The effect of a fixed vertical barrier on surface waves in deep water

1947 ◽  
Vol 43 (3) ◽  
pp. 374-382 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Integral Equation ◽  
Surface Waves ◽  
Deep Water ◽  
Vertical Plane ◽  
Standing Waves ◽  
Simple Type ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Vertical Barrier ◽  
The Mean

In this paper the two-dimensional reflection of surface waves from a vertical barrier in deep water is studied theoretically.It can be shown that when the normal velocity is prescribed at each point of an infinite vertical plane extending from the surface, the motion on each side of the plane is completely determined, apart from a motion consisting of simple standing waves. In the cases considered here the normal velocity is prescribed on a part of the vertical plane and is taken to be unknown elsewhere. From the condition of continuity of the motion above and below the barrier an integral equation for the normal velocity can be derived, which is of a simple type, in the case of deep water. We begin by considering in detail the reflection from a fixed vertical barrier extending from depth a to some point above the mean surface.

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.

The transmission of surface waves through a gap in a vertical barrier

1972 ◽  
Vol 71 (2) ◽  
pp. 411-421 ◽  
Author(s):  
D. Porter
Keyword(s):  
Surface Waves ◽  
Velocity Potential ◽  
Hilbert Problem ◽  
Fluid Motion ◽  
Two Dimensional ◽  
Reduction Procedure ◽  
Vertical Barrier ◽  
Integral Equation Formulation ◽  
Gap Width ◽  
Riemann Hilbert Problem

AbstractThe two-dimensional configuration is considered of a fixed, semi-infinite, vertical barrier extending downwards from a fluid surface and having, at some depth, a gap of arbitrary width. A train of surface waves, incident on the barrier, is partly transmitted and partly reflected. The velocity potential of the resulting fluid motion is determined by a reduction procedure and also by an integral equation formulation. It is shown that the two methods lead to the same Riemann–Hilbert problem. Transmission and reflexion coefficients are calculated for several values of the ratio gap width/mean gap depth.

Lifting Surface Theory for the Problem of an Arbitrarily Yawed Sinusoidal Gust Incident on a Thin Aerofoil in Incompressible Flow

1970 ◽  
Vol 21 (2) ◽  
pp. 182-198 ◽  
Author(s):  
J. M. R. Graham
Keyword(s):  
Integral Equation ◽  
Kernel Function ◽  
Incompressible Flow ◽  
Velocity Component ◽  
Normal Velocity ◽  
Two Dimensional ◽  
Surface Theory ◽  
Equation Solution ◽  
Chebyshev Expansion ◽  
Dimensional Integral

SummaryThe solution to the problem of the loading generated on a two-dimensional thin aerofoil by an incompressible flow whose normal velocity component is of the general form exp [i(λx+/μy — ωt)] is calculated. The method used extends the two-dimensional integral equation solution for the induced vorticity by means of a Chebyshev expansion of part of the kernel function. Thin aerofoil approximations are made throughout, but no collocation procedure, as such, is required.

Surface waves on deep water in the presence of a submerged circular cylinder. I

1950 ◽  
Vol 46 (1) ◽  
pp. 141-152 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Circular Cylinder ◽  
Surface Waves ◽  
Deep Water ◽  
Small Amplitude ◽  
Linear Equations ◽  
Normal Velocity ◽  
System Of Linear Equations ◽  
Reflexion Coefficient ◽  
Method Of Solution ◽  
Submerged Cylinder

ABSTRACTA method is given for the calculation of the surface waves of small amplitude generated on deep water by a normal velocity distribution of period 2π/σ prescribed over a submerged circular cylinder. The method of solution involves a system of linear equations in an infinite number of unknowns; this system always possesses a solution. The unknowns may be obtained as power series in a parameter Ka, convergent for sufficiently small values of the parameter. When the parameter is not small, the equations can be solved by infinite determinants. It is shown that the reflexion coefficient of waves incident on a fixed circular cylinder vanishes, as was first shown by Dean. The pulsations of a submerged cylinder are discussed when the normal velocity is the same at all points of the cylinder at any given time.

Double boundary layers in standing interfacial waves

1976 ◽  
Vol 76 (4) ◽  
pp. 819-828 ◽  
Author(s):  
B. D. Dore
Keyword(s):  
Boundary Layer ◽  
Mass Transport ◽  
Layer Structure ◽  
Standing Waves ◽  
Boundary Layer Theory ◽  
Two Dimensional ◽  
Transport Velocity ◽  
Oscillatory Velocity ◽  
Mass Transport Velocity ◽  
The Mean

The double-boundary-layer theory of Stuart (1963, 1966) and Riley (1965, 1967) is employed to investigate the mass transport velocity due to two-dimensional standing waves in a system comprising two homogeneous fluids of different densities and viscosities. The most important double-boundary-layer structure occurs in the neighbourhood of the oscillating interface, and the possible existence of jet-like motions is envisaged at nodal positions, owing to the nature of the mean flows in the layers. In practice, the magnitude of the mass transport velocity can be a significant fraction of that of the primary, oscillatory velocity.

Water waves in a deep square basin

1995 ◽  
Vol 302 ◽  
pp. 65-90 ◽  
Author(s):  
Peter J. Bryant ◽  
Michael Stiassnie
Keyword(s):  
Deep Water ◽  
Water Waves ◽  
Three Dimensional ◽  
Standing Waves ◽  
Two Dimensional ◽  
Initial State ◽  
Dimensional Properties ◽  
Periodic Standing Waves ◽  
The Waves ◽  
Small Disturbances

The form and evolution of three-dimensional standing waves in deep water are calculated analytically from Zakharov's equation and computationally from the full nonlinear bounddary value problem. The water is contained in a basin with a square cross-cection, when three-dimensional properties to pairs of sides are the same. It is found that non-periodic standing waves commonly follow forms of cyclic recurrence over times. The two-dimensional Stokes type of periodic standing waves (dominated by the fundamental harmonic) are shown to be unstable to three dimensional disturbances, but over long times the waves return cyclically close to their initial state. In contrast, the three-dimensional Stokes type of periodic standing waves are found to be stabel to small disturbances. New two-dimensional periodic standing waves with amplitude maxima at other than the fundamental harmonic have been investigated recently (Bryant & Stiassnie 1994). The equivalent three-dimensional standing waves are described here. The new two-dimensional periodic standing waves, like the two-dimensional Stokes standing waves, are found to be unstable to three-dimensional disturbances, and to exhibit cyclic recurrence over long times. Only some of the new three-dimensional periodic standing waves are found to be stable to small disturbances.

scholarly journals Partial match queries in two-dimensional quadtrees: a probabilistic approach

2011 ◽  
Vol 43 (01) ◽  
pp. 178-194 ◽  
Author(s):  
Nicolas Curien ◽  
Adrien Joseph
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Markov Chains ◽  
Probabilistic Approach ◽  
Two Dimensional ◽  
Partial Match ◽  
The Mean ◽  
Mean Cost

We analyze the mean cost of the partial match queries in random two-dimensional quadtrees. The method is based on fragmentation theory. The convergence is guaranteed by a coupling argument of Markov chains, whereas the value of the limit is computed as the fixed point of an integral equation.

Surface waves on deep water in the presence of a submerged circular cylinder. II

1950 ◽  
Vol 46 (1) ◽  
pp. 153-158 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Circular Cylinder ◽  
Surface Waves ◽  
Deep Water ◽  
Trivial Solution ◽  
Velocity Potential ◽  
Angular Variable ◽  
Normal Velocity ◽  
Wave Potential ◽  
Uniqueness Theory

ABSTRACTIn the first part of this paper a method was given for constructing a wave potential when the normal velocity is a prescribed function of the angular variable on a submerged circular cylinder. It was shown that the method breaks down for values of the parameters Ka and Kf for which a certain infinite determinant vanishes. The vanishing of this determinant implies the existence of a non-trivial velocity potential, such that the normal velocity vanishes on the cylinder and both velocity components vanish at infinity. In this part of the paper it is shown that there can be no non-trivial solution of this kind; in other words the infinite determinant does not vanish. In the absence of a general uniqueness theory for surface waves it seems worth while to establish this particular result.

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