2. On the Theory of Waves. Part II

1845 ◽  
Vol 1 ◽  
pp. 319-321
Author(s):  
Kelland
Keyword(s):  
General Problem ◽  
Wave Motion ◽  
Constant Width ◽  
Square Root ◽  
Variable Depth ◽  
Constant Depth ◽  
Variation Of Parameters ◽  
End To End

The present memoir is a continuation of that which the author presented to the Society in April 1839. It is divided into four sections. In the first (which is marked Section 4, in continuation of the preceding memoir) the general problem of wave motion is treated of, and the equations to the surface are obtained by two different processes, giving results which agree with those obtained in the former memoir. In the second, the problem of wave motion in a canal of constant width and constant depth in the direction of the width, but of variable depth in the direction of motion, is treated by the method of the variation of parameters. The following are the approximate results:—1. The length of the wave diminishes directly as the depth diminishes. 2. That the velocity of transmission at any point is directly proportional to the square-root of the depth at that point. 3. That, in a channel uniformly and gradually shelving, the whole time of transmission of the wave from end to end, is exactly double what it would be if the depth were uniform; and, 4. That the elevation of the crest of the wave is inversely as the depth of the fluid.

Internal-inertia waves in a fluid of variable depth

1973 ◽  
Vol 73 (1) ◽  
pp. 205-213 ◽  
Author(s):  
W. D. McKee
Keyword(s):  
Exact Solutions ◽  
Surface Waves ◽  
Internal Waves ◽  
Vertical Velocity ◽  
General Problem ◽  
Three Dimensional ◽  
Variable Depth ◽  
Internal Inertia ◽  
Perturbation Pressure ◽  
Rotating Stratified Fluid

AbstractWaves in a rotating, stratified fluid of variable depth are considered. The perturbation pressure is used throughout as the dependent variable. This proves to have some advantages over the use of the vertical velocity. Some previous three-dimensional solutions for internal waves in a wedge are shown to be incorrect and the correct solutions presented. A WKB analysis is then performed for the general problem and the results compared with the exact solutions for a wedge. The WKB solution is also applied to long surface waves on a rotating ocean.

Standardization of Three Efficient Footbridges: Von Mises, Monocontentio and Bicontentio Prototypes.

2021 ◽  
Author(s):  
Mario Guisasola
Keyword(s):  
Boundary Conditions ◽  
Structural Characteristics ◽  
Structural Behavior ◽  
Bending Moments ◽  
Variable Depth ◽  
Constant Depth ◽  
Simply Supported ◽  
Vertical Walls ◽  
Basic Concepts ◽  
Von Mises

<p>The Von Mises, Monocontentio and Bicontentio footbridges are three parameterized metal bridge whose main structural characteristics are their variable depth depending on the applied stress and the embedding of abutments. Its use is considered suitable for symmetrical or asymmetrical topographies with slopes or vertical walls on one or both edges. The footbridges include spans spaced apart by 20 to 66 meters, and are between 2 to 4.5 meters wide.</p><p>Its design is based on five basic concepts: integration in the geometry of the environment; continuous search for simplicity; design based on a geometry that emanates from structural behavior; unitary and round forms; and long- lasting details.</p><p>The structural behavior of these prototypes has been compared with three types of constant-depth metal beams: the bridge simply supported, and the bridge embedded on one or both sides.</p><p>The embedding of abutments, and the adoption of a variation of depth adapted to the bending moments diagrams, allow for more efficient and elegant forms which are well-adapted to the boundary conditions.</p>

scholarly journals Surface and internal waves in a liquid of variable depth

1969 ◽  
Vol 1 (1) ◽  
pp. 29-46 ◽  
Author(s):  
D. G. Hurley ◽  
J. Imberger
Keyword(s):  
Free Surface ◽  
Gravity Wave ◽  
Internal Waves ◽  
Infinite Number ◽  
Variable Depth ◽  
Constant Depth ◽  
Stratified Liquid

Consider a stably stratified liquid, whose density varies exponentially with the vertical co-ordinate, that is bounded above by a free surface and below by a bed whose height depends on only one of the horizontal co-ordinates. Suppose that a gravity wave, that may be either a surface or an internal one, is travelling in a direction normal to the lines of constant depth. It is shown that if the frequency is below a certain value an infinite number of waves, all of the same frequency but having differing wave lengths, are generated and expressions for their amplitude are given in terms of the changes in depth which are assumed to be small.

Frictional two-layer exchange flow

2003 ◽  
Vol 474 ◽  
pp. 339-354 ◽  
Author(s):  
LILLIAN J. ZAREMBA ◽  
G. A. LAWRENCE ◽  
R. PIETERS
Keyword(s):  
Numerical Model ◽  
Analytical Solution ◽  
Viscous Flow ◽  
Constant Width ◽  
Flow Regimes ◽  
Exchange Flow ◽  
Hydraulic Control ◽  
Laboratory Measurements ◽  
Constant Depth ◽  
Exchange Flows

A numerical model is developed to study the effects of friction on the steady exchange flow that evolves when a barrier is removed from a constriction separating two reservoirs of slightly different densities. The model has excellent agreement with an analytical solution and laboratory measurements of exchange flows through channels of constant width and depth. The model reveals three viscous flow regimes for a convergent–divergent contraction of constant depth, and three additional viscous flow regimes when an offset sill is introduced. Each regime is characterized by a different set of internal hydraulic control locations. Examination of the predicted interface profiles reveals that it is not possible to distinguish between different flow regimes on the basis of these profiles alone.

Variable-depth streamer acquisition: Broadband data for imaging and inversion

Geophysics ◽  
2013 ◽  
Vol 78 (2) ◽  
pp. WA27-WA39 ◽  
Author(s):  
Robert Soubaras ◽  
Yves Lafet
Keyword(s):  
Signal To Noise Ratio ◽  
Real Data ◽  
Variable Depth ◽  
Constant Depth ◽  
Data Set ◽  
Low Frequencies ◽  
Optimal Receiver ◽  
Amplitude Versus Offset ◽  
Elastic Inversion ◽  
And Inversion

Conventional marine acquisition uses a streamer towed at a constant depth. The resulting receiver ghost notch gives the maximum recoverable frequency. To push this limit, the streamer must be towed at a quite shallow depth, but this compromises the low frequencies. Variable-depth streamer (VDS) acquisition is an acquisition technique aimed at achieving the best possible signal-to-noise ratio at low frequencies by towing the streamer very deeply, but by using a depth profile varying with offset in order not to limit the high-frequency bandwidth by notches as in conventional constant-depth streamer acquisition. The idea is to use notch diversity, each receiver having a different notch, so that the final result, combining different receivers, will have no notches. The key step to process VDS acquisitions is the receiver deghosting. We found that the optimal receiver deghosting, instead of being a preprocessing step, should be done postimaging, by using a dual-input, migration and mirror migration, and a new joint deconvolution algorithm that produces a 3D real amplitude deghosted output. This method can be applied poststack, the inputs being the migration and mirror migration images and the output being the deghosted image. Using a multichannel joint deconvolution, the inputs are the migrated and mirror migrated image gathers and the outputs are the prestack deghosted image gathers. This method preserves the amplitude-versus-offset behavior, as the deghosted output can be seen on synthetic examples to be equal to a reference computed by migrating the data modeled without any reflecting water surface. A real data set was used to illustrate this method, and another one was used to check the possibility of performing prestack elastic inversion on the deghosted gathers.

Adjustment under gravity in a rotating channel

1976 ◽  
Vol 77 (3) ◽  
pp. 603-621 ◽  
Author(s):  
A. E. Gill
Keyword(s):  
Constant Width ◽  
Constant Depth ◽  
Final State ◽  
Left Hand ◽  
Gravitational Forces ◽  
Initial Discontinuity ◽  
The Right ◽  
Long Channel ◽  
Rotating Channel ◽  
A Current

What transient motions occur as a fluid responds to gravitational forces in a rotating channel, and what equilibrium does the fluid adjust to? This problem is studied to illustrate how boundaries affect the process of adjustment to a geostrophic equilibrium. In particular, linear solutions are found for an infinitely long channel of constant width and constant depth when there is an initial discontinuity in the level of the free surface. The results are summarized in the figures, and can be described in terms of Poinearé waves and Kelvin waves. When the channel is wide compared with the Rossby radius, the final state involves a current of that width which follows the left-hand boundary (for Northern-Hemisphere rotation) to the position of the initial discontinuity, then crosses the channel and continues downstream along the right-hand wall.

scholarly journals On oblique waves forcing by a porous cylindrical wall

1996 ◽  
Vol 19 (2) ◽  
pp. 351-362 ◽  
Author(s):  
M. S. Faltas
Keyword(s):  
Wave Motion ◽  
Scattering Problem ◽  
Porous Wall ◽  
Hydrodynamic Pressure ◽  
Constant Depth ◽  
Cylindrical Wall ◽  
Oblique Waves ◽  
Wave Trapping ◽  
Impermeable Wall ◽  
Incident Waves

The problem of oblique cylindrical linearized wave motion is considered for a fluid of infinite depth or finite constant depth in the presence of an impermeable cylindrical wall and coaxial porous wall immersed vertically in the fluid. The motion is generated once by the oscillations, which are periodic in time and inθ-direction, of the impermeable wall and next by the porous wall. The velocity potentials have been found in closed forms in the different regions of the fluid and then calculating the hydrodynamic pressure distribution on the porous wall and the profile of the free surface. The scattering problem of oblique waves is then considered. A wave trapping phenomenon is investigated. Numerical results are given to the case of radial incident waves and the case when the angle of incident waves is30°to the radial direction.

On the forced surface waves due to a vertical wave-maker in the presence of surface tension

1971 ◽  
Vol 70 (2) ◽  
pp. 323-337 ◽  
Author(s):  
P. F. Rhodes-Robinson
Keyword(s):  
Surface Tension ◽  
Surface Waves ◽  
Wave Motion ◽  
Velocity Potential ◽  
Vertical Plane ◽  
Finite Depth ◽  
Cylindrical Wave ◽  
Constant Depth ◽  
Classical Wave ◽  
Wave Maker

AbstractThe classical wave-maker problem to determine the forced two-dimensional wave motion with outgoing surface waves at infinity generated by a harmonically oscillating vertical plane wave-maker immersed in water was solved long ago by Sir Thomas Havelock. In this paper we reinvestigate the problem, making allowance for the presence of surface tension which was excluded before, and obtain a solution of the boundary-value problem for the velocity potential which is made unique by prescribing the free surface slope at the wave-maker. The cases of both infinite and finite constant depth are treated, and it is essential to employ a method which is new to this problem since the theory of Havelock cannot be extended in the latter case of finite depth. The solution of the corresponding problem concerning the axisymmetric wave motion due to a vertical cylindrical wave-maker is deduced in conclusion.

The solitary wave in water of variable depth

1970 ◽  
Vol 42 (3) ◽  
pp. 639-656 ◽  
Author(s):  
R. Grimshaw
Keyword(s):  
Solitary Wave ◽  
Differential Equations ◽  
Asymptotic Solution ◽  
Wave Amplitude ◽  
Finite Amplitude ◽  
Long Waves ◽  
Two Dimensional ◽  
Variable Depth ◽  
Constant Depth

Equations are derived for two-dimensional long waves of small, but finite, amplitude in water of variable depth, analogous to those derived by Boussinesq for water of constant depth. When the depth is slowly varying compared to the length of the wave, an asymptotic solution of these equations is obtained which describes a slowly varying solitary wave; also differential equations for the slow variations of the parameters describing the solitary wave are derived, and solved in the case when the solitary wave evolves from a region of uniform depth. For small amplitudes it is found that the wave amplitude varies inversely as the depth.

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