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 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.

scholarly journals A unified approach to problems of scattering of surface water waves by vertical barriers

1997 ◽  
Vol 39 (1) ◽  
pp. 93-103 ◽  
Author(s):  
A. Chakrabarti ◽  
Sudeshna Banerjea ◽  
B. N. Mandal ◽  
T. Sahoo
Keyword(s):  
Surface Water ◽  
Water Waves ◽  
Water Wave ◽  
Wave Scattering ◽  
Multiple Integral ◽  
Cauchy Type ◽  
Unified Approach ◽  
Water Wave Scattering ◽  
Vertical Barriers ◽  
Surface Water Waves

AbstractA unified analysis involving the solution of multiple integral equations via a simple singular integral equation with a Cauchy type kernel is presented to handle problems of surface water wave scattering by vertical barriers. Some well known results are produced in a simple and systematic manner.

Approximate Solution Of A Singular Integral Equation With A Multiplicative Cauchy Kernel In The Half-Plane

2008 ◽  
Vol 8 (2) ◽  
pp. 143-154 ◽  
Author(s):  
P. KARCZMAREK
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Half Plane ◽  
Singular Integral ◽  
Trigonometric Polynomials ◽  
Cauchy Kernel

AbstractIn this paper, Jacobi and trigonometric polynomials are used to con-struct the approximate solution of a singular integral equation with multiplicative Cauchy kernel in the half-plane.

Approximate Solution of a Singular Integral Equation with a Cauchy Kernel on the Euclidean Plane

2018 ◽  
Vol 18 (4) ◽  
pp. 741-752
Author(s):  
Dorota Pylak ◽  
Paweł Karczmarek ◽  
Paweł Wójcik
Keyword(s):  
Integral Equation ◽  
Approximate Solution ◽  
Singular Integral Equation ◽  
Singular Integral ◽  
Euclidean Plane ◽  
Singular Integral Equations ◽  
Cauchy Kernel ◽  
Challenging Problem ◽  
One Dimensional ◽  
Interpolating Polynomials

AbstractMultidimensional singular integral equations (SIEs) play a key role in many areas of applied science such as aerodynamics, fluid mechanics, etc. Solving an equation with a singular kernel can be a challenging problem. Therefore, a plethora of methods have been proposed in the theory so far. However, many of them are discussed in the simplest cases of one–dimensional equations defined on the finite intervals. In this study, a very efficient method based on trigonometric interpolating polynomials is proposed to derive an approximate solution of a SIE with a multiplicative Cauchy kernel defined on the Euclidean plane. Moreover, an estimation of the error of the approximated solution is presented and proved. This assessment and an illustrating example show the effectiveness of our proposal.

scholarly journals A convergence theorem for singular integral equations

1981 ◽  
Vol 22 (4) ◽  
pp. 539-552 ◽  
Author(s):  
David Elliott
Keyword(s):  
Integral Equation ◽  
Singular Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Convergence Theory ◽  
Cauchy Kernel ◽  
Algebraic Equations ◽  
Linear Algebraic Equations

AbstractThe principal result of this paper states sufficient conditions for the convergence of the solutions of certain linear algebraic equations to the solution of a (linear) singular integral equation with Cauchy kernel. The motivation for this study has been the need to provide a convergence theory for a collocation method applied to the singular integral equation taken over the arc (−1, 1). However, much of the analysis will be applicable both to other approximation methods and to singular integral equations taken over other arcs or contours. An estimate for the rate of convergence is also given.

scholarly journals Torsional Impact Response of a Penny-Shaped Interface Crack in Bonded Materials With a Graded Material Interlayer

2002 ◽  
Vol 69 (3) ◽  
pp. 303-308 ◽  
Author(s):  
C. Li ◽  
Z. Duan ◽  
Z. Zou
Keyword(s):  
Integral Equation ◽  
Stress Intensity ◽  
Singular Integral Equation ◽  
Interface Crack ◽  
Singular Integral ◽  
Dynamic Stress ◽  
Mixed Boundary ◽  
Dynamic Stress Intensity Factor ◽  
Cauchy Kernel ◽  
Dynamic Stress Intensity Factors

In this paper, the dynamic response of a penny-shaped interface crack in bonded dissimilar homogeneous half-spaces is studied. It is assumed that the two materials are bonded together with such a inhomogeneous interlayer that makes the elastic modulus in the direction perpendicular to the crack surface is continuous throughout the space. The crack surfaces are assumed to be subjected to torsional impact loading. Laplace and Hankel integral transforms are applied combining with a dislocation density function to reduce the mixed boundary value problem into a singular integral equation with a generalized Cauchy kernel in Laplace domain. By solving the singular integral equation numerically and using a numerical Laplace inversion technique, the dynamic stress intensity factors are obtained. The influences of material properties and interlayer thickness on the dynamic stress intensity factor are investigated.

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.

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