Linear Water Wave Theory Research Articles

Sufficient conditions on the existence of trapped modes in problems of the linear theory of surface waves

A new method is proposed for searching for trapped modes in several problems of linear water wave theory. In the case of submerged bodies, the method gives simple proofs of known results, and the condition obtained is completely new in the case of surface piercing bodies. Bibliography: 24 titles.

The semiclassical Maupertuis-Jacobi correspondence and applications to linear water wave theory

T d = R/2πZ, J = (J1, . . . , Jd) ∈ M . Here M is a domain in Rd, and φ = (φ1, . . . , φd), φj ∈ S1, are angle coordinates on Λ(J) defining the half-density |dφ1 ∧ · · · ∧ dφd|. Let γj , j = 1, . . . , d, be basic cycles on Λ(J). They determine the action variables given in the canonical coordinates (x, p) by the formulas Ij = ∮ γj p dx and also the Maslov indicesmj . Ij = ∮ γj pdx, andmj . We shall always assume that the actions Ij coincide with the parameters Jj and that the actions J and the angles φ onΛ(J) are are conjugate symplectic coordinates (in general, local). Since theΛ(J) smoothly depend on the parameters, we can assume that the cycles γj smoothly depend on J ∈ M , and hence the Maslov indices mj prove to be the same for the cycles γj with a fixed j. Let us fix certain (distinguished) points r(J) on Λ(J). Further, assume that we are given smooth functions χ(φ, J) on Λ(J), which also depend smoothly on J . Let h > 0 be a small parameter. For each fixed h, consider the setMq ⊂ M of vectors J(k, h), k = (k1, . . . , kd) ∈ Zn, with components Jj(k, h), satisfying the Bohr–Sommerfeld–Maslov quantization rule

Explicit Representation for the Infinite-Depth Two-Dimensional Free-Surface Green's Function in Linear Water-Wave Theory

In this paper we derive an explicit representation for the two-dimensional free-surface Green's function in water of infinite depth, based on a finite combination of complex-valued exponential integrals and elementary functions. This representation can easily and accurately be evaluated in a numerical manner, and its main advantage over other representations lies in its simplicity and in the fact that it can be extended towards the complementary half-plane in a straightforward manner. It seems that this extension has not been studied rigorously until now, and it is required when boundary integral equations are extended in the same way. For the computation of the Green's function, the limiting absorption principle and a partial Fourier transform along the free surface are used. Some of its properties are also discussed, and an expression for its far field is developed, which allows us to state appropriately the involved radiation condition. This Green's function is then used to solve the two-dimensional infinite-depth water-wave problem by developing a corresponding boundary integral equation, whose solution is determined by means of the boundary element method. To validate the computations, a benchmark problem based on a half-circle is presented and solved numerically.

Asymptotic reflection of linear water waves by submerged horizontal porous plates

On the basis of linear water-wave theory, an explicit expression is presented for the reflection coefficient R ∞ when a plane wave is obliquely incident upon a semi-infinite porous plate in water of finite depth. The expression, which correctly models the singularity in velocity at the edge of the plate, does not rely on knowledge of any of the complex-valued eigenvalues or corresponding vertical eigenfunctions in the region occupied by the plate. The solution R ∞ is the asymptotic limit of the reflection coefficient R as a → ∞, for a plate of finite length a bounded by a rigid vertical wall, and forms the basis of a rapidly convergent expansion for R over a wide range of values of a. The special case of normal incidence is relevant to the design of submerged wave absorbers in a narrow wave tank. Modifications necessary to account for a finite submerged porous plate in a fluid extending to infinity in both horizontal directions are discussed.

Wave radiation by a sphere submerged in a two-layer ocean with an ice-cover

Using linear water-wave theory, we consider wave radiation (both heave and sway) by a sphere submerged in a two-layer ocean consisting of a layer of fresh water of finite depth with an ice-cover and an infinite layer of salt water. The sphere is submerged in either of the two layers. Employing the method of multipoles each problem is reduced to an infinite system of linear equations which are solved numerically by standard techniques after truncation. The added-mass and damping coefficients for a heaving and a swaying sphere are obtained and depicted graphically against the wave number for various values of depth of the submerged sphere and flexural rigidity of the ice-cover to demonstrate the effect of the presence of ice-cover on these quantities. When the flexural rigidity is taken to be very small, the numerical results for these quantities almost coincide with those for a two-layer ocean with a free-surface.

Propagation of oblique waves over small bottom undulation in an ice-covered two-layer fluid

Scattering of oblique incident waves by small bottom undulation in a two-layer fluid, where the upper layer has a thin ice-cover while the lower one has the undulation, is investigated within the framework of linearized water wave theory. The ice-cover is being modeled as an elastic plate of very small thickness. There exist two modes of time-harmonic waves–one with lower wave number propagating along the ice-cover (ice-cover mode) and the other with higher wave number along the interface (interfacial mode). A perturbation analysis is employed to solve the corresponding boundary value problem governed by modified Helmholtz equation and thereby evaluating the reflection and transmission coefficients approximately up to first order for both modes. A patch of sinusoidal ripples, having two different wave numbers over two consecutive stretches, is considered as an example and the related coefficients are determined. It is observed that when the wave is incident on the ice-cover surface we always find energy transfer to the interface, but for interfacial incident waves there are parameter ranges for which no energy transfer to the ice-cover surface is possible. Also it is observed that for small angles of incidence, the reflected energy is more as compared to the other angles of incidence. These results are demonstrated in graphical form. From the derived results, the solutions for problems with free surface can be obtained as particular cases.

Scattering of internal waves in a two-layer fluid flowing through a channel with small undulations

Using two-dimensional linear water wave theory, we consider the problem of normal water wave (internal wave) propagation over small undulations in a channel flow consisting of a two-layer fluid in which the upper layer is bounded by a fixed wall, an approximation to the free surface, and the lower one is bounded by a bottom surface that has small undulations. The effects of surface tension at the surface of separation is neglected. Assuming irrotational motion, a perturbation analysis is employed to calculate the first-order corrections to the velocity potentials in the two-layer fluid by using Green’s integral theorem in a suitable manner and the reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. Two special forms of the shape function are considered for which explicit expressions for reflection and transmission coefficients are evaluated. For the specific case of a patch of sinusoidal ripples having the same wave number throughout, the reflection coefficient up to the first order is an oscillatory function in the quotient of twice the interface wave number and the ripple wave number. When this quotient approaches one, the theory predicts a resonant interaction between the bed and the interface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of the incident wave energy occurs if this number is large. Again, when a patch of sinusoidal ripples having two different wave numbers for two consecutive stretches is considered, the interaction between the bed and the interface near resonance attains in the neighborhood of two (singular) points along the x-axis (when the ripple wave number of the bottom undulation become approximately twice as large as the interface wave number). The theoretical observations are presented in graphical form.

A direct relationship between bending waves and transition conditions of floating plates

Ocean waves travel deep into ice fields in the polar regions, both affecting the formation of sea-ice and causing its break-up. Recently, it has been shown that a relatively simple linear water and bending wave theory can predict the decay rate of the wave energy travelling through fractured ice sheets and floes at the geophysically important wave periods of 6–15 s. That work used simple free-edge conditions. A possible improvement to the current model is to better represent the effective connection due to partially frozen cracks that occur in practice. The Wiener–Hopf technique gives explicit formulae for the velocity potential and surface deflection, expressed as series expansions over the modes of the elastic plate floating on water of finite depth, with the coefficients in the expansion given as functions of four constants. These constants are determined by a system of four linear equations, represented by a 4-by-4 matrix and a four-element vector. The elements of the matrix are given as explicit functions of relationship between edge conditions. General connections between ice sheets may be interpreted as a vertical and a rotational spring providing transition conditions for the shear force and the bending moment. The reflection and the transmission of waves can then be simply calculated as direct functions of the connection conditions. Conversely, reflected and transmitted waves allow complete characterization of the effective connection conditions at a material discontinuity.

Effect of Tsunamis generated in the Manila Trench on the Gulf of Thailand

Tsunamis generated in the Manila Trench can be a threat to Thailand. Besides runup of tsunamis along the eastern coast, infrastructures in the Gulf of Thailand, for example, gas pipelines and platforms can be affected by tsunamis. In this study, the simulation of tsunamis in the Gulf of Thailand is conducted. Six cases of fault ruptures in the Manila trench are considered for earthquakes with magnitudes of 8.0, 8.5, and 9.0. The linear shallow water wave theory in spherical coordinate system is used for tsunami simulation in the large area covering Southeast Asia while the nonlinear shallow water wave theory in Cartesian coordinate system is used for tsunami simulation in the Gulf of Thailand. It is found that tsunamis reach the southern part of Thailand in 13 h after an earthquake and reach Bangkok in 19 h. The tsunami amplitude is largest in the direction towards the Philippines and Vietnam. The southern part of China is also severely affected. The Gulf of Thailand is affected by the diffraction of tsunamis around the southern part of Vietnam and Cambodia. The tsunami amplitude at the southernmost coastline is about 0.65 m for the M w 9.0 earthquake. The current velocity in the Gulf of Thailand due to the M w 9.0 earthquake is generally less than 0.2 m/s.

Concentration of the point spectrum on the continuous one in problems of linear water-wave theory

For the linear theory of water waves, we find out families of submerged or surface-piercing bodies in an infinite three-dimensional canal, which depend on a small parameter ε > 0 and have the following property: for any positive d and natural J, there exists ε(d, J) > 0 such that, for ε ∈ (0, ε(d, J)], the segment [0, d] of the continuous spectrum of the problem contains at least J eigenvalues. These eigenvalues are associated with trapped modes, i.e., solutions of the homogeneous problem, which decay exponentially at infinity and possess finite energy. Bibliography: 27 titles.

Water wave interaction with a sphere in a two-layer fluid flowing through a channel of finite depth

Using linear water wave theory, we consider a three-dimensional problem involving the interaction of waves with a sphere in a fluid consisting of two layers with the upper layer and lower layer bounded above and below, respectively, by rigid horizontal walls, which are approximations of the free surface and the bottom surface; these walls can be assumed to constitute a channel. The effects of surface tension at the surface of separation is neglected. For such a situation time-harmonic waves propagate with one wave number only, unlike the case when one of the layers is of infinite depth with the waves propagating with two wave numbers. Method of multipole expansions is used to find the particular solutions for the problems of wave radiation and scattering by a submerged sphere placed in either of the upper or lower layer. The added-mass and damping coefficients for heave and sway motions are derived and plotted against various values of the wave number. Similarly the exciting forces due to heave and sway motions are evaluated and presented graphically. The features of the results find good agreement with previously available results from the point of view of physical interpretation.

Scattering of surface waves over an uneven sea-bed

This work is concerned with the scattering of a train of progressive waves by a small undulation of the bottom of a laterally unbounded sea, using two-dimensional linear water wave theory. Assuming irrotational motion, a simplified perturbation analysis is employed to obtain the first-order corrections to the velocity potential by using the Green’s integral theorem in a suitable manner, and the reflection and transmission coefficients in terms of integrals involving the shape of the function c ( x ) representing the bottom undulation. Particular forms of the shape function are considered and the integrals for reflection and transmission coefficients are evaluated for these shape functions. For a varying sinusoidal sea bottom when Bragg resonance occurs, the reflection coefficient becomes a multiple of the number of ripples, and high reflection of the incident wave energy occurs if this number is large. We present the result in graphical form for one, three and five ripples in the patch. The results find absolute agreement with available results.

Particle Trajectories in Linearized Irrotational Shallow Water Flows

We investigate the particle trajectories in an irrotational shallow water flow over a flat bed as periodic waves propagate on the water’s free surface. Within the linear water wave theory, we show that there are no closed orbits for the water particles beneath the irrotational shallow water waves. Depending on the strength of underlying uniform current, we obtain that some particle trajectories are undulating path to the right or to the left, some are looping curves with a drift to the right and others are parabolic curves or curves which have only one loop.

Performance Prediction of OWC Type Small Size Wave Power Device with Impulse Turbine

This paper investigates a small size wave power device with an impulse turbine installed in the breakwater near Niigata Port, Japan. The device consists of an air chamber, a turbine, a generator and pressure-relief valves. This study reveals the characteristics of each component in this system with impulse turbine and a direct current dynamo the power of which is consumed by a constant resistor. In this paper special features of the impulse turbine are found, and the system characteristics are briefly represented. The overall plant performance was analyzed using mathematical model of an oscillating water column (OWC) based on linear water wave theory and the special features of the impulse turbine.

Oblique surface wave propagation over a small undulation on the bottom of an ocean

The problem of oblique water wave diffraction by small undulation of the bottom of a laterally unbounded ocean is considered using linear water wave theory. A perturbation analysis is employed to obtain the velocity potential, the reflection and the transmission coefficients up to the first order in terms of integrals involving the shape functions c(x) representing the bottom undulation. Finite cosine transform is used to find the first order potential, and this potential is utilised in obtaining the first order reflection and transmission coefficients. Some particular forms of the shape function representing an exponentially damped undulation, a single hump and a patch of sinusoidal ripples are considered and the integrals for the reflection and transmission coefficients are evaluated. For the exponentially damped undulation, it is observed that the reflection ceases much before transmission while for the single hump, reflection and transmission go hand in hand up to a certain value of the wavenumber, after which they vanish. For the patch of sinusoidal ripples having the same wavenumber, the reflection coefficient up to the first order is found to be an oscillatory function in the quotient of twice the component of the wavenumber along x-axis and the ripple wavenumber. When this quotient becomes one, the theory predicts a resonant interaction between the bed and free surface, and the reflection coefficient becomes a multiple of the number of ripples. High reflection of the incident wave energy occurs if this number is large. Also, when a patch of ripples having different wavenumbers is considered the same result follows. Known results for the normal incidence are recovered as special cases for the patch of sinusoidal ripples. The theoretical observations are shown computationally.

Reflection and transmission coefficients for water wave scattering by a sea‐bed with small undulation

AbstractThe scattering of a train of progressive waves by a small undulation of the bottom of a laterally unbounded sea is considered with two dimensional linear water wave theory. Assuming irrotational motion, a simplified perturbation analysis is employed to obtain the first order corrections to the velocity potential by using the Green's integral theorem in a suitable manner and also to the reflection and transmission coefficients in terms of integrals involving the shape of the function $c(x)$ representing the bottom undulation. Two particular forms of the shape function are considered and the integrals for reflection and transmission coefficients are evaluated for these shape functions. Out of these two cases, we focus mainly on a patch of sinusoidal ripples at the bottom and find that the reflection coefficient up to the first order is an oscillatory function of β which is twice the ratio of the wave numbers of the wave and the ripple bed. When this ratio becomes 0.5, i.e., when β = 1 (the situation when Bragg resonance occurs), the reflection coefficient becomes a multiple of the number of ripples, and high reflection of the incident wave energy occurs if this number is large. We present the result in graphical form for one, two, and six ripple(s) in the patch. The results find absolute agreement with available results. We also show the graphical comparison of the reflection coefficient against the wave number for one, three, and five ripple(s).

Particle trajectories in linear deep-water waves

Using phase plane analysis we show that within the framework of linear water wave theory the particle paths in a deep-water wave are not closed: there is a forward drift over a period, which decreases with greater depth.

A note on the effects of a thin visco-elastic mud layer on small amplitude water-wave propagation

In this note we investigated the effects of a thin visco-elastic mud layer on wave propagation. Within the framework of linear water-wave theory, analytical solutions are obtained for damping rate, dispersion relation between wave frequency and wave number, and velocity components in the water column and mud layer. The wave attenuation rate reaches a maximum value when the mud layer thickness is about the same as the mud boundary layer thickness. Heavier mud has a weaker effect on the wave damping. However, the wave attenuation rate does not always decrease as the elastic shear modulus increases. In the range of small values for elastic shear modulus, the wave attenuation can be amplified quite significantly. The current solutions are compared with experimental data with different wave conditions and mud properties. In general, good agreements are observed.

Water wave diffraction by a small deformation of the ocean bottom for oblique incidence

In this paper, the problem of oblique water wave diffraction by a small deformation of the bottom of a laterally unbounded ocean is considered using linear water wave theory. It is assumed that the fluid is incompressible and inviscid, and the flow irrotational. A perturbation analysis is employed to obtain the velocity potential, reflection and transmission coefficients up to the first order in terms of integrals involving the shape functions c(x) representing the bottom deformation by using Green's integral theorem. Two particular forms of the shape function are considered, and the integrals for the reflection and transmission coefficients are evaluated for these two different functions. Among those cases, for the particular case of a patch of sinusoidal ripples at the bottom, the reflection coefficient up to the first order is found to be an oscillatory function in the quotient of twice the wave number along the x-axis and the ripple wave number. When this quotient becomes one, the theory predicts a resonant interaction between the bed and free surface, and the reflection coefficient becomes a multiple of the number of ripples, and high reflection of the incident wave energy occurs if this number is large. Known results for the normal incidence are recovered as special cases. The numerical solutions for the reflection and transmission coefficients are also evaluated against wave numbers and angles of incidence.

2510 空気タービン型波浪エネルギー変換システムの浮体特性と最適形状(S47-1 自然の流体エネルギー利用技術(波力,潮流),S47 自然の流体エネルギー利用技術)

A floating type wave power generating system has an Oscillating Water Column (OWC). The device captures the wave energy using the heaving, pitching and surging motion of the device and the heaving motion of OWC. The investigations are motivated by the experiments and the numerical analyses to find another possibility different from OWC type floating device, e.g. Mighty Whale of Japan Marine Science and Technology Center. The numerical method for analyzing a floating body with OWC type wave energy conversion device is introduced where the eigenfunction expansion method is described under the condition that the linear water wave theory is applicable. The optimal profiles are eliminated according to the different conditions from the high efficiency, i.e. the minimum size per a unit power and so on.