Propagation of Gravity Waves Past Multiple Bottom-Standing Barriers

2014 ◽  
Vol 137 (1) ◽  
Author(s):  
D. Karmakar ◽  
C. Guedes Soares
Keyword(s):  
Gravity Waves ◽  
Water Waves ◽  
Wave Interaction ◽  
Offshore Structures ◽  
Least Square ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Edge Conditions ◽  
Linearized Theory ◽  
Flexible Barriers

The interaction of oblique surface gravity waves with multiple bottom-standing flexible porous breakwaters is analyzed based on the linearized theory of water waves. Using the method of eigenfunction expansion and the least square approximation, the wave propagation in the presence of single bottom-standing barriers is analyzed considering the upper edge to be: (i) free and (ii) moored, whereas the lower edge is considered to be clamped at the bottom. The wide-spacing approximation is used to analyze the wave interaction with multiple porous bottom-standing flexible barriers to understand the effect of the submerged flexible barriers as an effective breakwater. A brief comparison of both the upper edge conditions is carried out to analyze the effect of wave dissipation due to the presence of multiple barriers. The numerical results for the reflection and transmission coefficients along with the free surface vertical deflection are obtained for the case of two and three multiple bottom-standing barriers. The attenuation in the wave height due to the presence of porosity, change in barrier depth, and distance between the barriers are analyzed. The present study will be helpful in the analysis of proper functioning of porous bottom-standing barrier as an effective breakwater for the protection of offshore structures.

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.

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.

Travelling waves on a membrane: reflection and transmission at a corner of arbitrary angle. I

1995 ◽  
Vol 451 (1943) ◽  
pp. 657-683 ◽  
Keyword(s):  
Special Functions ◽  
Numerical Study ◽  
Travelling Waves ◽  
Wedge Angle ◽  
Complete Agreement ◽  
Arbitrary Angle ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Edge Conditions ◽  
Impedance Conditions

This article is the first part of a study into the reflection, transmission and scattering of waves on the semi-infinite faces of a compressible fluid wedge of arbitrary angle. This class of problems, in which the wedge surfaces are described by high order impedance conditions (that is, containing derivatives with respect to variables both normal and tangential to the boundary), is of great interest in structural acoustics and electromagnetism. Here, for mathematical convenience, the canonical problem of a fluid wedge with two plane membrane surfaces is examined. Forcing is taken as an unattenuated fluid coupled surface wave incident from infinity on one of the wedge faces. Explicit application of the edge-constraints allows the boundary value problem to be formulated as an inhomogeneous difference equation which is then solved in terms of Maliuzhinets special functions. An analytical solution is thus obtained for arbitrary wedge angle and the membrane wave reflection and transmission coefficients are deduced. The solution method is straightforward to apply and can easily be generalized to any boundary or edge conditions. Also in Part I, the solution obtained for the case of a wedge of angle 2π is compared with that determined by the Wiener-Hopf technique. The two methods are in complete agreement. In the second half of this work the reflection coefficients calculated here will be shown to confirm those given previously in the literature for certain specific wedge angles. A full numerical study, for a range of fluid-membrane parameter values, will also be presented in Part II.

Reflection properties of internal gravity waves incident upon a hyperbolic tangent shear layer

1982 ◽  
Vol 120 ◽  
pp. 505-521 ◽  
Author(s):  
Cornelis A. Van Duin ◽  
Hennie Kelder
Keyword(s):  
Shear Layer ◽  
Gravity Waves ◽  
Critical Level ◽  
Boussinesq Approximation ◽  
Internal Gravity Waves ◽  
Hyperbolic Tangent ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Heun’S Equation ◽  
Stable Flow

The properties of reflection and transmission of internal gravity waves incident upon a shear layer containing a critical level are investigated. The shear layer is modelled by a hyperbolic tangent profile. In the Boussinesq approximation, the differential equation governing the propagation of these waves can then be transformed into Heun's equation. For large Richardson numbers this equation can be approximated by an equation that has solutions in terms of hypergeometric functions. For these values of the Richardson number the reflection coefficient proves to be strongly dependent on the place of the critical level in the shear flow. If the Doppler-shifted frequency is an odd function of the height difference with respect to the critical level, the reflection and transmission coefficients can be evaluated in closed form.Over-reflection is possible for sufficiently small wavenumbers and Richardson numbers. It is pointed out that over-reflection and over-transmission cannot occur in a stable flow and that resonant over-reflection is not possible in our model.

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 NON-CONSERVATIVE WAVE INTERACTION WITH FIXED SEMI-IMMERSED RECTANGULAR STRUCTURES

1978 ◽  
Vol 1 (16) ◽  
pp. 133
Author(s):  
Robert B. Steimer ◽  
Charles K. Sollitt
Keyword(s):  
Wave Interaction ◽  
Wave Structure ◽  
Variable Structure ◽  
Dimensional Structure ◽  
Solution Technique ◽  
Vertical Force ◽  
Computationally Efficient ◽  
Reflection And Transmission ◽  
Single Wave ◽  
Transmission Coefficients

Previous attempts to analytically describe wave reflection and transmission at surface penetrating structures have neglected losses due to flow expansion, contraction, and skin drag along the structure boundaries (Black and Mei, 1970; Ijima, et al., 1972). The model described in this study includes these effects and allows for the inclusion of a dissipative medium such as rubble or closely spaced piles in the region beneath the structure. The problem of a fixed, two-dimensional structure in a train of monochromatic incident waves is modeled, as shown in Figure 1. The solution allows for 1) variable structure length and draft, 2) different depths in the regions fore, aft, and beneath the structure, 3) variable wave amplitude and period, and 4) turbulent and inertial damping in the region beneath the structure. An equivalent work technique is applied to linearize the damping beneath the structure, yielding a potential flow problem in all three regions. Amplitudes for the resulting series of eigenfunctions in each region are determined by matching pressure and horizontal mass flux at the region interfaces, orthogonalizing these expressions over the depth, and simultaneously solving the resulting equations to yield complex reflection and transmission coefficients. Complex horizontal and vertical force coefficients for the structure are also determined from the integrated Bernoulli equation. The solution technique is computationally efficient. In general, five modes in the eigen series provide satisfactory convergence for the various hydrodynamic parameters. Approximately six-tenths of a computer system second are required to solve for a single wave-structure condition. The results compare favorably with variational methods used by others.

The decay of flexural-gravity waves in long sea ice transects

2009 ◽  
Vol 465 (2109) ◽  
pp. 2785-2812 ◽  
Author(s):  
Gareth L. Vaughan ◽  
Luke G. Bennetts ◽  
Vernon A. Squire
Keyword(s):  
Sea Ice ◽  
Gravity Waves ◽  
Experimental Validation ◽  
Ocean Waves ◽  
Correct Identification ◽  
Period Range ◽  
Reflection And Transmission ◽  
Transmission Coefficients ◽  
Naturally Occurring ◽  
Flexural Gravity Waves

Flexural oscillations of floating sea ice sheets induced by ocean waves travelling at the boundary between the ice and the water below can propagate great distances. But, by virtue of scattering, changes of ice thickness and other properties encountered during the journey affect their passage, notwithstanding attenuation arising from several other naturally occurring agencies. We describe here a two-dimensional model that can simulate wave scattering by long (approx. 50 km) stretches of inelastic sea ice, the goal being to replicate heterogeneity accurately while also assimilating supplementary processes that lead to energy loss in sea ice at scales that are amenable to experimental validation. In work concerned with scattering from solitary or juxtaposed stylized features in the sea ice canopy, reflection and transmission coefficients are commonly used to quantify scattering, but on this occasion, we use the attenuation coefficient as we consider that it provides a more helpful description when dealing with long sequences of adjoining scatterers. Results show that scattering and viscosity both induce exponential decay and we observe three distinct regimes: (i) low period, where scattering dominates, (ii) high period, where viscosity dominates, and (iii) a transition regime. Each regime’s period range depends on the sea ice properties including viscosity, which must be included for the correct identification of decay rate.

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