On the heaving motion of a circular cylinder more or less than half immersed in the free surface of a fluid

1988 ◽  
Vol 419 (1856) ◽  
pp. 107-120
Keyword(s):  
Integral Equation ◽  
Free Surface ◽  
Circular Cylinder ◽  
Integral Equations ◽  
Water Waves ◽  
Singular Integral Equations ◽  
Two Dimensional ◽  
Existence Of A Solution ◽  
Linearized Theory ◽  
Explicit Proof

We consider a problem in the linearized theory of water waves. A smooth oscillating two-dimensional body meets the free surface at angles other than right-angles. In this paper we prove the existence of a solution for this problem by using integral equations. This problem has been considered by other authors; however, their attempts have resulted in singular integral equations. To show that Fredholm theory applies to these equations involves a great deal of generalized analysis. It is shown that it is possible to obtain a well-behaved integral equation by means of an explicit modification of the source potential used to derive this equation. To illustrate this method a circular cylinder that is more than or less than half immersed and undergoing a heaving motion is considered. This method is in terms of more elementary concepts than those used by previous authors. The explicit proof also indicates how the problem may be solved in practice, and it is hoped to report on the numerical solution later.

Water wave diffraction by a surface strip

2007 ◽  
Vol 571 ◽  
pp. 419-438 ◽  
Author(s):  
RUPANWITA GAYEN ◽  
B. N. MANDAL ◽  
A. CHAKRABARTI
Keyword(s):  
Integral Equations ◽  
Wave Diffraction ◽  
Water Waves ◽  
Numerical Solutions ◽  
Singular Integral Equations ◽  
Fredholm Integral Equations ◽  
Transmission Coefficients ◽  
Wide Strip ◽  
Linearized Theory ◽  
Carleman Type

The two-dimensional problem of wave diffraction by a strip of arbitrary width is investigated here in the context of linearized theory of water waves by reducing it to a pair of Carleman-type singular integral equations. These integral equations have been solved earlier by an iterative process which is valid only for a sufficiently wide strip. A new method is described here by which solutions of these integral equations are determined by solving a set of four Fredholm integral equations of the second kind, and the process is valid for a strip of arbitrary width. Numerical solutions of these Fredholm integral equations are utilized to obtain fairly accurate numerical estimates for the reflection and transmission coefficients. Previous numerical results for a wide strip are recovered from the present analysis. Additional results for the reflection coefficient are presented graphically for moderate values of the strip width which exhibit a less oscillatory nature of the curve than the case of a wide strip.

The effect of surface tension on the waves produced by a heaving circular cylinder

1968 ◽  
Vol 64 (3) ◽  
pp. 833-847 ◽  
Author(s):  
D. V. Evans
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Circular Cylinder ◽  
Water Waves ◽  
Wave Amplitude ◽  
Velocity Potential ◽  
Linearized Theory ◽  
The Waves ◽  
Energy Argument ◽  
Effect Of Surface

AbstractIn this paper the effect of surface tension on water waves is considered. The usual assumptions of the linearized theory are made. A uniqueness theorem is derived for the waves at infinity for a general class of bounded two-dimensional obstacles in a free surface by means of an energy argument. It is shown how the wave amplitude at infinity depends on the prescribed angle at which the free surface meets the normal to the obstacle. The particular case of a heaving half-immersed circular cylinder is considered in detail, and an expression obtained for the velocity potential in terms of a convergent infinite series, the coefficients of which may be computed.

scholarly journals Nonlinear two-dimensional free surface solutions of flow exiting a pipe and impacting a wedge

2021 ◽  
Vol 126 (1) ◽  
Author(s):  
Alex Doak ◽  
Jean-Marc Vanden-Broeck
Keyword(s):  
Surface Tension ◽  
Integral Equation ◽  
Free Surface ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Two Dimensional ◽  
Numerical Schemes ◽  
Additional Advantage ◽  
Infinite Wedge ◽  
Good Agreement

AbstractThis paper concerns the flow of fluid exiting a two-dimensional pipe and impacting an infinite wedge. Where the flow leaves the pipe there is a free surface between the fluid and a passive gas. The model is a generalisation of both plane bubbles and flow impacting a flat plate. In the absence of gravity and surface tension, an exact free streamline solution is derived. We also construct two numerical schemes to compute solutions with the inclusion of surface tension and gravity. The first method involves mapping the flow to the lower half-plane, where an integral equation concerning only boundary values is derived. This integral equation is solved numerically. The second method involves conformally mapping the flow domain onto a unit disc in the s-plane. The unknowns are then expressed as a power series in s. The series is truncated, and the coefficients are solved numerically. The boundary integral method has the additional advantage that it allows for solutions with waves in the far-field, as discussed later. Good agreement between the two numerical methods and the exact free streamline solution provides a check on the numerical schemes.

scholarly journals New exact relations for steady irrotational two-dimensional gravity and capillary surface waves

2017 ◽  
Vol 376 (2111) ◽  
pp. 20170220 ◽  
Author(s):  
Didier Clamond
Keyword(s):  
Free Surface ◽  
Gravity Waves ◽  
Water Waves ◽  
Two Dimensional ◽  
Physical Plane ◽  
Theme Issue ◽  
Nonlinear Water Waves ◽  
Dimensional Gravity ◽  
Irrotational Motion ◽  
Exact Relations

Steady two-dimensional surface capillary–gravity waves in irrotational motion are considered on constant depth. By exploiting the holomorphic properties in the physical plane and introducing some transformations of the boundary conditions at the free surface, new exact relations and equations for the free surface only are derived. In particular, a physical plane counterpart of the Babenko equation is obtained. This article is part of the theme issue ‘Nonlinear water waves’.

A method for the exact solution of a class of two-dimensional diffraction problems

1951 ◽  
Vol 205 (1081) ◽  
pp. 286-308 ◽  
Keyword(s):  
Integral Equation ◽  
Plane Wave ◽  
Integral Equations ◽  
Scattered Field ◽  
Plane Waves ◽  
Angular Spectrum ◽  
Diffraction Theory ◽  
General Interest ◽  
Two Dimensional ◽  
Dual Integral Equations

In the last few years Copson, Schwinger and others have obtained exact solutions of a number of diffraction problems by expressing these problems in terms of an integral equation which can be solved by the method of Wiener and Hopf. A simpler approach is given, based on a representation of the scattered field as an angular spectrum of plane waves, such a representation leading directly to a pair of ‘dual’ integral equations, which replaces the single integral equation of Schwinger’s method. The unknown function in each of these dual integral equations is that defining the angular spectrum, and when this function is known the scattered field is presented in the form of a definite integral. As far as the ‘radiation’ field is concerned, this integral is of the type which may be approximately evaluated by the method of steepest descents, though it is necessary to generalize the usual procedure in certain circumstances. The method is appropriate to two-dimensional problems in which a plane wave (of arbitrary polarization) is incident on plane, perfectly conducting structures, and for certain configurations the dual integral equations can be solved by the application of Cauchy’s residue theorem. The technique was originally developed in connexion with the theory of radio propagation over a non-homogeneous earth, but this aspect is not discussed. The three problems considered are those for which the diffracting plates, situated in free space, are, respectively, a half-plane, two parallel half-planes and an infinite set of parallel half-planes; the second of these is illustrated by a numerical example. Several points of general interest in diffraction theory are discussed, including the question of the nature of the singularity at a sharp edge, and it is shown that the solution for an arbitrary (three-dimensional) incident field can be derived from the corresponding solution for a two-dimensional incident plane wave.

Solution of a Linear Integral Equation of Third Kind

2002 ◽  
Vol 9 (1) ◽  
pp. 179-196
Author(s):  
D. Shulaia
Keyword(s):  
Integral Equation ◽  
Integral Equations ◽  
Singular Integral ◽  
Sufficient Conditions ◽  
Singular Integral Equations ◽  
Fredholm Theory ◽  
Linear Integral Equation ◽  
Hölder Functions ◽  
Linear Integral ◽  
Necessary And Sufficient

Abstract The aim of this paper is to study, in the class of Hölder functions, a nonhomogeneous linear integral equation with coefficient cos 𝑥. Necessary and sufficient conditions for the solvability of this equation are given under some assumptions on its kernel. The solution is constructed analytically, using the Fredholm theory and the theory of singular integral equations.

EFFECTS OF SURFACE TENSION ON TRAPPED MODES IN A TWO-LAYER FLUID

2015 ◽  
Vol 57 (2) ◽  
pp. 189-203 ◽  
Author(s):  
S. SAHA ◽  
S. N. BORA
Keyword(s):  
Surface Tension ◽  
Free Surface ◽  
Circular Cylinder ◽  
Boundary Value Problems ◽  
Finite Depth ◽  
Trapped Modes ◽  
Trapped Waves ◽  
Linearized Theory ◽  
Horizontal Circular Cylinder ◽  
Effect Of Surface

We consider a two-layer fluid of finite depth with a free surface and, in particular, the surface tension at the free surface and the interface. The usual assumptions of a linearized theory are considered. The objective of this work is to analyse the effect of surface tension on trapped modes, when a horizontal circular cylinder is submerged in either of the layers of a two-layer fluid. By setting up boundary value problems for both of the layers, we find the frequencies for which trapped waves exist. Then, we numerically analyse the effect of variation of surface tension parameters on the trapped modes, and conclude that realistic changes in surface tension do not have a significant effect on the frequencies of these.

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.

Note on the scattering of water waves by a vertical barrier

1966 ◽  
Vol 62 (3) ◽  
pp. 507-509 ◽  
Author(s):  
W. E. Williams
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Small Amplitude ◽  
Two Dimensional ◽  
Vertical Barrier ◽  
Alternative Approach

Introduction. In this note an alternative approach is presented to the problem of the scattering of small amplitude two-dimensional water waves by a fixed barrier, one edge of the barrier lying in the free surface of the water. This problem was first solved by Ursell ((1)) and generalizations of the problem have been considered by John ((2)) and Lewin ((3)).

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