scholarly journals On the propagation of a disturbance in a fluid under gravity

1910 ◽  
Vol 83 (563) ◽  
pp. 347-356 ◽  
Keyword(s):  
Incompressible Fluid ◽  
Gravity Wave ◽  
Integral Form ◽  
Finite Depth ◽  
Infinite Depth ◽  
Poisson Problem

In a recent paper Lord Rayleigh has called attention to the fact of the instantaneous propagation of a limited disturbance over the surface of heavy incompressible fluid. This instantaneity occurs in spite of the fact that the velocity of any simple-harmonic gravity wave is finite, the depth being finite and constant. As the points thereby raised are of some delicacy, and are not completely settled in the paper quoted, some further remarks on the phenomenon may not be superfluous. 2. Solution of the Cauchy-Poisson Problem for Finite Depth . In his treatment of this case Lord Rayleigh used the integral form of solution. There are, however, great difficulties raised in this way on account of lack of convergence at the surface. I proceed to obtain a solution in the form of a series similar in type to the known serial solution of the problem for infinite depth.

scholarly journals Investigation of symmetry breaking in periodic gravity–capillary waves

2016 ◽  
Vol 811 ◽  
pp. 622-641 ◽  
Author(s):  
T. Gao ◽  
Z. Wang ◽  
J.-M. Vanden-Broeck
Keyword(s):  
Conformal Mapping ◽  
Symmetry Breaking ◽  
Incompressible Fluid ◽  
Numerical Method ◽  
Finite Depth ◽  
Capillary Waves ◽  
Fully Nonlinear ◽  
Infinite Depth ◽  
Breaking Points ◽  
Continuation Parameter

In this paper, fully nonlinear non-symmetric periodic gravity–capillary waves propagating at the surface of an inviscid and incompressible fluid are investigated. This problem was pioneered analytically by Zufiria (J. Fluid Mech., vol. 184, 1987c, pp. 183–206) and numerically by Shimizu & Shōji (Japan J. Ind. Appl. Maths, vol. 29 (2), 2012, pp. 331–353). We use a numerical method based on conformal mapping and series truncation to search for new solutions other than those shown in Zufiria (1987c) and Shimizu & Shōji (2012). It is found that, in the case of infinite-depth, non-symmetric waves with two to seven peaks within one wavelength exist and they all appear via symmetry-breaking bifurcations. Fully exploring these waves by changing the parameters yields the discovery of new types of non-symmetric solutions which form isolated branches without symmetry-breaking points. The existence of non-symmetric waves in water of finite depth is also confirmed, by using the value of the streamfunction at the bottom as the continuation parameter.

Reflexion of water waves by a permeable barrier

1979 ◽  
Vol 95 (1) ◽  
pp. 141-157 ◽  
Author(s):  
C. Macaskill
Keyword(s):  
Water Waves ◽  
Water Wave ◽  
Finite Depth ◽  
Infinite Depth ◽  
Permeable Barrier ◽  
Impermeable Barrier ◽  
Linearized Problem ◽  
Special Case ◽  
Thin Barrier ◽  
Standard Techniques

The linearized problem of water-wave reflexion by a thin barrier of arbitrary permeability is considered with the restriction that the flow be two-dimensional. The formulation includes the special case of transmission through one or more gaps in an otherwise impermeable barrier. The general problem is reduced to a set of integral equations using standard techniques. These equations are then solved using a special decomposition of the finite depth source potential which allows accurate solutions to be obtained economically. A representative range of solutions is obtained numerically for both finite and infinite depth problems.

scholarly journals Local uniqueness in boundary problems

1970 ◽  
Vol 17 (1) ◽  
pp. 23-36
Author(s):  
M. H. Martin
Keyword(s):  
Incompressible Fluid ◽  
Unit Circle ◽  
Finite Amplitude ◽  
Local Uniqueness ◽  
Boundary Problems ◽  
Infinite Depth ◽  
Image Position

The study of periodic, irrotational waves of finite amplitude in an incompressible fluid of infinite depth was reduced by Levi-Civita (1) to the determination of a functionregular analytic in the interior of the unit circle ρ = 1 and which satisfies the condition

scholarly journals Gravitational and Dynamical Instabilities of a Decelerating Plane-Parallel Slab of Finite Thickness

1988 ◽  
Vol 101 ◽  
pp. 509-512
Author(s):  
G. Mark Voit
Keyword(s):  
Star Formation ◽  
Interstellar Medium ◽  
Incompressible Fluid ◽  
Gravity Wave ◽  
Finite Thickness ◽  
Blast Waves ◽  
Plane Parallel ◽  
The Stability ◽  
Dynamical Instabilities ◽  
Symmetric And Antisymmetric Modes

AbstractIn order to explore how supernova blast waves might catalyze star formation, we investigate the stability of a slab of decelerating gas of finite thickness. We examine the early work in the field by Elmegreen and Lada and Elmegreen and Elmegreen and demonstrate that it is flawed. Contrary to their claims, blast waves can indeed accelerate the rate of star formation in the interstellar medium. Also, we demonstrate that in an incompressible fluid, the symmetric and antisymmetric modes in the case of zero acceleration transform continuously into Rayleigh-Taylor and gravity-wave modes as acceleration grows more important.

The instability theory of drumlin formation applied to Newtonian viscous ice of finite depth

2010 ◽  
Vol 466 (2121) ◽  
pp. 2673-2694 ◽  
Author(s):  
A. C. Fowler
Keyword(s):  
Transfer Functions ◽  
Three Dimensional ◽  
Analytic Form ◽  
Finite Depth ◽  
Pressure Gradients ◽  
Interfacial Instabilities ◽  
Infinite Depth ◽  
Ice Surface ◽  
Instability Theory ◽  
Ice Flows

The Hindmarsh instability theory of drumlin formation is applied to the study of interfacial instabilities, which may arise when ice flows viscously over deformable sediments. Here, the analytic form of this theory is extended to the case where the ice is Newtonian viscous and of finite depth, and where the basal till can be both sheared by the ice and squeezed by basal effective pressure gradients: previous authors assumed infinitely deep ice, based on the assumption that the developing waveforms had wavelength much less than ice depth. The previous infinite depth theory only allowed transverse instabilities to occur, and these have been associated with the formation of ribbed moraine; one of the purposes of extending the analysis to finite depth is to see whether three-dimensional instabilities, which might be associated with the formation of drumlins or mega-scale glacial lineations, can occur: we find that they do not. A second purpose is to calculate under what circumstances the infinite depth theory provides accurate prediction of bedform development in ice of finite depth d i . We find that this is the case if the waveforms have a wavelength less than approximately 1.2 d i . Finally, the finite depth theory allows us to compute, for the first time, the response of the ice surface to the developing unstable bedforms. We find that this response is rapid, and we give explicit recipes for the surface perturbation transfer functions in terms of the perturbations to the basal stress and the basal topography.

Exact large amplitude capillary waves on sheets of fluid

1976 ◽  
Vol 77 (2) ◽  
pp. 229-241 ◽  
Author(s):  
William Kinnersley
Keyword(s):  
Exact Solution ◽  
Large Amplitude ◽  
Elliptic Functions ◽  
Finite Depth ◽  
Capillary Waves ◽  
Infinite Depth ◽  
The Waves

We generalize Crapper's exact solution for capillary waves on fluid of infinite depth. We find two finite-depth solutions involving elliptic functions. We show they can also be interpreted as large amplitude symmetrical and antisymmetrical waves on a fluid sheet. Particularly interesting are the waves obtained from our solution in the limit when the fluid sheet is extremely thin.

scholarly journals Waves due to initial disturbances at the inertial surface in a stratified fluid of finite depth

2000 ◽  
Vol 24 (4) ◽  
pp. 265-276 ◽  
Author(s):  
Prity Ghosh ◽  
Uma Basu ◽  
B. N. Mandal
Keyword(s):  
Fourier Transform ◽  
Laplace Transform ◽  
Mathematical Analysis ◽  
Large Time ◽  
Finite Depth ◽  
Integral Theorem ◽  
Poisson Problem ◽  
Inertial Surface ◽  
Green’S Integral Theorem ◽  
Stratified Ocean

This paper is concerned with a Cauchy-Poisson problem in a weakly stratified ocean of uniform finite depth bounded above by an inertial surface (IS). The inertial surface is composed of a thin but uniform distribution of noninteracting materials. The techniques of Laplace transform in time and either Green's integral theorem or Fourier transform have been utilized in the mathematical analysis to obtain the form of the inertial surface in terms of an integral. The asymptotic behaviour of the inertial surface is obtained for large time and distance and displayed graphically. The effect of stratification is discussed.

scholarly journals The theory of ship waves

1926 ◽  
Vol 111 (759) ◽  
pp. 491-529 ◽  
Keyword(s):  
Finite Depth ◽  
Wave Disturbance ◽  
Present Communication ◽  
Ship Waves ◽  
Infinite Depth ◽  
The Subject ◽  
Considerable Detail

The theory of ship waves, when the sea is considered to be of infinite depth, has been the subject of many researches. When the sea is of finite depth the integrals involved are more complicated, but in this case also the theory has been worked out in considerable detail. The main object of the present communication is to add to the number of cases which have been solved, or, to be more precise, which have been exactly formulated, a certain series in which the depth is variable. Of subsidiary interest, but coming under the title of the paper, are some considerations relating to the wave disturbance when the depth is finite. These are dealt with briefly in section 5.

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