scholarly journals Scattering of water waves by a submerged nearly circular cylinder

1995 ◽  
Vol 36 (3) ◽  
pp. 372-380 ◽  
Author(s):  
B. N. Mandal ◽  
Sudeshna Banerjea
Keyword(s):  
Reflection Coefficient ◽  
Circular Cylinder ◽  
Cross Section ◽  
Water Waves ◽  
Wave Train ◽  
Integral Theorem ◽  
Water Wave Scattering ◽  
First Order ◽  
Transmission Coefficients ◽  
Surface Water Waves

AbstractThe problem of scattering of surface water waves by a horizontal circular cylinder totally submerged in deep water is well studied in the literature within the framework of linearised theory with the remarkable conclusion that a normally incident wave train experiences no reflection. However, if the cross-section of the cylinder is not circular then it experiences reflection in general. The present paper studies the case when the cylinder is not quite circular and derives expressions for reflection and transmission coefficients correct to order ∈, where ∈ is a measure of small departure of the cylinder cross-section from circularity. A simplified perturbation analysis is employed to derive two independent boundary value problems (BVP) up to first order in ∈. The first BVP corresponds to the problem of water wave scattering by a submerged circular cylinder. The reflection coefficient up to first order and the first order correction to the transmission coefficient arise in the second BVP in a natural way and are obtained by a suitable use of Green' integral theorem without solving the second BVP. Assuming a Fourier expansion of the shape function, these are evaluated approximately. It is noticed that for some particular shapes of the cylinder, these vanish. Also, the numerical results for the transmission coefficients up to first order for a nearly circular cylinder for which the reflection coefficients up to first order vanish, are given in tabular form. It is observed that for many other smooth cylinders, the result for a circular cylinder that the reflection coefficient vanishes, is also approximately valid.

scholarly journals Water wave scattering by two submerged nearly vertical barriers

2006 ◽  
Vol 48 (1) ◽  
pp. 107-117 ◽  
Author(s):  
B. N. Mandal ◽  
Soumen De
Keyword(s):  
Water Wave ◽  
Wave Scattering ◽  
Shape Function ◽  
Integral Theorem ◽  
Water Wave Scattering ◽  
First Order ◽  
Transmission Coefficients ◽  
Definite Integrals ◽  
Vertical Barriers ◽  
Two Barriers

AbstractThe problem of surface water wave scattering by two thin nearly vertical barriers submerged in deep water from the same depth below the mean free surface and extending infinitely downwards is investigated here assuming linear theory, where configurations of the two barriers are described by the same shape function. By employing a simplified perturbational analysis together with appropriate applications of Green's integral theorem, first-order corrections to the reflection and transmission coefficients are obtained. As in the case of a single nearly vertical barrier, the first-order correction to the transmission coefficient is found to vanish identically, while the correction for the reflection coefficient is obtained in terms of a number of definite integrals involving the shape function describing the two barriers. The result for a single barrier is recovered when two barriers are merged into a single barrier.

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.

Water Wave Scattering by Two Circular-Arc-Shaped Thin Plates With Non-Uniform Permeability

2019 ◽  
Author(s):  
R. Gayen ◽  
Sourav Gupta
Keyword(s):  
Approximate Solution ◽  
Integral Equations ◽  
Potential Function ◽  
Water Waves ◽  
Thin Plates ◽  
Integral Theorem ◽  
Water Wave Scattering ◽  
Circular Arc ◽  
Transmission Coefficients ◽  
Different Depths

Abstract The interaction of small amplitude water waves with a pair of circular-arc-shaped thin porous plates is studied under the assumption of linear theory. The plates are submerged at different depths in deep water and the permeability of the plates varies along the circumference of the plates. Applying Green’s integral theorem to the fundamental potential function and the scattered potential function and using the condition on the porous plates, the problem is reduced to that of solving a set of coupled integral equations for the potential difference functions across the plates. Two kernels of these integral equations are hypersingular and the other two kernels are regular. An expansion-cum-collocation method involving Chebyshev polynomials is employed to obtain the approximate solution of the integral equations. The numerical estimates for the reflection and the transmission coefficients are computed utilizing the approximate solution of the aforesaid integral equations. The numerical results for the reflection and the transmission coefficients are depicted graphically for several values of various parameters. Known results for dual symmetric circular-arc-shaped porous plates are recovered as special case.

scholarly journals Wave Scattering by Small Undulation on the Porous Bottom of an Ocean in the Presence of Surface Tension

2013 ◽  
Vol 2013 ◽  
pp. 1-6 ◽  
Author(s):  
Srikumar Panda ◽  
Sudhanshu Shekhar Samantaray ◽  
S. C. Martha
Keyword(s):  
Surface Tension ◽  
Water Waves ◽  
Bottom Topography ◽  
Wave Theory ◽  
Reflection And Transmission ◽  
First Order ◽  
Transmission Coefficients ◽  
Bottom Undulation ◽  
Surface Water Waves ◽  
The Fourier Transform

The scattering of incident surface water waves due to small bottom undulation on the porous bed of a laterally unbounded ocean in the presence of surface tension at the free surface is investigated within the framework of two-dimensional linearized water wave theory. Perturbation analysis in conjunction with the Fourier transform technique is employed to derive the first-order reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. One special type of bottom topography is considered as an example and the related coefficients are determined in detail. These coefficients are presented in graphical forms. The theoretical observations are validated computationally. The results for the problem involving scattering of water waves by bottom deformations on an impermeable ocean bed are obtained as a particular case.

On a uniformly valid model for surface wave interaction

1993 ◽  
Vol 247 ◽  
pp. 589-601 ◽  
Author(s):  
Yehuda Agnon
Keyword(s):  
Surface Wave ◽  
Water Waves ◽  
Wave Train ◽  
Wave Interaction ◽  
Shallow Water Waves ◽  
First Order ◽  
Near Resonance ◽  
Wave Trains ◽  
Velocity Changes ◽  
Forced Waves

Nonlinear interaction of surface wave trains is studied. Zakharov's kernel is extended to include the vicinity of trio resonance. The forced wave amplitude and the wave velocity changes are then first order rather than second order. The model is applied to remove near-resonance singularities in expressions for the change of speed of one wave train in the presence of another. New results for Wilton ripples and the drift current and setdown in shallow water waves are readily derived. The ideas are applied to the derivation of forced waves in the vicinity of quartet and quintet resonance in an evolving wave field.

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.

scholarly journals On waves due to small oscillations of a vertical plate submerged in deep water

1991 ◽  
Vol 32 (3) ◽  
pp. 296-303 ◽  
Author(s):  
B. N. Mandal
Keyword(s):  
Surface Water ◽  
Deep Water ◽  
Water Waves ◽  
Line Source ◽  
Vertical Plate ◽  
Integral Theorem ◽  
Small Oscillations ◽  
Surface Water Waves ◽  
Green’S Integral Theorem

AbstractThis paper is concerned with surface water waves produced by small oscillations of a thin vertical plate submerged in deep water. Green's integral theorem in the fluid region is used in a suitable manner to obtain the amplitude for the radiated waves at infinity. Particular results for roll and sway of the plate, and for a line source in the presence of a fixed vertical plate, are deduced.

The transmission of surface waves under surface obstacles

1961 ◽  
Vol 57 (3) ◽  
pp. 638-668 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Transmission Coefficient ◽  
Cross Section ◽  
Water Waves ◽  
Wave Train ◽  
Circular Cross Section ◽  
Great Distance ◽  
The Body ◽  
Radius Of Curvature

ABSTRACTA train of surface waves (water waves under gravity) is normally incident on a cylinder with horizontal generators fixed near the free surface, and is partially transmitted and partially reflected. At a great distance behind the cylinder the wave motion tends to a regular wave train travelling towards infinity; the ratio of its amplitude to the amplitude of the incident wave is the transmission coefficient . The transmission coefficient is studied when the wavelength is short compared to the dimensions of the body; physically (though not for engineering applications) this is the most interesting range of wavelengths, which corresponds to the range of shadow formation and ray propagation in optics and acoustics. The waves are then confined to a thin layer near the free surface, and the transmission under a partially immersed obstacle is then small. In the calculation the boundary condition at the free surface is linearized, viscosity is neglected, and the motion is assumed to be irrotational.At present the transmission coefficient is known only for a few configurations, all of them relating to infinitely thin plane barriers. A method is now given which is applicable to cylinders of finite cross-section and which is worked out in detail for a half-immersed cylinder of circular cross-section. The solution of the problem is made to depend on the solution of an integral equation which is solved by iteration. Only the first two terms can be obtained with any accuracy, and it appears at first that this is not sufficient to give the leading term in the transmission coefficient at short wavelengths; this difficulty is characteristic of transmission problems. By various mathematical devices which throw light on the mechanism of wave transmission, it is, nevertheless, found possible to prove that the transmission coefficient for waves of short wavelength λ and period 2π/ω incident on a half-immersed circular cylinder of radius a is asymptotically given bywhen N = 2πα/λ = ω2α/g is large. Earlier evidence had pointed towards an exponential law. It is suggested that transmission coefficients of order N−4 are typical for obstacles having vertical tangents and finite non-zero radius of curvature at the points where they meet the horizontal mean free surface. For obstacles having both front and rear face plane vertical to a depth a, is probably of order e−2N approximately; if only one of the two faces is plane vertical, is probably of order e−N approximately. Thus is seen to depend critically on the details of the cross-section.

scholarly journals A plane vertical submerged barrier in surface water waves

1987 ◽  
Vol 10 (4) ◽  
pp. 815-820 ◽  
Author(s):  
U. Basu ◽  
B. N. Mandal
Keyword(s):  
Surface Water ◽  
Deep Water ◽  
Water Waves ◽  
Line Source ◽  
Vertical Plane ◽  
Simple Application ◽  
Integral Theorem ◽  
Surface Water Waves ◽  
Plane Barrier ◽  
Green’S Integral Theorem

By a simple application of Green's integral theorem, amplitude of the radiated waves at infinity due to a line source in the presence of a fixed vertical plane barrier completely submerged in deep water is obtained.

scholarly journals On the solution of the problem of scattering of surface water waves by a sharp discontinuity in the surface boundary conditions

2000 ◽  
Vol 42 (2) ◽  
pp. 277-286 ◽  
Author(s):  
A. Chakrabarti
Keyword(s):  
Surface Water ◽  
Boundary Conditions ◽  
Water Waves ◽  
Surface Boundary ◽  
Transmission Coefficients ◽  
Sharp Discontinuity ◽  
Surface Boundary Conditions ◽  
Method Of Solution ◽  
Surface Water Waves ◽  
Carleman Type

AbstractClosed-form analytical expressions are derived for the reflection and transmission coefficients for the problem of scattering of surface water waves by a sharp discontinuity in the surface-boundary-conditions, for the case of deep water. The method involves the use of the Havelock-type expansion of the velocity potential along with an analysis to solve a Carleman-type singular integral equation over a semi-infinite range. This method of solution is an alternative to the Wiener-Hopf technique used previously.

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