Topic

# Standing wave

About: Standing wave is a research topic. Over the lifetime, 8476 publications have been published within this topic receiving 126227 citations. The topic is also known as: stationary wave.

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31 Oct 1964

TL;DR: In this article, measurements have been made of the time average and unsteady pressures acting on the roof and behind a series of rectangular cavities set in the roof of the 2 ft x 1 tr ft transonic tunnel.

Abstract: Measurements have been made of the time average and unsteady pressures acting on the roof and behind a series of rectangular cavities set in the roof of the 2 ft x 1tr ft transonic tunnel.
It was found that the unsteady pressures contain both random and periodic components. The random component predominates in the shallower cavities (length/depth ratio> 4) and is most intense near the rear wall. The periodioc component predominates in the deeper cavities (length/depth ratio < 4) and may form standing wave patterns. It is suggested that the periodic component is due to an acoustic resonance within the cavity excited by a phenomenon similar to th at causing edge-tones. It may be suppressed by fixing a small spoiler at the front of the cavity.

1,123 citations

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TL;DR: In this paper, a general theoretical treatment of a new class of long stationary waves with finite amplitude is presented, which differ in important respects from long waves of more familiar kinds, and their character is discussed on the basis of a comparison with solitary-wave and cnoidal-wave theories on customary lines.

Abstract: This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these waves, the density of the (incompressible) fluid varies only within a layer whose thickness h is much smaller than the total depth, and it is h rather than the total depth that must be considered as the fundamental scale against which wave amplitude and length are to be measured. Internal-wave motions supported by the oceanic thermocline appear to be the most promising field of practical application for the theory, although applications to atmospheric studies are also a possibility.The waves in question differ in important respects from long waves of more familiar kinds, and in § 1 their character is discussed on the basis of a comparison with solitary-wave and cnoidal-wave theories on customary lines, such as apply to internal waves in fluids of limited depth. A summary of some simple experiments is included at the end of § 1. Then, in § 2, the comparatively easy example of two-fluid systems is examined, again to illustrate principles and to prepare the way for the main analysis in § 3. This proceeds to a second stage of approximation in powers of wave amplitude, and its leading result is an equation (3·51) determining, for arbitrary specifications of the density distribution, the form of the streamlines in the layer of heterogeneous fluid. Periodic solutions of this equation are obtained, and their properties are discussed with regard to the interpretation of internal bores and wave-resistance phenomena. Solutions representing solitary waves are then obtained in the form
\[
f(x) = a\lambda^2/(x^2+\lambda^2),
\]
where xis the horizontal co-ordinate and where aΛ = O(h2). (The latter relation between wave amplitude and length scale contrasts with the customary one, aΛ2 = O(h3)). The main analysis is developed with particular reference to systems in which the heterogeneous layer lies on a rigid horizontal bottom below an infinite expanse of homogeneous fluid; but in § 4 ways are given to apply the results to various other systems, including ones in which the heterogeneous layer is uppermost and is bounded by a free surface. Finally, in §5, three specific examples are treated: the density variation with depth is taken, respectively, to have a discontinuous, an exponential and a ‘tanh’ profile.

979 citations

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Yale University

^{1}TL;DR: Interference of atomic de Broglie waves tunneling from a vertical array of macroscopically populated traps has been observed, closely related to the ac Josephson effect observed in superconducting electronic systems.

Abstract: Interference of atomic de Broglie waves tunneling from a vertical array of macroscopically populated traps has been observed. The traps were located in the antinodes of an optical standing wave and were loaded from a Bose-Einstein condensate. Tunneling was induced by acceleration due to gravity, and interference was observed as a train of falling pulses of atoms. In the limit of weak atomic interactions, the pulse frequency is determined by the gravitational potential energy difference between adjacent potential wells. The effect is closely related to the ac Josephson effect observed in superconducting electronic systems.

965 citations

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TL;DR: In this paper, it was shown that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller compared to that of a standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5.

Abstract: We prove, for the energy critical, focusing NLS, that for data whose energy is smaller than that of the standing wave, and whose homogeneous Sobolev norm H^1 is smaller than that of the standing wave and which is radial, we have global well-posedness and scattering in dimensions 3, 4 and 5. This is sharp since if the data is in the inhomogeneous Sobolev space H^1, of energy smaller than the standing wave but of larger homogeneous H^1 norm, we have blow-up in finite time. The result follows from a general method that we introduce into this type of critical problem. By concentration-compactness we produce a critical element, which modulo the symmetries of the equation is compact, has minimal energy among those which fail to have the conclusion of our theorem. In addition, we show that the dilation parameter in the symmetry, for this solution, can be taken strictly positive.We then establish a rigidity theorem that shows that no such compact, modulo symmetries, object can exist. It is only at this step that we use the radial hypothesis.The same analysis, in a simplified form, applies also to the defocusing case, giving a new proof of results of Bourgain and Tao.

945 citations

01 Jan 1964

TL;DR: In this paper, measurements have been made of the time average and unsteady pressures acting on the roof and behind a series of rectangular cavities set in the roof of the 2 ft x 1 tr ft transonic tunnel.

Abstract: Measurements have been made of the time average and unsteady pressures acting on the roof and behind a series of rectangular cavities set in the roof of the 2 ft x 1tr ft transonic tunnel.
It was found that the unsteady pressures contain both random and periodic components. The random component predominates in the shallower cavities (length/depth ratio> 4) and is most intense near the rear wall. The periodioc component predominates in the deeper cavities (length/depth ratio < 4) and may form standing wave patterns. It is suggested that the periodic component is due to an acoustic resonance within the cavity excited by a phenomenon similar to th at causing edge-tones. It may be suppressed by fixing a small spoiler at the front of the cavity.

882 citations