Surface and internal waves in a liquid of variable depth

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.

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Internal-inertia waves in a fluid of variable depth

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100047617 ◽

1973 ◽

Vol 73 (1) ◽

pp. 205-213 ◽

Cited By ~ 10

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.

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Internal waves in a viscous atmosphere

Journal of Fluid Mechanics ◽

10.1017/s0022112075003278 ◽

1975 ◽

Vol 72 (4) ◽

pp. 773-786 ◽

Cited By ~ 6

Author(s):

W. L. Chang ◽

T. N. Stevenson

Keyword(s):

Energy Source ◽

Incompressible Fluid ◽

Internal Waves ◽

Infinite Number ◽

Similarity Solution ◽

Small Amplitude ◽

Horizontal Plane ◽

Two Dimensional ◽

The Waves ◽

Isothermal Atmosphere

The way in which internal waves change in amplitude as they propagate through an incompressible fluid or an isothermal atmosphere is considered. A similarity solution for the small amplitude isolated viscous internal wave which is generated by a localized two-dimensional disturbance or energy source was given by Thomas & Stevenson (1972). It will be shown how summations or superpositions of this solution may be used to examine the behaviour of groups of internal waves. In particular the paper considers the waves produced by an infinite number of sources distributed in a horizontal plane such that they produce a sinusoidal velocity distribution. The results of this analysis lead to a new small perturbation solution of the linearized equations.

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

10.24904/footbridge2022.106 ◽

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

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.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.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.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.

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Free-surface flows with two stagnation points

Journal of Fluid Mechanics ◽

10.1017/s0022112096007975 ◽

1996 ◽

Vol 324 ◽

pp. 393-406 ◽

Cited By ~ 1

Author(s):

J.-M. Vanden-Broeck ◽

F. Dias

Keyword(s):

Free Surface ◽

Infinite Number ◽

Wave Breaking ◽

Free Surface Flows ◽

Number Of Solutions ◽

Surface Flows ◽

Stagnation Points ◽

Surface Profiles ◽

Countably Infinite

Symmetric suction flows are computed. The flows are free-surface flows with two stagnation points. The configuration is related to the modelling of wave breaking at the bow of a ship. It is shown that there is a countably infinite number of solutions and that the free-surface profiles are characterized by waves.

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Generation of internal waves in a two-layered stratified system with a free surface by coupled mass sources

Ocean Engineering ◽

10.1016/j.oceaneng.2021.110234 ◽

2022 ◽

Vol 243 ◽

pp. 110234

Author(s):

Dongyi Yan ◽

Dongming Liu ◽

Jijian Lian

Keyword(s):

Free Surface ◽

Internal Waves ◽

Mass Sources

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2. On the Theory of Waves. Part II

Proceedings of the Royal Society of Edinburgh ◽

10.1017/s0370164600039031 ◽

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.

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