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Journal ArticleDOI

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

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

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.
Topics: Inertial wave (64%), Dispersion (water waves) (59%), Gravity wave (57%), Internal wave (57%), Love wave (56%)
Citations
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Journal ArticleDOI
Abstract: Motion due to fundamental line and point singularities with time-dependent strengths submerged in water with an elastic solid cover is investigated assuming linear theory. The cover is of very small thickness and uniform area density. The problems are formulated as initial value problems and the Laplace transform technique is employed to obtain their solutions. For time-harmonic source strength, the steady-state development of the potential functions demonstrates the existence of outgoing progressive waves of any frequency under the solid cover. Alternative forms of the functions for their numerical evaluations are also obtained. Graphs of some particular integrals appearing in the representations of the potential functions are presented, which confirms 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.

19 citations


Journal ArticleDOI
B. N. Mandal1Institutions (1)
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.

16 citations


Journal ArticleDOI
Yat Sun Poon1, Andrew Swann2Institutions (2)
Abstract: We describe the potential functions for = 4B supersymmetric quantum mechanics with D(2,1;α) symmetry.

10 citations


Journal ArticleDOI
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.

9 citations


Journal ArticleDOI
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.

9 citations


References
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Journal ArticleDOI
R. C. Thorne1, F. UrsellInstitutions (1)
01 Oct 1953-
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.

167 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....

    [...]


Journal ArticleDOI
A. B. Finkelstein1Institutions (1)

73 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....

    [...]

  • ...References [1] A. B. Finkelstein, "The initial-value problem for transient water waves", Comm. Pure Appl....

    [...]

  • ...X) 1 — cosfit D(k) sinh kh k (3.8) This coincides with the expression obtained by Finkelstein [1] for e = 0....

    [...]


Journal ArticleDOI
Y. S. Yu1, F. Ursell2Institutions (2)
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.

70 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....

    [...]

  • ...Following Yu and Ursell [6], we note that the integral /•«> coshk(h -...

    [...]


Journal ArticleDOI
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.

32 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....

    [...]

  • ...[4] P. F. Rhodes-Robinson, "Note on the effect of surface tension on water waves at an inertial surface", J. Fluid Mech....

    [...]

  • ...[3] P. F. Rhodes-Robinson, "On the generation of water waves at an inertial surface", J. Austral....

    [...]

  • ...[2] P. F. Rhodes-Robinson, "Fundamental singularities in the theory of water waves with surface tension", Bull Austral....

    [...]


Journal ArticleDOI
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

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....

    [...]

  • ...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) ....

    [...]

  • ...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....

    [...]

  • ...Rhodes-Robinson [3] considered a two-dimensional wave source submerged in a liquid of infinite depth with an inertial surface....

    [...]


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