incident wave length and VLFS-wave length is derived. The effects of water depth and rigidity of VLFS on wave length (or dispersion relation) of VLFS-wave are investigated. Next, the reflection wave due to the difference of dispersion relation between the incident wave and VLFS-wave is discussed under some particular assumptions. Finally, a simple method for the response analysis of VLFS is proposed. Using this method, the dynamic response analysis is carried out numerically, and its validity is shown as compared with the results of some other exact analyses. From these results, we can identify the following properties of VLFS : (1) VLFS-wave propagates faster than the incident wave. (2) VLFS-wave length is longer than the incident wave length. (3) When the water depth or the rigidity of VLFS increases, VLFS-wave length also increases. (4) When the incident wave number is sufficiently smaller than the characteristic wave number of VLFS, the wave length or the wave height of VLFS-wave becomes equal to those of the incident wave.

The Wave Resistance of a Two-Dimensional Body Moving Forward in a Two-Layer Fluid

A two-dimensional body moves forward with constant velocity in an inviscid, incompressible fluid under gravity. The fluid consists of two layers having dierent densities, and the body is totally submerged in one of them. The resulting fluid motion is assumed to be steady state in a coordinate system attached to the body. The boundary value problem for velocity potential is considered in the framework of linearized water wave theory. The asymptotics of solution at infinity is obtained with the help of integral representation, based on the explicitly known Green function. The theorem of unique solvability is announced, and the method applied for its proving is briefly explained (the detailed proof is given in another work). An explicit formula for the wave resistance is derived and discussed. An example of numerical calculation of the wave resistance serves to illustrate the so-called "dead water" phenomenon.

Diffraction of water waves by slender barriers

Linear water-wave theory is used to tackle the problem of diffraction of surface waves by a fixed slender barrier in deep water for two basic situations: (i) when the barrier is partially immersed, and (ii) when the barrier is completely submerged. Analytical expressions for the first-order corrections to the reflection and transmission coefficients are derived in terms of integrals involving the shape functions describing the two sides of the slender barrier. A relatively straightforward perturbation technique is used along with the application of Green's theorem in the fluid region. Corresponding analytical expressions representing the reflection and transmission coefficients are also deduced, (i) for a nearly vertical barrier and (ii) for a vertically symmetric slender barrier, as special cases for both the problems. For a nearly vertical barrier it is observed, analytically, that there is no first-order correction to the transmitted wave at any frequency. Computations for the reflection and transmission coefficients up to O(ɛ), where ɛ; is a small nondimensional number, are also performed and presented here.

Sloshing characteristics in rectangular tanks with a submerged block

The effects of a bottom-mounted rectangular block on the sloshing characteristics of the fluid in rectangular tanks are investigated using the linear water wave theory. For the sloshing analysis the velocity potential is decomposed into those for the wall-induced waves and the reflected, transmitted and scattered waves by the block. The reflection and transmission coefficients related to the block are determined using the continuity conditions of mass flux and energy flux on the common vertical boundaries of the fluid regions and the boundary conditions on both sides of the block. Verification of the present analysis method is carried out by comparing the results with those by other methods. The analysis results indicate that the size and location of the block significantly influence the sloshing frequencies and mode shapes. In general, sloshing frequencies reduce as the block becomes tall and wide The variations of the sloshing frequencies and mode shapes due to the block are found to be more sensitive in broad tanks than in tall tanks. The hydrodynamic pressure exerted on the tank wall and the block changes significantly as the block becomes large and moves toward the wall.

Hydrodynamic characteristics of a thin rolling plate in finite depth of water

The two-dimensional problem of the small-amplitude forced rolling motion of a thin vertical plate in finite water depth is considered using linear water wave theory. A range of geometric cases is considered, allowing the plate to be surface-piercing, attached to the bottom, unmounted and in each case either isolated or in the presence of a vertical wall. The method of solution uses two complementary formulations in conjunction with an accurate and efficient Galerkin method which are shown to give results that are in good agreement for all cases.

Relation of vertical ground reaction forces to walking speed

The influence of viscoelastic ocean beds on the characteristics of surface waves passing through mangrove forests is analyzed under the assumption of linearized water wave theory in two dimensions. The trunks of the mangroves are assumed to be in the upper-layer inviscid fluid domain, whilst the roots are inside the viscoelastic bed. The associated equation of motion is obtained by coupling the Voigt’s model for flow within the viscoelastic medium with the equation of motion in the presence of mangroves. The modified dynamic conditions are coupled with the kinematic conditions to obtain the boundary condition at the free surface and the interface of the two fluids consisting of the upper layer inviscid fluid and the viscoelastic fluid bed. To understand the effects of bed viscosity as well as elasticity on energy dissipation, the complex dispersion relation associated with the plane progressive wave is derived and analyzed. Effect of physical parameters associated with mangroves and viscoelastic bed on wave motion in surface and internal modes are computed and analyzed to understand their roles in attenuating wave effects. The present model will be useful in the better understanding of wave propagation through mangroves in the coastal zone having muddy seabed.

Wave scattering by periodic arrays of breakwaters

Oblique incidence of plane waves upon an infinite array of in-line periodic screens or breakwaters in finite water depth is considered using linear water-wave theory. The number of reflected or transmitted waves is a function of the angle of incidence and the ratio of wavelength to array periodicity. A simple matrix formulation is provided for all the reflection and transmission coefficients arising from a particular set of parameters, using a formulation based either on the unknown velocity through a gap or on the unknown pressure difference across a breakwater screen. Integral properties of functions related to these unknowns form the basis of the matrix structure, the functions themselves satisfying a set of integral equations which are solved using a Galerkin approximation that gives highly accurate approximations with very few terms in the expansion. The problem is extended to consider two identical parallel arrays and it is shown that there exists zeros of both reflection and transmission. Finally, a wide-spacing approximation is derived for two arrays based on the accurate results found from the single array problem, where the two arrays do not have to be identical, but must have the same periodicity.

Numerical Methods to Predict Characteristics of Air Chamber for Fixed Terminator Type of Wave Energy Converter.

The numerical methods for analyzing a fixed terminator type of wave energy conversion device are described under the condition that the linear water wave theory is applicable. Two methods are proposed in order to calculate the device characteristics of an air chamber. One uses the flow rate and the gage pressure in the air chamber directly, because the interaction between the oscillating water column and the turbine is found to be controlled only by the flow rate and the pressure drop through the turbine in this system. The other method uses the equation of the floating body motion in a manner similar to the equivalent floating body approximation. The relations between these two methods are also examined. The hydrodynamic performance with frequency from zero to infinity is required in the simulation of irregular waves. However, resonances occur when the air chamber breadth is equal to an integral multiple of a half wavelength. Therefore the impulse response function is modified in order to eliminate the effects of the resonance frequency. Finally, it is confirmed that these solutions give good agreement with the experimental data.

Scattering by periodic array of rectangular blocks

Scattering properties of an incident field upon a periodic array of identical rectangular barriers, each extending throughout the water depth, are calculated based on a Galerkin approximation to an integral representation of the problem derived using the linear theory of water waves. The method incorporates full multi-modal scattring using the linear theory of water waves. The method incorporates full multi-modal scattering using a matrix formulation and is equivalent to a corresponding two-dimensional acoustics problem also discussed.

The interaction of waves with horizontal cylinders in two-layer fluids

We consider two-dimensional problems based on linear water wave theory concerning the interaction of waves with horizontal cylinders in a fluid consisting of a layer of finite depth bounded above by a free surface and below by an infinite layer of fluid of greater density. For such a situation time-harmonic waves can propagate with two different wavenumbers K and k. In a single-layer fluid there are a number of reciprocity relations that exist connecting the various hydrodynamic quantities that arise. These relations are systematically extended to the two-fluid case. It is shown that for symmetric bodies the solutions to scattering problems where the incident wave has wavenumber K and those where it has wavenumber k are related so that the solution to both can be found by just solving one of them. The particular problems of wave scattering by a horizontal circular cylinder in either the upper or lower layer are then solved using multipole expansions.

Edge waves along periodic coastlines. Part 2

Numerical evidence of the existence of edge waves travelling along a periodic coastline consisting of a straight and vertical cliff face from which protrudes an infinite number of identical rectangular barriers, each extending throughout the water depth, is given based on a Galerkin approximation to an integral representation of the problem derived using the linear theory of water waves.

Hydrodynamic characteristics of an oscillating water column device

We consider a wave energy device consisting of a thin vertical surface-piercing barrier next to a vertical wall in finite depth water. Power is extracted due to a normally incident wave forcing the free surface of the fluid between the barrier and the wall to oscillate, in turn pumping the volume of air above the free surface through a uni-directional turbine housed at the opening of the device. Under the assumptions of linear water wave theory, the important hydrodynamic properties are expressible in terms of integral quantities of functions proportional to the fluid velocity under the barrier. These functions each satisfy integral equations, the solutions of which are approximated very accurately and efficiently using a Galerkin method as described in Porter and Evans [Porter, R. & Evans, D. V., Complementary approximations to wave scattering by vertical barriers. J. Fluid Mech., 294 (1995) 155–80].

Asymptotic approximations for functions defined by series, with some applications to the theory of guided waves

Mellin transforms are used here to find asymptotic approximations for functions defined by series. The simplest cases are those of the form Σ n=1 ∞ u(nx). Such series are called separable here, because the given function u is sampled at points whose variation with n and x is separated. Nonseparable series are analysed by first approximating them by separable series. Both types of series arise in the theory of electromagnetic waveguides and in the theory of linear water waves; several examples are worked out in detail

On step approximations for water-wave problems

The scattering of water waves by a varying bottom topography is considered using two-dimensional linear water-wave theory. A new approach is adopted in which the problem is first transformed into a uniform strip resulting in a variable free-surface boundary condition. This is then approximated by a finite number of sections on which the free-surface boundary condition is assumed to be constant. A transition matrix theory is developed which is used to relate the wave amplitudes at ±∞. The method is checked against examples for which the solution is known, or which can be computed by alternative means. Results show that the method provides a simple accurate technique for scattering problems of this type.

Solution of a singular integral equation involving two intervals arising in the theory of water waves

Using the well-known inverse of the Cauchy singular integral operator on a single interval along with certain standard results associated with singular integrals, the general solution of a singular integral equation in a double interval arising in the linear theory of water waves, whose kernel involves a logarithmic as well as a Cauchy type singularity, has been obtained. The solutions are found to agree with the known ones.

Detection of oil spills on the sea using radar measurements

We propose a simple approach to the evaluation of parameters of organic films on the sea surface from radar measurements. The approach is based on the perturbation technique for calculating the scattered field intensity and linear theory of water waves on the surface of a viscous fluid. As a result, we have obtained an evident expression for the intensity of the signal backscattered from the sea surface covered by a film, depending on the angle of incidence, surface tension, and elastic properties of the film. The obtained dependence is in qualitative agreement with the experimental data. The influence of the film on the Doppler shift of the scattered field frequency is also discussed.