Diffraction of water waves by a submerged vertical plate

1970 ◽  
Vol 40 (3) ◽  
pp. 433-451 ◽  
Author(s):  
D. V. Evans
Keyword(s):  
Free Surface ◽  
Boundary Value Problem ◽  
Incident Wave ◽  
Water Waves ◽  
Vertical Plate ◽  
Velocity Potential ◽  
Wave Train ◽  
Plane Waves ◽  
Two Dimensional ◽  
Special Case

A thin vertical plate makes small, simple harmonic rolling oscillations beneath the surface of an incompressible, irrotational liquid. The plate is assumed to be so wide that the resulting equations may be regarded as two-dimensional. In addition, a train of plane waves of frequency equal to the frequency of oscillation of the plate, is normally incident on the plate. The resulting linearized boundary-value problem is solved in closed form for the velocity potential everywhere in the fluid and on the plate. Expressions are derived for the first- and second-order forces and moments on the plate, and for the wave amplitudes at a large distance either side of the plate. Numerical results are obtained for the case of the plate held fixed in an incident wave-train. It is shown how these results, in the special case when the plate intersects the free surface, agree, with one exception, with results obtained by Ursell (1947) and Haskind (1959) for this problem.

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’.

On the subharmonic instabilities of steady three-dimensional deep water waves

1994 ◽  
Vol 262 ◽  
pp. 265-291 ◽  
Author(s):  
Mansour Ioualalen ◽  
Christian Kharif
Keyword(s):  
Gravity Waves ◽  
Deep Water ◽  
Water Waves ◽  
Transverse Direction ◽  
Plane Waves ◽  
Three Dimensional ◽  
Numerical Procedure ◽  
Two Dimensional ◽  
Wave Steepness ◽  
Oblique Angle

A numerical procedure has been developed to study the linear stability of nonlinear three-dimensional progressive gravity waves on deep water. The three-dimensional patterns considered herein are short-crested waves which may be produced by two progressive plane waves propagating at an oblique angle, γ, to each other. It is shown that for moderate wave steepness the dominant resonances are sideband-type instabilities in the direction of propagation and, depending on the value of γ, also in the transverse direction. It is also shown that three-dimensional progressive gravity waves are less unstable than two-dimensional progressive gravity waves.

scholarly journals A non-local formulation of rotational water waves

2011 ◽  
Vol 689 ◽  
pp. 129-148 ◽  
Author(s):  
A. C. L. Ashton ◽  
A. S. Fokas
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Velocity Potential ◽  
Vries Equation ◽  
Travelling Waves ◽  
Explicit Dependence ◽  
Local Equation ◽  
Constant Vorticity ◽  
Small Parameters ◽  
Non Local

AbstractThe classical equations of irrotational water waves have recently been reformulated as a system of two equations, one of which is an explicit non-local equation for the wave height and for the velocity potential evaluated on the free surface. Here, in the two-dimensional case: (a) we generalize the relevant formulation to the case of constant vorticity, as well as to the case where the free surface is described by a multivalued function; (b) in the case of travelling waves we derive an upper bound for the free surface; (c) in the case of constant vorticity we construct a sequence of nearly Hamiltonian systems which provide an approximation in the asymptotic limit of certain physical small parameters. In particular, the explicit dependence of the vorticity on the coefficients of the Korteweg–de Vries equation is clarified.

The radiation and scattering of surface waves by vertical barriers

1974 ◽  
Vol 63 (4) ◽  
pp. 625-634 ◽  
Author(s):  
D. Porter
Keyword(s):  
Boundary Value Problem ◽  
Surface Waves ◽  
Incident Wave ◽  
Small Amplitude ◽  
Fluid Motion ◽  
Vertical Plane ◽  
Wave Radiation ◽  
Two Dimensional ◽  
Radiation Problem ◽  
Vertical Barriers

A train of small-amplitude surface waves is incident normally on an arbitrary arrangement of thin barriers lying in a vertical plane in deep water. Each barrier is allowed to make small rolling or swaying oscillations of the same frequency as that of the incident wave. The boundary-value problem for the consequent fluid motion, assumed two-dimensional, is solved exactly using a technique which enables the amplitudes of the scattered waves far from the barriers to be readily determined. Reference is made to the associated wave radiation problem and to the calculation of forces and moments on the barriers.

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)).

The transmission of surface waves under surface obstacles

1961 ◽  
Vol 57 (3) ◽  
pp. 638-668 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Transmission Coefficient ◽  
Cross Section ◽  
Water Waves ◽  
Wave Train ◽  
Circular Cross Section ◽  
Great Distance ◽  
The Body ◽  
Radius Of Curvature

ABSTRACTA train of surface waves (water waves under gravity) is normally incident on a cylinder with horizontal generators fixed near the free surface, and is partially transmitted and partially reflected. At a great distance behind the cylinder the wave motion tends to a regular wave train travelling towards infinity; the ratio of its amplitude to the amplitude of the incident wave is the transmission coefficient . The transmission coefficient is studied when the wavelength is short compared to the dimensions of the body; physically (though not for engineering applications) this is the most interesting range of wavelengths, which corresponds to the range of shadow formation and ray propagation in optics and acoustics. The waves are then confined to a thin layer near the free surface, and the transmission under a partially immersed obstacle is then small. In the calculation the boundary condition at the free surface is linearized, viscosity is neglected, and the motion is assumed to be irrotational.At present the transmission coefficient is known only for a few configurations, all of them relating to infinitely thin plane barriers. A method is now given which is applicable to cylinders of finite cross-section and which is worked out in detail for a half-immersed cylinder of circular cross-section. The solution of the problem is made to depend on the solution of an integral equation which is solved by iteration. Only the first two terms can be obtained with any accuracy, and it appears at first that this is not sufficient to give the leading term in the transmission coefficient at short wavelengths; this difficulty is characteristic of transmission problems. By various mathematical devices which throw light on the mechanism of wave transmission, it is, nevertheless, found possible to prove that the transmission coefficient for waves of short wavelength λ and period 2π/ω incident on a half-immersed circular cylinder of radius a is asymptotically given bywhen N = 2πα/λ = ω2α/g is large. Earlier evidence had pointed towards an exponential law. It is suggested that transmission coefficients of order N−4 are typical for obstacles having vertical tangents and finite non-zero radius of curvature at the points where they meet the horizontal mean free surface. For obstacles having both front and rear face plane vertical to a depth a, is probably of order e−2N approximately; if only one of the two faces is plane vertical, is probably of order e−N approximately. Thus is seen to depend critically on the details of the cross-section.

Blockage with a Free Surface

1976 ◽  
Vol 20 (04) ◽  
pp. 199-203
Author(s):  
J. N. Newman
Keyword(s):  
Free Surface ◽  
Velocity Potential ◽  
Finite Depth ◽  
Wave Resistance ◽  
Steady Flows ◽  
Two Dimensional ◽  
Present Context ◽  
Two Dimensional Flow ◽  
Field Radiation ◽  
Radiation Conditions

The occurrence of blockage, or a jump in the velocity potential between the upstream and downstream infinities, is well known for steady two-dimensional flow past a body in a rigid channel. This paper considers the analogous situation where there is a free surface, as in the wave resistance problem for submerged two-dimensional bodies in a fluid of finite depth. It is shown that blockage occurs in spite of the free surface, taking values which depend not only on the dipole moment but also upon the Froude number based on depth. The occurrence of blockage, in the present context, has a bearing primarily upon the correct formulation of far-field radiation conditions for steady flows with finite depth.

Calculation of the Streamlines About a Ship Assuming a Linearized Free-Surface Boundary Condition

1973 ◽  
Vol 17 (03) ◽  
pp. 140-146
Author(s):  
Bruce H. Adee
Keyword(s):  
Boundary Condition ◽  
Fluid Flow ◽  
Free Surface ◽  
Boundary Value Problem ◽  
Velocity Potential ◽  
Potential Surface ◽  
Inviscid Fluid ◽  
Surface Boundary ◽  
Free Surface Boundary Condition ◽  
Surface Boundary Condition

The problem of computing the inviscid-fluid flow about a ship hull is investigated. A boundary- value problem, including a linearized free-surface boundary condition, is posed for the velocity potential. Singularities distributed over the hull surface are used to determine this potential. Surface streamlines are computed by numerically integrating a set of differential equations along the hull surface. A sample calculation for a Series 60, block coefficient 0.60 hull at a Froude number of 0.259 is included.

scholarly journals The Froude number for solitary water waves with vorticity

2015 ◽  
Vol 768 ◽  
pp. 91-112 ◽  
Author(s):  
Miles H. Wheeler
Keyword(s):  
Free Surface ◽  
Shear Flow ◽  
Froude Number ◽  
Simple Proof ◽  
Upper Bound ◽  
Water Waves ◽  
Wave Speed ◽  
Critical Value ◽  
Arbitrary Distribution ◽  
Two Dimensional

We consider two-dimensional solitary water waves on a shear flow with an arbitrary distribution of vorticity. Assuming that the horizontal velocity in the fluid never exceeds the wave speed and that the free surface lies everywhere above its asymptotic level, we give a very simple proof that a suitably defined Froude number $F$ must be strictly greater than the critical value $F=1$. We also prove a related upper bound on $F$, and hence on the amplitude, under more restrictive assumptions on the vorticity.

scholarly journals Hamiltonian long–wave expansions for water waves over a rough bottom

2005 ◽  
Vol 461 (2055) ◽  
pp. 839-873 ◽  
Author(s):  
Walter Craig ◽  
Philippe Guyenne ◽  
David P. Nicholls ◽  
Catherine Sulem
Keyword(s):  
Free Surface ◽  
Water Waves ◽  
Kdv Equation ◽  
Bottom Topography ◽  
Multiple Scale ◽  
Three Dimensions ◽  
Length Scale ◽  
Two Dimensional ◽  
Long Wave ◽  
Starting Point

This paper is a study of the problem of nonlinear wave motion of the free surface of a body of fluid with a periodically varying bottom. The object is to describe the character of wave propagation in a long–wave asymptotic regime, extending the results of R. Rosales & G. Papanicolaou (1983 Stud. Appl. Math. 68 , 89–102) on periodic bottoms for two–dimensional flows.We take the point of view of perturbation of a Hamiltonian system dependent on a small scaling parameter, with the starting point being Zakharov's Hamiltonian (V. E. Zakharov 1968 J. Appl. Mech. Tech. Phys. 9, 1990–1994) for the Euler equations for water waves. We consider bottom topography which is periodic in horizontal variables on a short length–scale, with the amplitude of variation being of the same order as the fluid depth. The bottom may also exhibit slow variations at the same length–scale as, or longer than, the order of the wavelength of the surface waves. We do not take up the question of random bottom variations, a topic which is considered in Rosales & Papanicolaou (1983). In the two–dimensional case of waves in a channel, we give an alternate derivation of the effective Korteweg–de Vries (KdV) equation that is obtained in Rosales & Papanicolaou (1983). In addition, we obtain effective Boussinesq equations that describe the motion of bidirectional long waves, in cases in which the bottom possesses both short and long–scale variations. In certain cases we also obtain unidirectional equations that are similar to the KdV equation. In three dimensions we obtain effective three–dimensional long–wave equations in a Boussinesq scaling regime, and again in certain cases an effective Kadomtsev–Petviashvili (KP) system in the appropriate unidirectional limit. The computations for these results are performed in the framework of an asymptotic analysis of multiple–scale operators. In the present case this involves the Dirichlet–Neumann operator for the fluid domain which takes into account the variations in bottom topography as well as the deformations of the free surface from equilibrium.

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