A note on the singularities in the theory of water waves with an inertial surface

doi:10.1017/S0334270000005361

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A note on the singularities in the theory of water waves with an inertial surface

Journal Article•DOI•

A note on the singularities in the theory of water waves with an inertial surface

B. N. Mandal, Krishna Kundu¹•Institutions (1)

01 Oct 1986-The Journal of The Australian Mathematical Society. Series B. Applied Mathematics (Cambridge University Press)-Vol. 28, Iss: 02, pp 271-278

TL;DR: In this paper, the derivation of velocity potentials describing the generation of infinitesimal gravity waves in a motionless liquid with an inertial surface composed of uniformly distributed floating particles, due to fundamental line and point sources with time-dependent strengths submerged in a liquid of finite constant depth, is discussed.

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Abstract: This note is concerned with the derivation of velocity potentials describing the generation of infinitesimal gravity waves in a motionless liquid with an inertial surface composed of uniformly distributed floating particles, due to fundamental line and point sources with time-dependent strengths submerged in a liquid of finite constant depth.

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Journal Article•DOI•

Motion due to fundamental singularities in finite depth water with an elastic solid cover

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Rupanwita Gayen Chowdhury¹, B. N. Mandal¹•Institutions (1)

Indian Statistical Institute¹

01 Apr 2006-Fluid Dynamics Research

TL;DR: In this article, the Laplace transform technique is employed to obtain their solutions for initial value problems and Graphs of particular integrals appearing in the representations of the potential functions are presented, which confirm that the expressions thus derived are suitable for numerical computation for all possible combinations of the positions of the field point and the source point.

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19 citations

Journal Article•DOI•

Water waves generated by disturbance at an inertial surface

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B. N. Mandal¹•Institutions (1)

University of Calcutta¹

01 Mar 1988-Flow Turbulence and Combustion

TL;DR: In this paper, the Laplace transform technique is used to solve an initial value problem describing waves generated by a disturbance created at the surface of water covered by an inertial surface composed of a thin but uniform distribution of floating particles.

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Abstract: Laplace transform technique is used to solve an initial value problem describing waves generated by a disturbance created at the surface of water covered by an inertial surface composed of a thin but uniform distribution of floating particles. Green's integral theorem produces the transformed potential function from which the form of the inertial surface is obtained as an infinite integral after taking Laplace inversion. The method of stationary phase is then employed to evaluate this integral approximately for large time and distance.

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16 citations

Journal Article•DOI•

Potential functions of HKT spaces

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Yat Sun Poon¹, Andrew Swann²•Institutions (2)

University of California, Riverside¹, University of Southern Denmark²

07 Nov 2001-Classical and Quantum Gravity

TL;DR: In this article, the potential functions for = 4B supersymmetric quantum mechanics with D(2,1,α) symmetry were described for the case of 4B quantum mechanics.

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Abstract: We describe the potential functions for = 4B supersymmetric quantum mechanics with D(2,1;α) symmetry.

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10 citations

Journal Article•DOI•

Water waves generated at an inertial surface by an axisymmetric initial surface disturbance

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B. N. Mandal, S. Mukherjee

01 Sep 1989-International Journal of Mathematical Education in Science and Technology

TL;DR: In this article, the problem of generating surface waves in a liquid with an inertial surface due to an initial axially symmetric surface disturbance is discussed, and the form of the inertial surfaces is obtained asymptotically for large values of time and distance.

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Abstract: The problem of generation of surface waves in a liquid with an inertial surface due to an initial axially symmetric surface disturbance is discussed. The form of the inertial surface is obtained asymptotically for large values of time and distance.

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9 citations

Journal Article•DOI•

Problems in the dynamics of flotation liquids

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S. A. Gabov, A. G. Sveshnikov

01 Apr 1991-Journal of Mathematical Sciences

TL;DR: In this article, the authors present new mathematical results in the theory of linear and nonlinear waves on the surface of a flotation liquid, and prove the existence of nonlinear standing waves within the framework of an exact physical model.

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Abstract: This survey presents new mathematical results in the theory of linear and nonlinear waves on the surface of a flotation liquid. A flotation liquid is a liquid on whose surface heavy particles are floating; the particles may consist of arbitrary materials or may be particles of frozen liquid. The first part of the article considers initial- and boundary-value problems in the theory, their solvability, and the behavior of the solutions over long periods. In the second part of the survey, theorems are proved on the existence of nonlinear standing waves within the framework of an exact physical model, and both internal and free waves are considered. Also, the fundamental equations for shallow flotation waves are derived and examined.

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9 citations

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Journal Article•DOI•

Multipole expansions in the theory of surface waves

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R. C. Thorne¹, F. Ursell•Institutions (1)

University of Cambridge¹

01 Oct 1953

TL;DR: In this article, the authors considered the problem of the generation of surface waves by a submerged circular cylinder and used a series of complex potential functions arising from multipoles at the centre of the cylinder, but the velocity potential of the motion could have been described, without the introduction of the stream function.

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Abstract: Problems dealing with the generation of surface waves in water involve the consideration of singularities of different types in the liquid. In the case when bodies are present in the liquid, waves may be either generated by the movement of the body, or reflected from the body. The two cases are essentially equivalent, and the resulting motion can be described by a series of singularities placed within the body. The boundary conditions on the surface of the body give equations from which the exact form of the potential can be obtained. Ursell (10) has solved in this manner the problem, earlier discussed by Dean(1), of the generation of surface waves by a submerged circular cylinder. In this two-dimensional problem he used a series of complex potential functions arising from multipoles at the centre of the cylinder, but the velocity potential of the motion could have been described, without the introduction of the stream function, in terms of the velocity potentials of the multipoles.

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176 citations

"A note on the singularities in the ..." refers background in this paper

...Thorne [5] listed the different forms of velocity potentials due to different kinds of time-harmonic singularities present in the liquid with a free surface and Rhodes-Robinson [2] modified these results by taking into account the effect of surface tension of the free surface....
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Journal Article•DOI•

The initial value problem for transient water waves

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A. B. Finkelstein¹•Institutions (1)

Pratt Institute¹

01 Jan 1957-Communications on Pure and Applied Mathematics

79 citations

"A note on the singularities in the ..." refers methods in this paper

...8) This coincides with the expression obtained by Finkelstein [1] for e = 0....
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...References [1] A. B. Finkelstein, "The initial-value problem for transient water waves", Comm. Pure Appl....
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...X) 1 — cosfit D(k) sinh kh k (3.8) This coincides with the expression obtained by Finkelstein [1] for e = 0....
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Journal Article•DOI•

Surface waves generated by an oscillating circular cylinder on water of finite depth: theory and experiment

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Y. S. Yu¹, F. Ursell²•Institutions (2)

University of Kansas¹, University of Cambridge²

01 Dec 1961-Journal of Fluid Mechanics

TL;DR: In this article, the mean position of the cylinder is in the mean free surface, and the cylinder oscillates vertically (heaving) with a simple harmonic motion of small amplitude, both theoretically and experimentally.

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Abstract: The mean position of the cylinder is in the mean free surface, and the cylinder is oscillating vertically (heaving) with a simple harmonic motion of small amplitude. The problem is studied both theoretically and experimentally. The present paper may be regarded as a sequel to an earlier paper (Ursell, Dean & Yu 1959) on the waves due to a piston wave-maker.

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74 citations

"A note on the singularities in the ..." refers background in this paper

...[6] Y. S. Yu and F. Ursell, "Surface waves generated by an oscillating circular cylinder on water of finite depth: theory and experiment", J. Fluid Mech., 11 (1961), 529-551....
[...]
...Following Yu and Ursell [6], we note that the integral /•«> coshk(h - Y)coshk(h - y)exp(ik\x - X\) (k sinh kh — K* cosh kh ) cosh kh along the real k axis (indented at k = k$, where the integrand has a simple pole) is equal to the integral along arg/c = w/4, and thus clearly tends to zero as \x - X\ -» oo....
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...Following Yu and Ursell [6], we note that the integral /•«> coshk(h -...
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Journal Article•DOI•

Fundamental singularities in the theory of water waves with surface tension

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P. F. Rhodes-Robinson

01 Jun 1970-Bulletin of The Australian Mathematical Society

TL;DR: In this paper, the authors obtained the forms for the harmonic potential functions describing the fundamental wave-source and multipole singularities which pertain to the study of infinitesimal time-harmonic waves on the free surface of water when the effect of surface tension is included.

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Abstract: In this paper the forms are obtained for the harmonic potential functions describing the fundamental wave-source and multipole singularities which pertain to the study of infinitesimal time-harmonic waves on the free surface of water when the effect of surface tension is included. Line and point singularities are considered for both the cases of infinite and finite constant depth of water. The method used is an extension of that which has been used to obtain these potentials in the absence of surface tension.

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33 citations

"A note on the singularities in the ..." refers background in this paper

...Thorne [5] listed the different forms of velocity potentials due to different kinds of time-harmonic singularities present in the liquid with a free surface and Rhodes-Robinson [2] modified these results by taking into account the effect of surface tension of the free surface....
[...]
...Rhodes-Robinson [3] considered a two-dimensional wave source submerged in a liquid of infinite depth with an inertial surface....
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...[4] P. F. Rhodes-Robinson, "Note on the effect of surface tension on water waves at an inertial surface", J. Fluid Mech....
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...[3] P. F. Rhodes-Robinson, "On the generation of water waves at an inertial surface", J. Austral....
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...[2] P. F. Rhodes-Robinson, "Fundamental singularities in the theory of water waves with surface tension", Bull Austral....
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Journal Article•DOI•

On the generation of water waves at an inertial surface

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P. F. Rhodes-Robinson

01 Jan 1984-The Journal of The Australian Mathematical Society. Series B. Applied Mathematics

TL;DR: On developpe une methode de transformee de Laplace as mentioned in this paper for resoudre les problemes de valeur initiale, le potentiel de vitesse decrivant la generation d'ondes de gravite capillaires infinitesimales dans un liquide au repos avec une surface inertielle composee de particules flottantes distribuees uniformement.

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Abstract: On developpe une methode de transformee de Laplace pour resoudre les problemes de valeur initiale, le potentiel de vitesse decrivant la generation d'ondes de gravite capillaires infinitesimales dans un liquide au repos avec une surface inertielle composee de particules flottantes distribuees uniformement

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26 citations

"A note on the singularities in the ..." refers background or result in this paper

...As discussed in [3] 9 can be shown to satisfy the initial conditions 9 - edq>/dy = 0, 3/3f(<p - edtp/dy) = 0 on y = 0 at t = 0, (2....
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...Hence as t -» oo, following [3], we obtain after simplification, coshkjh - Y)coshk{h - y) G - sinaf lln-^ - f , k sinh kh - K* cosh kh , exp(/rft) ....
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...Thorne [5] listed the different forms of velocity potentials due to different kinds of time-harmonic singularities present in the liquid with a free surface and Rhodes-Robinson [2] modified these results by taking into account the effect of surface tension of the free surface....
[...]
...It is believed that the potential functions derived here and in [3] due to different types of singularities will be of use when problems associated with obstacles of different geometrical shapes present in a liquid with an inertial surface are considered....
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...Rhodes-Robinson [3] considered a two-dimensional wave source submerged in a liquid of infinite depth with an inertial surface....
[...]