On the Secondary Bifurcation of Three Dimensional Standing Waves.

1985 ◽  
Author(s):  
Thomas J. Bridges
Keyword(s):  
Three Dimensional ◽  
Standing Waves ◽  
Secondary Bifurcation

Secondary bifurcation and change of type for three-dimensional standing waves in finite depth

1987 ◽  
Vol 179 ◽  
pp. 137-153 ◽  
Author(s):  
Thomas J. Bridges
Keyword(s):  
Periodic Solutions ◽  
Natural Frequencies ◽  
Rectangular Cross Section ◽  
Three Dimensional ◽  
Standing Waves ◽  
Finite Depth ◽  
Double Eigenvalue ◽  
Double Eigenvalues ◽  
Secondary Bifurcation ◽  
Change Of Type

The nonlinear periodic free oscillations of irrotational surface waves in a three-dimensional basin with a rectangular cross-section and finite depth are considered. A previous work by Verma & Keller (1962) has analysed the case when the linear natural frequencies are non-commensurate. For particular values of the parameters, however, strong internal resonance occurs (two natural frequencies are equal). Instead of the usual loss of stability and exchange of energy, it is found that the double eigenvalue generates a higher multiplicity of periodic solutions. Eight solution branches are found to be emitted by the double eigenvalues. It is also shown that perturbing the double eigenvalue results in a secondary bifurcation of periodic solutions. The direction of the branches for the multiple and secondary bifurcation changes with the depth. Finally it is shown that the formal solutions obtained are not uniformly valid and an additional expansion in the Boussinesq regime shows that the wave field changes type. One of the solutions in this regime is a field of three-dimensional cnoidal standing waves.

The two-sided lid-driven cavity: experiments on stationary and time-dependent flows

2002 ◽  
Vol 450 ◽  
pp. 67-95 ◽  
Author(s):  
CH. BLOHM ◽  
H. C. KUHLMANN
Keyword(s):  
Three Dimensional ◽  
Standing Waves ◽  
Reynolds Numbers ◽  
Time Dependent ◽  
Flow State ◽  
Incompressible Fluid Flow ◽  
Velocity Measurements ◽  
Driven Cavity ◽  
Rectangular Container ◽  
Hot Film

The incompressible fluid flow in a rectangular container driven by two facing sidewalls which move steadily in anti-parallel directions is investigated experimentally for Reynolds numbers up to 1200. The moving sidewalls are realized by two rotating cylinders of large radii tightly closing the cavity. The distance between the moving walls relative to the height of the cavity (aspect ratio) is Γ = 1.96. Laser-Doppler and hot-film techniques are employed to measure steady and time-dependent vortex flows. Beyond a first threshold robust, steady, three-dimensional cells bifurcate supercritically out of the basic flow state. Through a further instability the cellular flow becomes unstable to oscillations in the form of standing waves with the same wavelength as the underlying cellular flow. If both sidewalls move with the same velocity (symmetrical driving), the oscillatory instability is found to be tricritical. The dependence on two sidewall Reynolds numbers of the ranges of existence of steady and oscillatory cellular flows is explored. Flow symmetries and quantitative velocity measurements are presented for representative cases.

Three-dimensional standing waves on an obliquely flowing film

1988 ◽  
Vol 28 (4) ◽  
pp. 618-624
Author(s):  
S. V. Alekseenko ◽  
S. I. Shtork
Keyword(s):  
Three Dimensional ◽  
Standing Waves

scholarly journals Two-Layer Tidal Circulation in a Frictional, Rotating Basin

2010 ◽  
Vol 40 (6) ◽  
pp. 1390-1404 ◽  
Author(s):  
Clinton D. Winant
Keyword(s):  
Three Dimensional ◽  
Standing Waves ◽  
Internal Tides ◽  
Surface Pattern ◽  
Layer Model ◽  
Interface Deformation ◽  
Tidal Circulation ◽  
Density Anomaly ◽  
Two Layer Model

Abstract The three-dimensional tidal circulation in an elongated basin of arbitrary depth is described with a coupled barotropic and baroclinic two-layer model on the f plane. As long as friction is not dominant, near-standing waves are present on the interface as well as on the surface. The surface pattern is principally determined by the product of the tidal barotropic wavenumber by the basin length. The interface deformation is determined by a baroclinic equivalent, usually a much larger number. As a result, the shape of the interface is characterized by horizontally smaller features than the surface. If the product of the tidal baroclinic wavenumber by the basin width is greater than one, both lateral and axial modes can be excited at the interface. If these modes are near resonant, large internal tides can be forced directly by the co-oscillating surface tide at the basin entrance. The amplitude and phase of the baroclinic component are sensitive functions of the density anomaly and the interface depth. As a result, the phase and amplitude of the interface vary by large amounts with comparatively small changes in those parameters. The model behavior is qualitatively consistent with observations in fjords and straits.

Three-Dimensional Containment of Charged Particles by Orthogonal Standing Waves

1963 ◽  
Vol 131 (2) ◽  
pp. 495-500 ◽  
Author(s):  
A. R. Shapiro ◽  
W. K. R. Watson
Keyword(s):  
Three Dimensional ◽  
Charged Particles ◽  
Standing Waves

Two- and three-dimensional standing waves in a reaction-diffusion system

2012 ◽  
Vol 86 (4) ◽  
Author(s):  
Tamás Bánsági ◽  
Vladimir K. Vanag ◽  
Irving R. Epstein
Keyword(s):  
Three Dimensional ◽  
Standing Waves ◽  
Reaction Diffusion ◽  
Reaction Diffusion System ◽  
Diffusion System

Parameters for polarization gradients in three-dimensional electromagnetic standing waves

1997 ◽  
Vol 56 (5) ◽  
pp. 4012-4022 ◽  
Author(s):  
S. A. Hopkins ◽  
A. V. Durrant
Keyword(s):  
Three Dimensional ◽  
Standing Waves

Separation of Traveling and Standing Waves in a Rigid-Walled Circular Duct Containing an Intermediate Impedance Discontinuity

2017 ◽  
Vol 139 (6) ◽  
Author(s):  
Yongxiong Xiao ◽  
Antoine Blanchard ◽  
Yao Zhang ◽  
Huancai Lu ◽  
D. Michael McFarland ◽  
...  
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Three Dimensional ◽  
Standing Waves ◽  
Side Branch ◽  
Element Analysis ◽  
Circular Duct ◽  
Dimensional Finite Element ◽  
Three Dimensional Finite Element ◽  
Impedance Discontinuity

In this paper, we study the phenomenon of separation of traveling and standing waves in a one-dimensional rigid-walled circular duct. The underlying mechanism for separation, mode complexity, is linear and introduced here by a damped side branch representing an impedance discontinuity. The left end of the duct is driven at a single frequency by a harmonic acoustic source, and the right end is a rigid termination. The position and impedance of the side branch are independent parameters in the analysis. Sufficient conditions for acoustic wave separation in the duct are derived analytically and employed in a three-dimensional finite element analysis to verify the theoretical result. A physical experiment, consisting of a circular duct with a damped side branch, was constructed based on analytical predictions, the physical parameters were measured or identified, and its performance was documented. These experimental parameters were employed in a second three-dimensional finite element analysis to obtain a direct comparison with experimental results. The comparison reveals the extent to which higher-order (unmodeled) effects degrade the separation phenomenon. It is demonstrated that an intermediate damped side branch used as a nonresonant device can be predictively designed to achieve nearly ideal separation of traveling and standing waves in a rigid-walled circular duct in order to direct and control acoustic energy transmission through the duct system.

An experimental study of standing waves

1953 ◽  
Vol 218 (1132) ◽  
pp. 44-59 ◽  
Keyword(s):  
Small Amplitude ◽  
Wave Length ◽  
Three Dimensional ◽  
Standing Waves ◽  
Close Approximation ◽  
Frequency Curve ◽  
Free Standing ◽  
Experimental Conditions ◽  
The Moment ◽  
Good Agreement

The experiments here described were designed to test experimentally some conclusions about free standing waves recently reached analytically by Penney & Price. A close approximation to free oscillations was produced in a tank by wave makers operating with small amplitude and at frequencies where great amplification occurred, owing to resonance. The amplitude-frequency curve proved to consist of two non-intersecting branches, a result which can be explained theoretically. A striking prediction made by Penney & Price was that when the height of the crests of standing waves reaches about 0·15 wave-length they will become pointed, in the form of a 90° ridge. Higher waves were expected to be unstable because the downward acceleration of the free surface near the crest would exceed that of gravity. The experimental conditions necessary for producing a crest in the form of an angled ridge were found and the wave photographed in this condition. Good agreement was found with the calculated form of the profile of the highest wave, which had an angle very near to 90°. The predicted instability for two-dimensional waves was found to begin at the moment the crest became a sharp ridge. It rapidly assumed a three-dimensional character which was revealed by two photographic techniques. Even when the amplitude of oscillation of the wave makers was only 0·85°, violent types of instability developed which produced effects that are here recorded.

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