VIBRATIONS ON POWER LINES IN A STEADY WIND: VI. FORCED VIBRATIONS OF UNIFORM STRINGS AND OF STRINGS WITH STRENGTHENED ENDS UNDER THE ACTION OF UNIFORM PERIODIC FORCES

A study of the complete equation expressing the action of a driving force, periodic in time but constant throughout the length of the string and opposed by a damping force proportional to the velocity, leads to formulae suitable for the practical calculation of the shape of standing waves that are produced by a plane wave of sound or by a steady wind. At resonance the amplitude at the midpoint of a uniform string set into a plane wave of sound is proportional to the diameter, to the square root of the intensity of the wave (G erg per sq. cm. per sec.), and inversely proportional to the order of the overtone and to the square root of the frequency. Damping causes the lag between force and motion to differ from point to point, particularly near the nodes, so that even at resonance the wave pattern is not rigorously stationary. On the average, the lag increases from the value zero, obtained when the ratio v/v0 between applied frequency and fundamental frequency is zero, to ±π/2 when v/v0 = 1, increases again from −π/2 through 0 at v/v0 = 2, to π/2 at v/v0 = 3, and so on.

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VIBRATIONS ON POWER LINES IN A STEADY WIND: V. RESONANCE OF STRINGS WITH STRENGTHENED ENDS

Canadian Journal of Research ◽

10.1139/cjr38a-022 ◽

1938 ◽

Vol 16a (12) ◽

pp. 215-225

Author(s):

R. Ruedy

Keyword(s):

Total Mass ◽

Linear Density ◽

Unit Length ◽

Standing Waves ◽

Power Lines ◽

Resonance Frequencies ◽

The Law

The resonance frequencies, and in particular all the overtones of a string along which the linear density varies according to the law ρ(1 + λx/L)m, are slightly higher than the frequencies of a uniform string of the same total mass when the ratio between the mass of an element at the end and a corresponding element at the centre is varied between 1 and 25. In order to bring a string with strengthened ends into resonance it is necessary not only that the force acting on unit length of the string be of the same frequency as one of the resonance frequencies, and that its strength varies along the string in proportion to the amplitudes of the corresponding standing waves, but it must also be proportional to the mass of each element. It is therefore more difficult to produce true resonance in a string with strengthened ends than in a uniform string.

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Plane wave, pattern subtraction, range compensation

IEEE Transactions on Antennas and Propagation ◽

10.1109/8.982468 ◽

2001 ◽

Vol 49 (12) ◽

pp. 1843-1851 ◽

Cited By ~ 32

Author(s):

D.A. Leatherwood ◽

E.B. Joy

Keyword(s):

Plane Wave ◽

Wave Pattern

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Imaging systems based on generalized lenses

Computer Optics ◽

10.18287/2412-6179-2019-43-5-789-795 ◽

2019 ◽

Vol 43 (5) ◽

pp. 789-795

Author(s):

V.S. Vasilev ◽

R.V. Skidanov ◽

S.V. Ganchevskaya

Keyword(s):

Plane Wave ◽

Incident Wave ◽

Small Deviation ◽

Imaging Systems ◽

Optical Systems ◽

High Quality ◽

Point Mapping ◽

High Quality Image ◽

Quality Image ◽

Point To Point

We consider the possibility of using generalized lenses in imaging optical systems. We propose using two generalized lenses in a system similar to a Fourier-correlator to obtain a point-to-point mapping, where the second lens is set so that a plane wave is also formed at the output of the system when the incident wave is plane. We show that for generalized lenses with a small deviation of their degree from 2, such a system forms a fairly high-quality image. We use a standard radial photographic chart to test the system, which allows us to determine its key optical characteristics.

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Theory and observations of waves on hollow-core vortices

Journal of Fluid Mechanics ◽

10.1017/s0022112080000730 ◽

1980 ◽

Vol 99 (3) ◽

pp. 495-511 ◽

Cited By ~ 16

Author(s):

Jakob J. Keller ◽

M. P. Escudier

Keyword(s):

Vortex Tube ◽

Capillary Wave ◽

Standing Waves ◽

Wave Pattern ◽

Interesting Case ◽

Hollow Core ◽

Core Size ◽

Viscous Effects ◽

Wave Problem ◽

The Core

A linear non-homogeneous analysis is presented for the standing waves produced on the hollow core of an irrotational vortex by an arbitrary obstacle on the wall of the tube containing the vortex. The group-velocity criterion based upon Kelvin's corresponding dispersion relation predicts whether a certain asymptotic wave pattern appears upstream or downstream of the obstacle. The analysis leads to amplitude singularities for the standing waves at certain critical radii of the core. The particularly interesting case of a counter-helix for which the wave energy is propagating upstream appears for a first-mode angular disturbance. For this situation it seems to be possible that the helix ends in a hydraulic jump and is continued by a counter-helix downstream, as the core size gradually diminishes due to the deceleration of the flow caused by viscous effects (not included in the analysis). The capillary-wave pattern produced by surface tension is also considered. A brief outline for the analogous wave problem is given for the case where the fluid rotates like a rigid body.Photographic observations of hollow-core vortices in water flow are presented which confirm the qualitative predictions of the analysis, both for the response to an axisymmetric area contraction and also to a 90° bend at the downstream end of the vortex tube.

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Sum of squares in discrete physical spaces

Revista Brasileira de Ensino de Física ◽

10.1590/s1806-11172008000400002 ◽

2008 ◽

Vol 30 (4) ◽

pp. 4302.1-4302.5

Author(s):

R. De Luca

Keyword(s):

Number Theory ◽

Radiation Spectrum ◽

Three Dimensional ◽

Standing Waves ◽

General Criterion ◽

Square Root ◽

Introductory Courses ◽

Physical Spaces ◽

Physical Quantities ◽

The Continuum

Discretization of three-dimensional physical spaces can induce, on observable physical quantities, effects which are not present in the continuum. Consider, as an example, the problem of the radiation spectrum of a blackbody, studied in introductory courses in quantum mechanics. One sees that the Rayleigh assumption of continuous and uniform frequency distribution of standing waves inside a cubic cavity with perfectly reflecting inner walls can be validated by a heuristic type of reasoning. However, by means of number theory, one sees that there might exist frequencies for which it is not possible to have standing waves inside the cavity. Nevertheless, within the same context, one can argue that a more general criterion can be adopted to validate the hypothesis of continuity of the observables which are expressed as the square root of the sum of three integers of a threedimensional space On x n y n z.

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VIBRATIONS OF POWER LINES IN A STEADY WIND: III. THE FREE VIBRATIONS OF A HEAVY STRING

Canadian Journal of Research ◽

10.1139/cjr36a-002 ◽

1936 ◽

Vol 14a (1) ◽

pp. 16-24

Author(s):

R. Ruedy

Keyword(s):

Natural Frequency ◽

Finite Length ◽

Standing Waves ◽

Free Vibrations ◽

Power Lines ◽

Middle Portion ◽

Zero Values ◽

Hermitian Polynomials ◽

The Waves

When a long heavy wire is fastened to two supports under such a tension that it is nearly straight, the span behaves in the same manner as a stretched string, but when there is appreciable sag, standing waves form mainly in the middle portion; apart from damping, their shape is described by Hermitian polynomials, which in the case of high overtones resemble sine waves. The change in the tension along the line causes attenuation of the waves. By keeping the ratio [Formula: see text] less than about unity the span is rendered vibration-proof, at least near the ends. The natural frequency of strings of finite length depends on the zero values of the Hermitian polynomials.

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Acoustic radiation force and motion of a free cylinder in a viscous fluid with a boundary defined by a plane wave incident at an arbitrary angle

Journal of Applied Physics ◽

10.1063/5.0005866 ◽

2020 ◽

Vol 128 (4) ◽

pp. 044902

Author(s):

Yupei Qiao ◽

Xiaowei Zhang ◽

Menyang Gong ◽

Haibin Wang ◽

Xiaozhou Liu

Keyword(s):

Viscous Fluid ◽

Plane Wave ◽

Acoustic Radiation ◽

Radiation Force ◽

Acoustic Radiation Force ◽

Wave Incident ◽

Arbitrary Angle ◽

Force And Motion ◽

Plane Wave Incident

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Improving the reliability of combined overhead and cable power lines by organizing smart autoreclosing

E3S Web of Conferences ◽

10.1051/e3sconf/201913901062 ◽

2019 ◽

Vol 139 ◽

pp. 01062

Author(s):

Pavel Pelevin ◽

Anton Loskutov ◽

Timur Sharafeev

Keyword(s):

Power System ◽

Pattern Classification ◽

Traveling Waves ◽

Traveling Wave ◽

Wave Pattern ◽

Power Lines ◽

Power System Protection ◽

System Protection ◽

Special Procedure

The application of combined overhead and cable power lines (OCPLs) of 110 kV voltage rate and higher call for new features for power system protection and automation devices. One of the most urgent is the problem of autoreclosing (AR) of OCPL. One or another solution to this problem significantly affects the reliability and efficiency of the OCPL operating. In many countries, both non-selective AR and selective AR are used. A method of selective AR of OCPL based on traveling waves is proposed, which increases the reliability of OCPL protection and automation. The method uses a special procedure of traveling wave pattern classification for faulted section identification.

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