scholarly journals USE OF GALERKIN TECHNIQUE TO THE ROLLING OF A PLATE IN DEEP WATER

2021 ◽  
Vol 26 (2) ◽  
pp. 209-222
Author(s):  
Swagata Ray ◽  
Soumen De ◽  
B. N. Mandal
Keyword(s):  
Integral Equation ◽  
Deep Water ◽  
Water Waves ◽  
Vertical Plate ◽  
Fluid Velocity ◽  
Galerkin Approximation ◽  
Horizontal Component ◽  
Wave Number ◽  
Small Oscillations ◽  
Surface Water Waves

The classical problems of surface water waves produced by small oscillations of a thin vertical plate partially immersed as well as submerged in deep water are reinvestigated here. Each problem is reduced to an integral equation involving horizontal component of velocity across the vertical line outside the plate. The integral equations are solved numerically using Galerkin approximation in terms of simple polynomials multiplied by an appropriate weight function whose form is dictated by the behaviour of the fluid velocity near the edge(s) of the plate. Fairly accurate numerical estimates for the amplitude of the radiated wave at infinity due to rolling and also for swaying of the pate in each case are obtained and these are depicted graphically against the wave number for various cases.

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.

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.

scholarly journals Effect of Capillarity on Fourth Order Nonlinear Evolution Equation for Two Stokes Wave Trains in Deep Water in the Presence of Air Flowing Over Water

2015 ◽  
Vol 20 (2) ◽  
pp. 267-282
Author(s):  
A.K. Dhar ◽  
J. Mondal
Keyword(s):  
Fourth Order ◽  
Deep Water ◽  
Water Waves ◽  
Nonlinear Evolution ◽  
Wave Train ◽  
Nonlinear Evolution Equations ◽  
Wave Number ◽  
Stokes Wave ◽  
Instability Wave ◽  
Starting Point

Abstract Fourth order nonlinear evolution equations, which are a good starting point for the study of nonlinear water waves, are derived for deep water surface capillary gravity waves in the presence of second waves in which air is blowing over water. Here it is assumed that the space variation of the amplitude takes place only in a direction along which the group velocity projection of the two waves overlap. A stability analysis is made for a uniform wave train in the presence of a second wave train. Graphs are plotted for the maximum growth rate of instability wave number at marginal stability and wave number separation of fastest growing sideband component against wave steepness. Significant improvements are noticed from the results obtained from the two coupled third order nonlinear Schrödinger equations.

scholarly journals Scattering of water waves by a submerged thin vertical wall with a gap

1998 ◽  
Vol 39 (3) ◽  
pp. 318-331 ◽  
Author(s):  
Sudeshna Banerjea ◽  
B. N. Mandal
Keyword(s):  
Integral Equation ◽  
Surface Water ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Water Waves ◽  
Water Wave ◽  
Vertical Wall ◽  
Multiple Integral ◽  
Cauchy Kernel ◽  
Surface Water Waves

AbstractA train of surface water waves normally incident on a thin vertical wall completely submerged in deep water and having a gap, experiences reflection by the wall and transmission through the gaps above and in the wall. Using Havelock's expansion of water wave potential, two different integral equation formulations of the problem are presented. While the first formulation involves multiple integral equations which are solved here by reducing them to a singular integral equation with Cauchy kernel in a double interval, the second formulation involves a first-kind singular integral equation in a double interval with a combination of logarithmic and Cauchy kernel, the solution of which is obtained by utilizing the solution of a singular integral equation with Cauchy kernel in (0, ∞) and also in a double interval. The reflection coefficient is evaluated by both the methods.

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 WATER WAVE SCATTERING BY A THIN VERTICAL SUBMERGED PERMEABLE PLATE

2021 ◽  
Vol 26 (2) ◽  
pp. 223-235
Author(s):  
Rupanwita Gayen ◽  
Sourav Gupta ◽  
Aloknath Chakrabarti
Keyword(s):  
Integral Equation ◽  
Water Waves ◽  
Water Wave ◽  
Wave Theory ◽  
Hypersingular Integral Equation ◽  
Cauchy Type ◽  
Hypersingular Integral ◽  
Permeable Plate ◽  
Linear Water Wave Theory ◽  
Surface Water Waves

An alternative approach is proposed here to investigate the problem of scattering of surface water waves by a vertical permeable plate submerged in deep water within the framework of linear water wave theory. Using Havelock’s expansion of water wave potential, the associated boundary value problem is reduced to a second kind hypersingular integral equation of order 2. The unknown function of the hypersingular integral equation is expressed as a product of a suitable weight function and an unknown polynomial. The associated hypersingular integral of order 2 is evaluated by representing it as the derivative of a singular integral of the Cauchy type which is computed by employing an idea explained in Gakhov’s book [7]. The values of the reflection coefficient computed with the help of present method match exactly with the previous results available in the literature. The energy identity is derived using the Havelock’s theorems.

scholarly journals Waves on a vortex: rays, rings and resonances

2018 ◽  
Vol 857 ◽  
pp. 291-311 ◽  
Author(s):  
Theo Torres ◽  
Antonin Coutant ◽  
Sam Dolan ◽  
Silke Weinfurtner
Keyword(s):  
Experimental Data ◽  
Black Hole ◽  
Surface Water ◽  
Ray Tracing ◽  
Deep Water ◽  
Water Waves ◽  
Water Regime ◽  
Recent Experimental Data ◽  
Small Depth ◽  
Surface Water Waves

We study the scattering of surface water waves with irrotational draining vortices. At small depth, this system is a mathematical analogue of a rotating black hole and can be used to mimic some of its peculiar phenomena. Using ray-tracing methods, we exhibit the existence of unstable orbits around vortices at arbitrary depth. These orbits are the analogue of the light rings of a black hole. We show that these orbits come in pairs, one co-rotating and one counter-rotating, at an orbital radius that varies with the frequency. We derived an explicit formula for this radius in the deep-water regime. Our method is validated by comparison with recent experimental data from a wavetank experiment. We finally argue that these rings will generate a discrete set of damped resonances that we characterize and that could possibly be observed in future experiments.

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 On a singular integral equation with log kernel and its application

2006 ◽  
Vol 2006 ◽  
pp. 1-12 ◽  
Author(s):  
Sudeshna Banerjea ◽  
Chiranjib Sarkar
Keyword(s):  
Integral Equation ◽  
Surface Water ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Unknown Function ◽  
Water Waves ◽  
Theoretic Method ◽  
Vertical Barrier ◽  
Time Harmonic ◽  
Surface Water Waves

We used function theoretic method to solve a singular integral equation with logarithmic kernel in two disjoint finite intervals where the unknown function satisfying the integral equation may be bounded or unbounded at the nonzero finite endpoints of the interval concerned. An appropriate solution of this integral equation is then applied to solve the problem of scattering of time harmonic surface water waves by a fully submerged thin vertical barrier with a single gap.

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