Diffraction Of Water Waves Research Articles

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.

On the radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas

The radiation and diffraction of linear water waves by an infinitely long rectangular structure submerged in oblique seas of finite depth is investigated. The analytical expressions for the radiated and diffracted potentials are derived as infinite series by use of the method of separation of variables. The unknown coefficients in the series are determined by the eigenfunction expansion matching method. The expressions for wave forces, hydrodynamic coefficients and reflection and transmission coefficients are given and verified by the boundary element method. Using the present analytical solution, the hydrodynamic influences of the angle of incidence, the submergence, the width and the thickness of the structure on the wave forces, hydrodynamic coefficients, and reflection and transmission coefficients are discussed in detail.

Second Order Diffraction of Water Waves by a Bottom Mounted Vertical Circular Cylinder and Some Related Numerical Problems

A Hankel transformation is used to obtain the second order diffraction solution of vertical cylinder of circular cross section. The improper integral over the free surface is tackled carefully. The singularity at the free surface is overcome effectively using a third order nonlinear transformation. Numerical results for free surface elevations compare well with the published data.

Radiation and diffraction of linear water waves by an infinitely long submerged rectangular structure parallel to a vertical wall

The radiation and the diffraction of linear water waves by an infinitely long floating rectangular structure submerged in water of finite depth with leeward boundary being a vertical wall are analyzed in this paper by using the method of separation of variables. Analytical expressions for the radiated and diffracted potentials are derived as infinite series with unknown coefficients determined by the eigenfunction expansion matching method. The expressions for wave forces and hydrodynamic coefficients are given. A comparison is made between the results obtained by the present analytical solution and those obtained by the boundary element method. By using the present analytical solution, the hydrodynamic influences of the submergence, the width, the thickness of the structure, and the distance between the structure and the wall on the wave forces and hydrodynamic coefficients are discussed in detail.

Second-order refraction and diffraction of surface water waves

The mild-slope equation is an effective approximation for treating the combined effects of refraction and diffraction of infinitesimal water waves, for it reduces the spatial dimension of the linear boundary-value problem from three to two. We extend this approximation to nonlinear waves up to the second order in wave steepness, in order to simplify the inherently three-dimensional task. Assuming that the geometrical complexity is restricted to a finite, though large, horizontal domain, the hybrid-element method designed earlier for linearized problems is modified for the two-dimensional elliptic boundary-value problems at the second order. In the special case of a semi-circular peninsula (or a vertical cylinder on a cliff) in a sea of constant depth, the solution is analytical. Effects of the angle of incidence are examined for the free-surface height along the cylinder. For a cylinder standing on a shoal of radially varying depth, numerical results are discussed.

Diffraction of Water Waves by a Segmented Permeable Breakwater

The linearized theory of water waves is used to examine the diffraction of an incident wave by a permeable breakwater that consists of a number of distinct elements. Under the assumption that the wavelength is much greater than the thickness, each element is replaced by a thin structure and the permeability is modeled by a suitable boundary condition applied on its surface. The diffracted wave field is obtained by the solution of an integral equation and results are presented to illustrate the effects of permeability and of the characteristics of the incident wave.

On the radiation and diffraction of linear water waves by a rectangular structure over a sill. Part I. Infinite domain of finite water depth

The effects of a bottom sill on the hydrodynamic coefficients, wave force, and reflection and transmission coefficients of a rectangular structure floating on the free surface are analyzed in this paper by use of a semi-analytical method. Linearized water wave theory and the method of separation of variables are applied to obtain analytical expressions for the diffracted and radiated wave potentials. The unknown coefficients in the obtained expressions are determined by use of the eigenfunction expansion matching method. To verify the obtained expressions and the proposed method, a comparison of the analytical results with those obtained by the boundary element method is made and very good agreement is achieved, which shows that the analytical expressions for the radiated and diffracted potentials are correct. By use of the present analytical solution, several results for the hydrodynamic coefficients, wave force, reflection and transmission coefficients are given.

Hydrodynamic coefficients for water-wave diffraction by spherical structures

Evaluation of hydrodynamic coefficients and loads on submerged or floating bodies is of great significance in designing these structures. Some special regular-shaped geometries such as those of cylindrical (circular, elliptic) and spherical (hemisphere, sphere, spheroid) structures are usually considered to obtain analytical solutions to wave diffraction and radiation problems. The work presented here is the result of water-wave interaction with submerged spheres. Analytical expressions for various hydrodynamic coefficients and loads due to the diffraction of water waves by a submerged sphere are obtained. The exciting force components due to surge and heave motions are derived by solving the diffraction problem. Theory of multipole expansions is used to express the velocity potentials in terms of an infinite series of associated Legendre polynomials with unknown coefficients and the orthogonality of the polynomials is utilized to simplify the expressions. Since the infinite series appearing in various expressions have excellent truncation properties, they are evaluated by considering only a finite number of terms. Gaussian quadrature is used to evaluate the integrals. Numerical estimates for the analytical expressions for the hydrodynamic coefficients and loads are presented for various depth to radius ratios. Consideration of more values for depth makes it easy to compare the results with those available. The results obtained match closely with those obtained earlier by Wang and Wu and their coworkers

04/02910 An iterative procedure for the diffraction of water waves by multiple cylinders: Yilmaz, O. Ocean Engineering, 2004, 31, (11–12), 1437–1446

04/02910 An iterative procedure for the diffraction of water waves by multiple cylinders: Yilmaz, O. Ocean Engineering, 2004, 31, (11–12), 1437–1446

On the radiation and diffraction of water waves by a rectangular structure with a sidewall

A radiation and diffraction boundary value problem is investigated. It arises from the interaction of linear water waves with a freely floating rectangular structure in a semi-infinite fluid domain of finite water depth with the leeward boundary being a vertical wall. Analytical expressions for the radiated potentials and the diffracted potential are obtained by use of the method of separation of variables and the eigenfunction expansion method. The added masses and damping coefficients for the structure heaving, swaying and rolling in calm water are obtained by use of the corresponding radiated potentials and the wave excitation forces are calculated by use of the diffracted potential. To verify the correctness of the method, a boundary element method is used. A comparison of the analytical results with those obtained by the boundary element method is made and good agreement is achieved, which shows that the analytical expressions for the radiated and diffracted potentials are correct. By use of the present analytical solution, the added mass, damping coefficients, wave excitation force, together with the hydrodynamic effects of the draft, width of the structure and the clearance between the structure and the sidewall are also investigated.

Wave diffraction by a small elevation of the bottom of an ocean with an ice-cover

Diffraction of water waves by a small cylindrical elevation of the bottom of a laterally unbounded ocean covered by an ice sheet is investigated by the perturbation analysis. The ice sheet is modelled as a thin elastic plate. The reflection and transmission coefficients are evaluated up to the first order in terms of integrals involving the shape function representing the bottom elevation. Three particular forms of the shape function are considered for which explicit expressions for these coefficients are obtained. For the particular case of a patch of sinusoidal undulations at the bottom, the reflection coefficient up to first order is found to be an oscillatory function of the ratio of the wavelength of the bottom undulations and that of the incident wave train. When this ratio approaches 0.5, the reflection coefficient becomes a multiple of the number of undulations and high reflection of the incident wave energy occurs if this number is large. Reflection coefficient is depicted graphically to visualize the effect of the presence of ice-cover and the number of undulations.

An iterative procedure for the diffraction of water waves by multiple cylinders

Diffraction of water waves by an array of vertical cylinders of circular cross section is studied. In order to account for first order interaction among the cylinders, the body boundary condition is satisfied for each cylinder considering the scattered wave field from other cylinders in an iterative way. After each iteration, coefficients in the partial wave decomposition of the wave potential are modified. Convergence is fast for the whole range of frequencies and for a large number of bodies, compared with exact algebraic methods of Linton and Evans [J. Fluid Mech. 46 (1990) 549] and Kagemoto and Yue [J. Fluid Mech. 166 (1) (1986) 189].

On the radiation and diffraction of water waves by a rectangular buoy

An analytical method is presented to analyze the radiation and diffraction of water waves by a rectangular buoy in an infinite fluid domain of finite water depth. Analytical expressions for the radiated potentials and the diffracted potentials are obtained by use of the method of separation of variables. The unknown coefficients in the expressions are determined by use of the eigenfunction expansion matching method. The added masses and damping coefficients for the buoy heaving, swaying and rolling in calm water are obtained by use of the corresponding radiated potentials. Wave excitation forces are calculated by two different approaches, one is by use of the radiated potentials through Haskind’s theorem and the other is by the diffracted potential. It can be seen that the latter approach for wave forces on a rectangular buoy is much simpler than the former. To verify the correctness of the method, two specific examples in the past references are recomputed and the obtained results are in good agreement with those by use of other methods, which shows that the present method is correct.

Finite elements using a plane-wave basis for scattering of surface water waves.

A generalized, frequency-independent, finite-element method is proposed for the numerical simulation of combined diffraction, refraction, radiation, reflection and absorption of water waves. Plane waves are used as an external function space, exploiting the standard set of approximation functions defined by the finite element method to enforce conformity and to impose boundary conditions in a straightforward manner by applying the method to the plane waves on each element. An integration method originally devised only for diffraction of waves by Ortiz and Sanchez in 2001 is extended to combined refraction and diffraction of high-frequency waves, retaining its dramatic savings in the number of operations for a given error compared with standard numerical integration methods.

A Potential/RANSE Approach for Regular Water Wave Diffraction about 2-d Structures

A new formulation is proposed for the simulation of viscous flows around structures in waves. It consists in modifying the Reynolds-averaged Navier-Stokes equations: velocity, pressure or free-surface elevation fields are split into incident and diffracted fields to compute the diffracted flow only. The incident flow may be explicitly given by a stream function theory for non-linear regular waves, or by a spectral method for irregular waves. This method avoids classical problems (large CPU time, poor quality of generated wave) of numerical generation of waves in a viscous flow solver. The 2D flow around an immersed square in regular waves demonstrates the effectiveness of the method.

Accurate refraction–diffraction equations for water waves on a variable-depth rough bottom

The extended mild-slope equation and the modified mild-slope equation have been used successfully to study refraction–diffraction of linear water waves by steep bottom roughness. Their consistency has been questioned. A systematic derivation of these model equations exposes and illuminates their rationale. Their good performance stems from an accurate representation of (Class I) Bragg resonance. As a benchmark test case, we consider scattering by a sloping bottom with random roughness. The rates of scattering found for the mean field in both of the approximate models agree exactly with the full theory for scattering by small roughness. This greatly improves the limited agreement which was found for the mild-slope equation, and establishes the validity of the above model equations. The study involves operator calculus, a powerful method for simplifying problems with variable coefficients. The augmented mild-slope equation serves to consistently derive accurate model equations.

A coupled-mode model for the refraction–diffraction of linear waves over steep three-dimensional bathymetry

A consistent coupled-mode model recently developed by Athanassoulis and Belibassakis [1], is generalized in 2+1 dimensions and applied to the diffraction of small-amplitude water waves from localized three-dimensional scatterers lying over a parallel-contour bathymetry. The wave field is decomposed into an incident field, carrying out the effects of the background bathymetry, and a diffraction field, with forcing restricted on the surface of the localized scatterer(s). The vertical distribution of the wave potential is represented by a uniformly convergent local-mode series containing, except of the ususal propagating and evanescent modes, an additional mode, accounting for the sloping bottom boundary condition. By applying a variational principle, the problem is reduced to a coupled-mode system of differential equations in the horizontal space. To treat the unbounded domain, the Berenger perfectly matched layer model is optimized and used as an absorbing boundary condition. Computed results are compared with other simpler models and verified against experimental data. The inclusion of the sloping-bottom mode in the representation substantially accelerates its convergence, and thus, a few modes are enough to obtain accurately the wave potential and velocity up to and including the boundaries, even in steep bathymetry regions. The present method provides high-quality information concerning the pressure and the tangential velocity at the bottom, useful for the study of oscillating bottom boundary layer, sea-bed movement and sediment transport studies.

Polynomial approximations for Fresnel integrals in diffraction analysis

The Fresnel integrals in the diffraction analyses of Penny and Price [Penny, W.G., Price, A.T., 1944. Diffraction of water waves by breakwaters. Misc. Weapons Development Technical History 26, Artificial Harbors, Sec. 3D; Philos. Trans. R. Soc. London, Ser. A, 244 (1952) 236] are replaced by polynomial approximations for positive arguments in the range of zero to infinity. The approximations over this range of arguments have an error that is less than, or equal to, 0.002. The value of using the approximations is twofold: first, the use of polynomials is computationally efficient; second, the relative simplicity of the polynomial approximations allows for field engineering computations by hand calculators in the determination of the theoretical values of diffraction coefficients and phase relationships. The diffraction coefficient values determined for demonstration purposes are those for diffraction caused by both totally reflective or totally absorbent breakwaters, the latter referred to as a “cushion” type by Penny and Price. For either case, the breakwaters can be finite or semi-infinite. Examples are presented.

Absorbing boundary condition on elliptic boundary for finite element analysis of water wave diffraction by large elongated bodies

AbstractIn a domain method of solution of exterior scalar wave equation, the radiation condition needs to be imposed on a truncation boundary of the modelling domain. The Bayliss, Gunzberger and Turkel (BGT) boundary dampers of first‐ and second‐orders, which require a circular cylindrical truncation boundary in the diffraction‐radiation problem of water waves, have been particularly successful in this task. However, for an elongated body, an elliptic cylindrical truncation boundary has the potential to reduce the modelling domain and hence the computational effort. Grote and Keller [On non‐reflecting boundary conditions. Journal of Computational Physics 1995; 122: 231–243] proposed extension of the first‐ and second‐order BGT dampers for the elliptic radiation boundary and used these conditions to the acoustic scattering by an elliptic scatterer using the finite difference method. In this paper, these conditions are implemented for the problem of diffraction of water waves using the finite element method. Also, it is shown that the proposed extension works well only for head‐on wave incidence. To remedy this, two new elliptic dampers are proposed, one for beam‐on incidence and the other for general wave incidence. The performance of all the three dampers is studied using a numerical example of diffraction by an elliptic cylinder. Copyright © 2001 John Wiley & Sons, Ltd.

Water wave diffraction by a cylinder array. Part 1. Regular waves

Diffraction of regular waves by arrays of vertical bottom-mounted circular cylinders is investigated using theoretical, computational, and experimental methods. Experiments in an offshore wave basin are designed to measure free surface elevation η at multiple locations in the vicinity of a multi-column structure subjected to regular waves of frequency 0.449 < ka < 0.524 and steepness 0.122 < kA < 0.261, where k is the wavenumber, a the cylinder radius and A the wave amplitude. Results from regular wave data analysis for first-order amplitudes are compared with those from analytical linear diffraction theory, which is shown to be accurate for predicting incident waves of low steepness. Second- and third-order terms are also estimated from the measured time series, and the effects near a second-order near-trapping frequency are compared to semi-analytical second-order diffraction theory. Linear diffraction theory is shown to be very accurate at predicting the global surface elevation features, even for waves of high steepness. However, violent events and significant nonlinear interactions, including breaking induced by wave scattering, have been observed. Furthermore, second-order near-trapping was observed to affect the magnitude of local free surface oscillations as well as scattered far-field radiation.