scholarly journals The method of images in some problems of surface waves

1927 ◽  
Vol 115 (771) ◽  
pp. 268-280 ◽  
Keyword(s):  
Free Surface ◽  
Circular Cylinder ◽  
Surface Waves ◽  
Unit Length ◽  
Wave Length ◽  
Fluid Motion ◽  
Degree Of Approximation ◽  
Image Point ◽  
Horizontal Line ◽  
Image System

1. When a circular cylinder is submerged in a uniform stream, the surface elevation may be calculated, to a first approximation, by a method due originally to Lamb for this case, and later extended to bodies of more general form: the method consists in replacing the cylinder by the equivalent doublet at its centre and then finding the fluid motion due to this doublet. In discussing the problem some years ago, I remarked that if the solution so obtained were interpreted in terms of an image system of sources, we should then be able to proceed to further approximations by the method of successive images, taking images alternately in the surface of the submerged body and in the free surface of the stream. This is effected in the following paper for two-dimensional fluid motion, and the method is applied to the circular cylinder. It provides, theoretically at least, a process for obtaining any required degree of approximation but, of course, the expressions soon become very complicated. It is, however, of interest to examine some cases numerically so as to obtain some idea of the degree of approximation of the first stage. An expression is first obtained for the velocity potential of the fluid motion due to a doublet at a given depth below the surface of a stream, the doublet being of given magnitude with its axis in any direction. A transformation of this expression then gives a simple interpretation in terms of an image system. This system consists of a certain isolated doublet at the image point above the free surface, together with a line distribution of doublets on a horizontal line to the rear of this point; the moment per unit length of the line distribution is constant, but the direction of the axis rotates as we pass along the line, the period of a revolution being equal to the wave-length of surface waves for the velocity of the stream. The contribution of each part of the image system to the surface disturbance is indicated.

scholarly journals The resistance of a ship among waves

1937 ◽  
Vol 161 (906) ◽  
pp. 299-308 ◽  
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Fluid Velocity ◽  
Wave Length ◽  
Steady Motion ◽  
Degree Of Approximation ◽  
Point Of View ◽  
Similar Degree ◽  
Horizontal Line ◽  
Free Surface Waves

1—The wave resistance of a ship advancing in still water may be calculated under certain assumptions, which amount to supposing the forced wave motion to be small so that squares of the fluid velocity may be neglected; moreover, the ship is supposed to advance with constant velocity in a horizontal line. It does not appear to have been noticed that we may super­ pose on the solution so obtained free surface waves of small amplitude, and that the addition to the resistance may be calculated, to a similar degree of approximation, as the horizontal resultant of the additional fluid pressures due to the free surface waves; this additional resistance, which may be negative, depends upon the position of the ship among the free waves. Various calculations are now made from this point of view. We consider first transverse following waves moving at the same speed as the ship, and then a ship moving in the waves left by another ship in advance moving at the same speed; finally, we examine the more general case of a ship moving through free transverse waves of any wave-length. All the cases are discussed with reference to such experimental results as are available. 2—We treat the problem first as one of steady motion with the ship at rest in a uniform stream of velocity c in the negative direction of Ox ; we take the origin O in the undisturbed water surface, and Oz vertically upwards. The velocity potential is given by ϕ = cx + ϕ 1 , (1)

Trapping modes in the theory of surface waves

1951 ◽  
Vol 47 (2) ◽  
pp. 347-358 ◽  
Author(s):  
F. Ursell
Keyword(s):  
Free Surface ◽  
Circular Cylinder ◽  
Total Energy ◽  
Surface Waves ◽  
Radiation Condition ◽  
Finite Width ◽  
Energy Trapping ◽  
Sloping Beach ◽  
The Waves

ABSTRACTIt is shown that a mass of fluid bounded by fixed surfaces and by a free surface of infinite extent may be capable of vibrating under gravity in a mode (called a trapping mode) containing finite total energy. Trapping modes appear to be peculiar to the theory of surface waves; it is known that there are no trapping modes in the theory of sound. Two trapping modes are constructed: (1) a mode on a sloping beach in a semi-infinite canal of finite width, (2) a mode near a submerged circular cylinder in an infinite canal of finite width. The existence of trapping modes shows that in general a radiation condition for the waves at infinity is insufficient for uniqueness.

scholarly journals Capillary damping of inviscid surface waves in a circular cylinder

2009 ◽  
Vol 627 ◽  
pp. 323-340 ◽  
Author(s):  
R. KIDAMBI
Keyword(s):  
Free Surface ◽  
Circular Cylinder ◽  
Surface Waves ◽  
Contact Line ◽  
Simple Procedure ◽  
Edge Condition ◽  
Complex Frequency ◽  
Moving Contact Line ◽  
Moving Contact ◽  
Static Contact

We consider the effect of a wetting condition at the moving contact line on the frequency and damping of surface waves on an inviscid liquid in a circular cylinder. The velocity potential φ and the free surface elevation η are sought as complex eigenfunction expansions. The φ eigenvalues are the classical ones whereas the η eigenvalues are unknown and have to be computed so as to satisfy the wetting condition on the contact line and the other free surface conditions – these turn out to be complex in general. A projection of the latter conditions on to an appropriate basis leads to an eigenvalue problem, for the complex frequency Ω, which has to be solved iteratively with the wetting condition. The variation of Ω with liquid depth h, Bond number Bo, capillary coefficient λ and static contact angle θc0 is explored for the (1, 0),(2, 0),(0, 1),(3, 0) and (4, 0) modes. The damping vanishes for λ = 0 (pinned-end edge condition) and λ = ∞ (free-end edge condition) with a maximum in the interior while the frequency decreases with increasing λ, approaching limiting values at the endpoints. A comparison with the analytic results of Miles (J. Fluid Mech., vol. 222, 1991, p. 197) for the no-meniscus case and the experimental results of Cocciaro, Faetti, & Festa (J. Fluid Mech., vol. 246, 1993, p. 43), where a meniscus is present, is good. The study provides a simple procedure for calculating the inviscid capillary damping associated with the moving contact line in a circular cylinder of finite depth with meniscus effects also being considered.

Motion Characteristics of Floating Bodies

1976 ◽  
Vol 20 (04) ◽  
pp. 181-189
Author(s):  
Choung M. Lee
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Linear Theory ◽  
Cross Sections ◽  
Wave Length ◽  
Added Mass ◽  
Exciting Force ◽  
Hydrodynamic Coefficients ◽  
Floating Bodies ◽  
Motion Characteristics

The motion characteristics of floating bodies, in and below a free surface, that are subject to surface waves are examined. The effects of various degrees of approximation of hydrodynamic coefficients on the results of the motion computations are investigated on the basis of linear theory. The behavior of hydrodynamic coefficients such as added mass, damping, and wave-exciting force with respect to frequency of oscillation (or wave length) and depth of submergence is also investigated. Numerical results are obtained for three cylindrical bodies of circular, rectangular, and rhombic cross sections. The results reveal that fairly crude approximations can be applied to compute the oscillatory motion of submerged bodies.

Stokes flow about a sphere attached to a slender body

1974 ◽  
Vol 64 (4) ◽  
pp. 817-826 ◽  
Author(s):  
N. J. De Mestre ◽  
D. F. Katz
Keyword(s):  
Stokes Flow ◽  
Unit Length ◽  
Interactive Effect ◽  
Slender Body ◽  
Slip Condition ◽  
Image Point ◽  
Image System ◽  
Slender Body Theory ◽  
Spherical Head ◽  
Body Theory

Stokes flow is analysed for a combination body, consisting of a sphere attached to a slender body, translating along its axis in an infinite and otherwise un-disturbed fluid. The cross-section of the after-body, or tail, is circular; the radius, while not necessarily constant, is small compared with the radius of the spherical head. The tail is represented by a distribution of Stokeslets of strength per unit length F(z), located and directed along its axis. The interactive effect of head-tail attachment is manifested by the presence of image singularities located within the sphere. The image system for a single tail Stokeslet must be such that the no-slip condition is satisfied on the surface of the sphere. It is shown that this system consists of a Stokeslet, a Stokes doublet (stresslet only) and a source doublet located a t the image point. The strength F(z) is obtained by applying the no-slip condition to the combination body. The solution follows the lines of traditional slender-body theory, an expansion being performed in ascending powers of the reciprocal of the logarithm of the aspect ratio. The integral force parameters and F(z) are obtained to second order. The interactive effect is assessed, and the results are discussed in the context of a sedimenting micro-organism, such as a spermatozoon. The drag on the combination body is shown to be less by around 10% than the sum of the drags on an isolated sphere and tail. This drag, for a sperm-shaped body, is divided approximately equally between head and tail.

Fluid-Particle Motion During Rotary Sloshing

1964 ◽  
Vol 31 (1) ◽  
pp. 123-130 ◽  
Author(s):  
R. E. Hutton
Keyword(s):  
Angular Momentum ◽  
Free Surface ◽  
Rigid Body ◽  
Surface Waves ◽  
Particle Motion ◽  
Fluid Motion ◽  
The Body ◽  
Fluid Particle ◽  
Free Surface Waves ◽  
Theoretical Predictions

This paper presents the results of a theoretical and experimental investigation of fluid-particle motion in a partially filled cylindrical tank. The body of fluid is assumed to have a steady-state motion consisting of first nonsymmetric sloshing mode only (lowest J1 mode), which gives rise to a free-surface wave that rotates around the tank at the sloshing natural frequency. It was found that theory predicts a net transport motion of the fluid particles in the direction of the free-surface waves when nonlinear terms are retained in differential equations that describe the fluid-particle displacements. Experimental measurements of the particle motion are compared with theoretical predictions. Fluid angular momentum was computed using the theoretical fluid motion and compared with the angular momentum that the fluid would possess if the fluid moved as a rigid body at the same rate as the free-surface waves. It was found that an upper bound to the ratio of the transport angular momentum to the rigid-body angular momentum was equal to 0.8 (η/a)2, where η is the peak wave height of the free surface waves and a is the tank radius.

scholarly journals Experiments on generation of surface waves by an underwater moving bottom

2015 ◽  
Vol 471 (2178) ◽  
pp. 20150069 ◽  
Author(s):  
Timothée Jamin ◽  
Leonardo Gordillo ◽  
Gerardo Ruiz-Chavarría ◽  
Michael Berhanu ◽  
Eric Falcon
Keyword(s):  
Free Surface ◽  
Surface Waves ◽  
Surface Deformation ◽  
Fluid Layer ◽  
Fluid Velocity ◽  
Pass Filter ◽  
Low Pass Filter ◽  
Low Pass ◽  
Spatio Temporal ◽  
Experimental Findings

We report laboratory experiments on surface waves generated in a uniform fluid layer whose bottom undergoes an upward motion. Simultaneous measurements of the free-surface deformation and the fluid velocity field are focused on the role of the bottom kinematics (i.e. its spatio-temporal features) in wave generation. We observe that the fluid layer transfers bottom motion to the free surface as a temporal high-pass filter coupled with a spatial low-pass filter. Both filter effects are often neglected in tsunami warning systems, particularly in real-time forecast. Our results display good agreement with a prevailing linear theory without any parameter fitting. Based on our experimental findings, we provide a simple theoretical approach for modelling the rapid kinematics limit that is applicable even for initially non-flat bottoms: this may be a key step for more realistic varying bathymetry in tsunami scenarios.

scholarly journals The Electrostatic Energy of a Two-Dimensional System

1959 ◽  
Vol 42 ◽  
pp. 1-2
Author(s):  
LL. G. Chambers
Keyword(s):  
Complex Variable ◽  
Complex Potential ◽  
Unit Length ◽  
Electrostatic Energy ◽  
Dimensional System ◽  
Two Dimensional ◽  
Image System ◽  
Actual System ◽  
Image Domain ◽  
Two Dimensional System

The use of the complex variable z( = x + iy) and the complex potential W(= U + iV) for two-dimensional electrostatic systems is well known and the actual system in the (x, y) plane has an image system in the (U, V) plane. It does not seem to have been noticed previously that the electrostatic energy per unit length of the actual system is simply related to the area of the image domain in the (U, V) plane.

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