Diffraction Of Oblique Waves Research Articles

Oblique Wave Diffraction by a Bottom-Standing Thick Barrier and a Pair of Partially Immersed Barriers

Abstract On the basis of the linearized theory of water waves, the present study has demonstrated a semi-analytical method to assess the hydrodynamic performance of a pair of partially immersed barriers just above a thick bottom-standing barrier. By means of the eigenfunction expansion method, a system of the first kind Fredholm-type integral equation involving a horizontal component of velocity as unknown functions is developed for the interaction of water waves with both types of barriers. The multiterm Galerkin approximation is adopted to determine these unknown functions having square root singularities at the submerged edge of the thin barriers and one-third singularities at the corners of the thick barrier. In order to overcome such types of singularities, Chebychev polynomials for half-singularities and ultra-spherical Gegenbauer polynomials for one-third singularities with suitable weight functions have been taken into consideration. The numerical examples of both reflection and transmission coefficients are presented to examine the hydrodynamic performance of breakwater. Some fascinating results like resonant frequencies are obtained for practical engineering. At the same time, reflection coefficients for the present breakwater agree reasonably for the limiting cases with the previously available results.

Analysis of oblique wave diffraction by rectangular thick barrier in the presence of surface tension

Analysis of oblique wave diffraction by rectangular thick barrier in the presence of surface tension

Mitigation of wave force and dissipation of energy by multiple arbitrary porous barriers

A model of oblique wave diffraction by multiple arbitrary porous barriers in infinitely deep water is proposed to study the role of the porous breakwater in mitigating wave effects and dissipating wave energy. The Havelock's expansion of water wave potential along with suitable matching conditions and the single term Galerkin approximation method are used to handle the mathematical boundary value problem. The role of the arbitrary porous barrier is studied by analyzing the reflection coefficient, dissipated energy, and wave forces on the barriers. Significant changes are found in wave reflection and forces due to the consideration of porous barriers as compared to rigid barriers in the fluid region. As the separation length between the vertical barriers increases, the reflection coefficient becomes oscillatory as a function of the wavenumber, which is due to multiple reflections by the barriers. It is seen that the wave load reduces significantly with an increase in the number of barriers. It is also found that more energy is dissipated if the permeability of the barriers is increased. The correctness of the present method is confirmed by comparing the results available in the literature.

Energy dissipation and oblique wave diffraction by three asymmetrically arranged porous barriers

ABSTRACT The problem of oblique water wave diffraction by three asymmetrically arranged vertical porous barriers with different widths in infinitely deep water is investigated. Reflection and transmission coefficients are evaluated, applying single-term Galerkin approximation. The amount of energy loss is derived employing Green's integral theorem. Wave force is determined using the linear Bernoulli equation of dynamic pressure jump, on the porous barriers. Forces and energy of the wave are studied to understand the effect of the dissipation of wave energy due to the presence porous barriers. Numerical results for the reflection coefficient, wave force, and wave energy dissipation are analysed graphically. The comparison of the numerical results are carried out for the validity of the derived solution. It is observed that porosity affects significantly on reflection coefficient and amount of energy dissipation. It is also found that more energy is dissipated if the permeability of the barriers is increased.

Effect of Porosity on Oblique Wave Diffraction by Two Unequal Vertical Porous Barriers

The diffraction of obliquely incident wave by two unequal barriers with different porosity in infinitely deep water is investigated by using two-dimensional linearized potential theory. Reflection and transmission coefficients are computed numerically using appropriate Galerkin approximations for two partially immersed and two submerged barriers. The amount of energy dissipation due to the permeable barriers is derived using Green’s integral theorem. The coefficient of wave force is determined using the linear Bernoulli equation of dynamic pressure jump on the porous barriers. The numerical results of hydrodynamics quantities are illustrated graphically.

Oblique wave diffraction by a flexible floating structure in the presence of a submerged flexible structure

Expansion formulae associated with the interaction of oblique surface gravity waves with a floating flexible plate in the presence of a submerged horizontal flexible structure are derived using Green’s integral theorem in water of finite and infinite water depths. The associated Green’s functions are derived using the fundamental solution associated with the reduced wave equation. The integral forms of the Green’s functions and the velocity potentials are advantageous over the eigenfunction expansion method in situation when the roots of the dispersion relation coalesce. As an application of the expansion formulae, diffraction of oblique waves by a finite floating elastic plate in the presence of a submerged horizontal flexible membrane is investigated in water of finite depth. The accuracy of the numerical computation is demonstrated by analysing the convergence of the complex amplitude of the reflected waves and the energy relation. Effect of the submerged membrane on the diffraction of surface waves is studied by analysing the reflection and transmission coefficients for various parametric values. Further, the derivation of long wave equation under shallow water approximation is derived in a direct manner in the appendix. The concept and methodology can be easily extended to deal with acoustic wave interaction with flexible structures and related problems of mathematical physics and engineering.

Diffraction of oblique waves by thick rectangular barriers

A complete analytical solution is presented for the linear diffraction of oblique waves by horizontal rectangular cylinders either fixed at the free surface or mounted on the sea bed in a finite-depth of water. Helmholtz equation is employed as the governing differential equation obtained by reducing the 3-D oblique wave scattering problem to a 2-D case. According to the method proposed, the fluid region is divided into three sub-regions in which the governing differential equation is solved by the separation of variables. The solutions for each region are then matched on the common boundaries of sub-regions to determine the unknowns of the eigen series expansions and Fourier series. Thus transmitted and reflected waves are obtained in the far-field, and forces and moments acting on the rectangular cylinder fixed at the free surface are also given. Comparisons are made in order to check the accuracy of the method.

Simulation of directional waves in a numerical basin by a desingularized integral equation approach

A numerical solution is developed to investigate the generation and propagation of small-amplitude water waves in a semi-infinite rectangular wave basin. The three-dimensional wave field is produced by the prescribed “snake-like” motion of an array of segmented wave generators located along the wall at one end of the tank. The solution technique is based on the boundary element approach and uses an appropriate three-dimensional Green function which explicitly satisfies the tank-wall boundary conditions. The Green function and its derivatives which appear in the integral equation formulation can be shown to be slowly convergent when the source and field points are in close proximity. Therefore, when computing the velocity potentials on the wave generators, the source points are chosen outside the fluid domain, thereby ensuring the rapid convergence of these functions and rendering the integral equations non-singular. Numerical results are shown which illustrate the influence of the various wavemaker and basin parameters on the generated wave field. Finally, the complete wave field produced by the diffraction of oblique waves by a vertical circular cylinder in a basin is presented.

Oblique Wave Diffraction by Parallel Thin Vertical Barriers with Gaps

The problem of oblique water wave diffraction by two equal thin, parallel, fixed vertical barriers with gaps present in uniform finite-depth water is investigated here. Three types of barrier configurations are considered. A one-term Galerkin approximation is used to evaluate upper and lower bounds for reflection and transmission coefficients for each configuration. These bounds are seen to be very close numerically for all wave numbers and as such their averages produce good numerical estimates for these coefficients. Only the bounds for the reflection coefficient are numerically computed. These are also numerically compared with the results obtained by using multiterm Galerkin approximations involving Chebyshev polynomials for a wide range of parameters. Numerical results for the reflection coefficients for the three barrier configurations are presented graphically. It is seen that total reflection occurs only for the surface-piercing barriers while total transmission occurs for all the three configurations considered here. It is also observed that the introduction of an equal second barrier to a submerged barrier increases the reflection coefficient considerably in some frequency bands and as such submerged double barrier configurations are preferable to a submerged single barrier for the purpose of reflecting more wave energy into the open sea.

Oblique wave diffraction by segmented offshore breakwaters

This paper presents a theoretical model to examine oblique wave diffraction by a detached breakwater system consisting of an infinite row of regularly-spaced thin, impermeable structures located in water of uniform depth. The fluid is assumed incompressible and inviscid and to undergo irrotational motion. Wave heights are assumed to be sufficiently small such that linear wave theory is applicable. The eigenfunction expansion solution of Dalrymple and Martin (1990) for normal wave incidence on this breakwater geometry is modified herein to study oblique wave effects. Numerical results, in the form of contour maps of the relative wave height behind the structure, or complex reflection coefficients, are presented for a range of wave and breakwater parameters. The accuracy of the present model is verified by a comparison with existing results for the limiting cases of an isolated detached breakwater, and a breakwater with a single gap. Also, for the multi-gap breakwater, the present solution is further verified for both normal and oblique wave incidence with results in the open literature.

Boundary element method for diffraction of oblique waves by an infinite cylinder

This paper presents the solutions of the boundary element method (BEM) for the potential flow problem associated with the diffraction of oblique waves by a horizontal cylinder of infinite length on a free surface. This physical model usually has its direct applications in engineering, such as in strip theory. The mathematical model for this problem is a 2D Helmholtz equation, whose BEM formulation involves kernels in the form of modified Bessel functions. The application to this problem is among the first for this method. The diffraction forces and moment for a fixed rectangular cylinder have been calculated and compared with the existing results of K.J. Bai, as an illustration of the effectiveness of the developed BEM code. The paper also compares results obtained using constant and linear elements.

Transmission of oblique waves through submerged apertures

Theoretical solutions are obtained for the diffraction of oblique waves by an infinite vertical barrier with a small slit in deep water. The method of matched asymptotic expansions is used to solve two problems: (1) thin barrier where the thickness of the barrier is small in comparison with wavelength, and (2) wide barrier where the wavelength and the thickness of the barrier are in the same order of magnitude. Approximate formula for the transmission coefficient and the reflection coefficient are presented for both cases.

An integral equation method for the diffraction of oblique waves by an infinite cylinder

AbstractA hybrid integral equation method is formulated to study the diffraction of oblique waves by an infinite cylinder. The water depth and the geometry of the floating cylinder are assumed to be uniform in the y‐direction (one of the horizontal axes). Numerical discretization and integrations are performed in the vertical plane. Analytical solutions are used in far fields such that radiation boundary conditions are satisfied. Numerical results are obtained for the case of wave scattering by a floating rectangular cylinder in a constant water depth. The accuracy and efficiency of present method are compared with those obtained by other numerical techniques.

Diffraction of oblique waves by an infinite cylinder

This paper presents a numerical method for solving linearized water-wave problems with oscillatory time dependence. Specifically it considers the diffraction problem for oblique plane waves incident upon an infinitely long fixed cylinder on the free surface. The numerical method is based on a variational principle equivalent to the linearized boundary-value problem. Finite-element techniques are used to represent the velocity potential; and the variational principle is used to determine the unknown coefficients in the solution throughout the fluid domain. To illustrate this method, reflexion and transmission coefficients and the diffraction forces and moment are computed for oblique waves incident upon a vertical flat plate, a horizontal flat plate and rectangular cylinders, where the comparison is made with the existing results by others. Also considered is the associated sinuous forced-motion problem, where comparison is made with the results for a circular cylinder obtained by Bolton & Ursell (1973).