scholarly journals WATER-WAVES SCATTERING BY PERMEABLE BOTTOM IN TWO-LAYER FLUID IN THE PRESENCE OF SURFACE TENSION

2017 ◽  
Vol 22 (6) ◽  
pp. 827-851 ◽  
Author(s):  
Srikumar Panda ◽  
Subash C. Martha
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Reflection And Transmission Coefficients ◽  
Wave Number ◽  
Physical Parameters ◽  
Fluid System ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Linearized Theory ◽  
Incident Waves

In the present paper, reflection and transmission phenomena of water waves due to undulating permeable bottom in a two-layer fluid system are investigated using two-dimensional linearized theory. The effect of surface tension on the free surface is included in this work. In two-layer fluid system, there exist waves with two different wave numbers (modes). When a wave of a particular wave number encounters the undulating bottom, reflection and transmission phenomena occur in both the layers. The reflection and transmission coefficients in both layers due to incident waves of both modes are analyzed with the aid of perturbation analysis along with Fourier transform technique. It is found that these coefficients are obtained in terms of integrals which depend on the shape function of the undulating bottom. Two different kinds of undulating bottoms are considered to determine these coefficients. For a particular undulating bottom, namely sinusoidal bottom undulation the effect of various physical parameters such as number of ripples, surface tension and porous effect parameters are demonstrated graphically. The study further elaborates the energy balance relations associated with the reflection and transmission coefficients to ascertain the correctness of all the computed results.

OBLIQUE WAVE SCATTERING BY A RECTANGULAR SUBMARINE TRENCH

2015 ◽  
Vol 56 (3) ◽  
pp. 286-298 ◽  
Author(s):  
RUMPA CHAKRABORTY ◽  
B. N. MANDAL
Keyword(s):  
Water Waves ◽  
Wave Scattering ◽  
Wave Train ◽  
Galerkin Approximation ◽  
Central Line ◽  
Wave Number ◽  
Reflection And Transmission ◽  
Oblique Wave ◽  
Transmission Coefficients ◽  
Linearized Theory

The problem of oblique wave scattering by a rectangular submarine trench is investigated assuming a linearized theory of water waves. Due to the geometrical symmetry of the rectangular trench about the central line $x=0$, the boundary value problem is split into two separate problems involving the symmetric and antisymmetric potential functions. A multi-term Galerkin approximation involving ultra-spherical Gegenbauer polynomials is employed to solve the first-kind integral equations arising in the mathematical analysis of the problem. The reflection and transmission coefficients are computed numerically for various values of different parameters and different angles of incidence of the wave train. The coefficients are depicted graphically against the wave number for different situations. Some curves for these coefficients available in the literature and obtained by different methods are recovered.

scholarly journals Scattering of water waves by rectangular thick barriers in presence of surface tension

2021 ◽  
Vol 23 (11) ◽  
pp. 30-55
Author(s):  
Gour Das ◽  
◽  
Rumpa Chakraborty ◽  
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Graphical Representation ◽  
Finite Depth ◽  
Approximation Technique ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Gegenbauer Polynomial ◽  
Incident Waves ◽  
Effect Of Surface

The influence of surface tension over an oblique incident waves in presence of thick rectangular barriers present in water of uniform finite depth is discussed here. Three different structures of a bottom-standing submerged barrier, submerged rectangular block not extending down to the bottom and fully submerged block extending down to the bottom with a finite gap are considered. An appropriate multi-term Galekin approximation technique involving ultraspherical Gegenbauer polynomial is employed for solving the integral equations arising in the mathematical analysis. The reflection and transmission coefficients of the progressive waves for two-dimensional time har- monic motion are evaluated by utilizing linearized potential theory. The theoretical result is validated numerically and explained graphically in a number of figures. The present result will almost match analytically and graphically with those results already available in the literature without considering the effect of surface tension. From the graphical representation, it is clearly visible that the amplitude of reflection coefficient decreases with increasing values of surface tension. It is also seen that the presence of surface tension, the change of width, and the height of the thick barriers affect the nature of the reflection coefficients significantly

scholarly journals Scattering of Oblique Water Waves by an Infinite Step

2018 ◽  
Vol 23 (2) ◽  
pp. 327-338
Author(s):  
P. Dolai ◽  
D.P. Dolai
Keyword(s):  
Surface Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Train ◽  
Reflection And Transmission Coefficients ◽  
Physical Parameters ◽  
Reflection And Transmission ◽  
Infinite Depth ◽  
Transmission Coefficients ◽  
Galerkin Approximations

AbstractThe present paper is concerned with the problem of scattering of obliquely incident surface water wave train passing over a step bottom between the regions of finite and infinite depth. Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the physical parameters reflection and transmission coefficients in terms of integrals. Appropriate multi-term Galerkin approximations involving ultra spherical Gegenbauer polynomials are utilized to obtain very accurate numerical estimates for reflection and transmission coefficients. The numerical results are illustrated in tables.

Characteristics of Bragg Reflection of Water Waves by Multiple Vertical Flexible Membranes

2018 ◽  
Author(s):  
Wei-Wei Ding ◽  
Zao-Jian Zou ◽  
Jing-Ping Wu
Keyword(s):  
Water Waves ◽  
Bragg Reflection ◽  
Least Square ◽  
Reflection And Transmission Coefficients ◽  
Effective Bandwidth ◽  
Linear Wave ◽  
Reflection And Transmission ◽  
Flexible Membrane ◽  
Transmission Coefficients ◽  
Flexible Membranes

Bragg reflection of water waves by multiple vertical flexible membranes in water of uniform depth is investigated based on the assumption of linear wave theory and small membrane deflection. The multiple vertical flexible membranes consist of several floating vertical flexible membranes which are installed with both ends fixed. First, a single vertical flexible membrane in water waves is considered, and the reflection and transmission coefficients are obtained based on the eigenfunction expansion method and the least square method. Then the interaction of water waves with the multiple vertical flexible membranes is studied. Using the reflection and transmission coefficients obtained for the single flexible membrane, the reflection and transmission coefficients of the multiple vertical flexible membranes are obtained based on the wide spacing approximation. The proposed method is proved to be efficient by comparing the calculated coefficients with the results published in literature. The characteristics of Bragg reflection, such as the occurring condition, the primary amplitude and the effective bandwidth, are systematically investigated under various factors including the number, the tension, the draft and the spacing of membranes. The results of the present study have certain reference value for design of multiple vertical flexible membranes as effective floating breakwaters.

Scattering of surface waves obliquely incident on a fixed half immersed circular cylinder

1984 ◽  
Vol 96 (2) ◽  
pp. 359-369 ◽  
Author(s):  
B. N. Mandal ◽  
S. K. Goswami
Keyword(s):  
Integral Equation ◽  
Surface Water ◽  
Circular Cylinder ◽  
Water Waves ◽  
Wave Number ◽  
Angle Of Incidence ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Obliquely Incident ◽  
Surface Water Waves

AbstractThe problem of scattering of surface water waves obliquely incident on a fixed half immersed circular cylinder is solved approximately by reducing it to the solution of an integral equation and also by the method of multipoles. For different values of the angle of incidence and the wave number the reflection and transmission coefficients obtained by both methods are evaluated numerically and represented graphically to compare the results obtained by the respective methods.

On the scattering of water waves by a submerged slender barrier

1998 ◽  
Vol 40 (2) ◽  
pp. 171-189
Author(s):  
P. K. Kundu ◽  
N. K. Saha
Keyword(s):  
Finite Length ◽  
Water Waves ◽  
Vertical Plate ◽  
Perturbation Technique ◽  
Reflection And Transmission Coefficients ◽  
Reflection And Transmission ◽  
First Order ◽  
Transmission Coefficients ◽  
Special Cases ◽  
Analytical Expressions

AbstractAn approximate analysis, based on the standard perturbation technique, is described in this paper to find the corrections, up to first order to the reflection and transmission coefficients for the scattering of water waves by a submerged slender barrier, of finite length, in deep water. Analytical expressions for these corrections for a submerged nearly vertical plate as well as for a submerged vertically symmetric slender barrier of finite length are also deduced, as special cases, and identified with the known results. It is verified, analytically, that there is no first order correction to the transmitted wave at any frequency for a submerged nearly vertical plate. Computations for the reflection and transmission coefficients up to O(ε), where ε is a small dimensionless quantity, are also performed and presented in the form of both graphs and tables.

The influence of surface tension on the reflection of water waves by a plane vertical barrier

1968 ◽  
Vol 64 (3) ◽  
pp. 795-810 ◽  
Author(s):  
D. V. Evans
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Related Problem ◽  
Velocity Potential ◽  
Surface Tension Force ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Reflected Wave ◽  
Vertical Barrier ◽  
Effect Of Surface

AbstractIn this paper the effect of surface tension is included in a well-known problem in the theory of two-dimensional infinitesimal water waves. The problem is that of the reflection of waves from a fixed vertical barrier immersed to a depth a into deep water. It is shown how the solution for the velocity potential may be determined uniquely when simple assumptions are made concerning the behaviour of the free surface near the barrier. In particular, expressions are derived for the reflection coefficient, defined as the ratio of the amplitude of the reflected wave to that of the incident wave, at infinity, and the transmission coefficient, defined similarly. It is shown how these coefficients, for small values of the surface tension force, tend to the values obtained by Ursell (4) when surface tension is ignored. The related problem of a completely immersed vertical barrier extending to a distance a from the surface may be solved in a similar manner. Expressions for the reflection and transmission coefficients for this case are given.

scholarly journals Effect of Mooring Lines on the Hydroelastic Response of a Floating Flexible Plate Using the BIEM Approach

2021 ◽  
Vol 9 (9) ◽  
pp. 941
Author(s):  
Sarat Chandra Mohapatra ◽  
C. Guedes Soares
Keyword(s):  
Shallow Water ◽  
Reflection And Transmission Coefficients ◽  
Physical Parameters ◽  
Flexible Plate ◽  
Approximation Model ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Deflection Amplitude ◽  
Shallow Water Approximation ◽  
Hydroelastic Response

A boundary integral equation method (BIEM) model for the problem of surface wave interaction with a moored finite floating flexible plate is presented. The BIEM solution is obtained by employing the free surface Greens function and Green’s theorem, and the expressions for the plate deflection, reflection, and transmission coefficients are derived from the integro-differential equation. Furthermore, the shallow water approximation model and its solution is obtained based on the matching technique in a direct manner. The accuracy of the present BIEM code is checked by comparing the results of deflection amplitude, reflection, and transmission coefficients with existing published results and experimental datasets as well as the shallow water approximation model. The hydroelastic response of the moored floating flexible plate is studied by analyzing the effects of the mooring stiffness, incidence angle, and flexural rigidity on the deflection amplitude, plate deformations, reflection, and transmission coefficients. The present analysis may be helpful in understanding the different physical parameters to model a wave energy conversion device with mooring systems over BIEM formulations.

scholarly journals Oblique Water Wave Diffraction by a Step

2017 ◽  
Vol 22 (1) ◽  
pp. 35-47 ◽  
Author(s):  
P. Dolai
Keyword(s):  
Surface Water ◽  
Water Wave ◽  
Numerical Estimate ◽  
Wave Train ◽  
Reflection And Transmission Coefficients ◽  
Physical Parameters ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Galerkin Approximations ◽  
Finite Step

Abstract This paper is concerned with the problem of diffraction of an obliquely incident surface water wave train on an obstacle in the form of a finite step. Havelock expansions of water wave potentials are used in the mathematical analysis to obtain the physical parameters reflection and transmission coefficients in terms of integrals. Appropriate multi-term Galerkin approximations involving ultraspherical Gegenbauer polynomials are utilized to obtain a very accurate numerical estimate for reflection and transmission coefficients which are depicted graphically. From these figures various interesting results are discussed.

16.—Waves over Obstacles on a Shallow Seabed

1973 ◽  
Vol 71 (3) ◽  
pp. 181-192 ◽  
Author(s):  
Alan Jeffrey ◽  
Saw Tin
Keyword(s):  
Shallow Water ◽  
Water Waves ◽  
Recurrence Relations ◽  
Reflection And Transmission Coefficients ◽  
Shallow Water Waves ◽  
Reflection And Transmission ◽  
Transmission Coefficients

SynopsisIn this paper we consider the effect of the passage of shallow water waves over vertical walled objects on a flat seabed. The effect of reflections at the successive walls is taken into account when determining the place of breaking of the wave. The results are obtained by the introduction of special reflection and transmission coefficients at each wall and recurrence relations are formulated connecting their values at successive walls. The effect of this is to reduce the calculation of the place of breaking of the wave over a stepped seabed to an equivalent one for a wave over a flat seabed, the result of which is well known.

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