定常波、定在波

関: standing wave

WordNet

an undulating curve (同)undulation
a persistent and widespread unusual weather condition (especially of unusual temperatures); "a heat wave"
set waves in; "she asked the hairdresser to wave her hair"
a movement like that of a sudden occurrence or increase in a specified phenomenon; "a wave of settlers"; "troops advancing in waves"
one of a series of ridges that moves across the surface of a liquid (especially across a large body of water) (同)moving ridge
(physics) a movement up and down or back and forth (同)undulation
the act of signaling by a movement of the hand (同)waving, wafture
a hairdo that creates undulations in the hair
something that rises rapidly; "a wave of emotion swept over him"; "there was a sudden wave of buying before the market closed"; "a wave of conservatism in the country led by the hard right"
standing still; "the car remained stationary with the engine running"
not capable of being moved; "stationary machinery"
a member of the womens reserve of the United States Navy; originally organized during World War II but now no longer a separate branch

PrepTutorEJDIC

(前後左右・上下に)『揺れる』,ひらめく,ひらひらする / (波形に)『起伏する』,うねる / (手・旗などを振って)『合図する』,手を振る;(…のために,…に向かって)手を振る《+in(to, at)+名》 / (に向かって)‘…を'振る,揺り動かす,ひらめかす《+名+at+名》 / 《方向を表す副詞[句]を伴って》(手・旗などを振って)〈人〉‘に'『合図する』 / ‘…を'起伏させる,うねらせる:〈頭髪〉‘を'ウエーブさせる / 『波』,波浪 / (…の)『波のような動き』,うねり《+of+名》 / (感情・景気などの一時的な)高まり,強まり;(寒暑の)急激な波《+of+名》 / (手・旗などを)振ること,振ってする合図;(…を)振ること《+of+名》 / (髪の)縮れ,ウエーブ / (光・音・電気などの)波
動かない,静止した / すえつけの(fixed) / 駐留の,常備の / 一定不変の,増減しない

Wikipedia preview

出典(authority):フリー百科事典『ウィキペディア（Wikipedia）』「2013/08/21 06:09:05」(JST)

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For other uses, see Hydraulic jump.

In physics, a standing wave – also known as a stationary wave – is a wave that remains in a constant position.

Two opposing waves combine to form a standing wave.

This phenomenon can occur because the medium is moving in the opposite direction to the wave, or it can arise in a stationary medium as a result of interference between two waves traveling in opposite directions. In the second case, for waves of equal amplitude traveling in opposing directions, there is on average no net propagation of energy.

In a resonator, standing waves occur during the phenomenon known as resonance.

Moving medium[edit source | edit]

As an example of the first type, under certain meteorological conditions standing waves form in the atmosphere in the lee of mountain ranges. Such waves are often exploited by glider pilots.

Standing waves and hydraulic jumps also form on fast flowing river rapids and tidal currents such as the Saltstraumen maelstrom. Many standing river waves are popular river surfing breaks.

Opposing waves[edit source | edit]

Standing waves

Standing wave in stationary medium. The red dots represent the wave nodes.
A standing wave (black) depicted as the sum of two propagating waves traveling in opposite directions (red and blue).

Electric force vector (E) and magnetic force vector (H) of a standing wave.
Standing waves in a string — the fundamental mode and the first 5 overtones.

A three-dimensional standing wave on a disk; this is the fundamental mode
A higher harmonic standing wave on a disk with two nodal lines crossing at the center.

As an example of the second type, a standing wave in a transmission line is a wave in which the distribution of current, voltage, or field strength is formed by the superposition of two waves of the same frequency propagating in opposite directions. The effect is a series of nodes (zero displacement) and anti-nodes (maximum displacement) at fixed points along the transmission line. Such a standing wave may be formed when a wave is transmitted into one end of a transmission line and is reflected from the other end by an impedance mismatch, i.e., discontinuity, such as an open circuit or a short.^[1] The failure of the line to transfer power at the standing wave frequency will usually result in attenuation distortion.

In practice, losses in the transmission line and other components mean that a perfect reflection and a pure standing wave are never achieved. The result is a partial standing wave, which is a superposition of a standing wave and a traveling wave. The degree to which the wave resembles either a pure standing wave or a pure traveling wave is measured by the standing wave ratio (SWR).^[2]

Another example is standing waves in the open ocean formed by waves with the same wave period moving in opposite directions. These may form near storm centres, or from reflection of a swell at the shore, and are the source of microbaroms and microseisms.

Mathematical description[edit source | edit]

In one dimension, two waves with the same frequency, wavelength and amplitude traveling in opposite directions will interfere and produce a standing wave or stationary wave. For example: a wave traveling to the right along a taut string and hitting the end will reflect back in the other direction along the string, and the two waves will superpose to produce a standing wave. The reflective wave has to have the same amplitude and frequency as the incoming wave.

If the string is held at both ends, forcing zero movement at the ends, the ends become zeroes or nodes of the wave. The length of the string then becomes a measure of which waves the string will entertain: the longest wavelength is called the fundamental. Half a wavelength of the fundamental fits on the string. Shorter wavelengths also can be supported as long as multiples of half a wavelength fit on the string. The frequencies of these waves all are multiples of the fundamental, and are called harmonics or overtones. For example, a guitar player can select an overtone by putting a finger on a string to force a node at the proper position between the ends of the string, suppressing all harmonics that do not share this node.

Harmonic waves travelling in opposite directions can be represented by the equations below:

$y_1\; =\; y_0\, \sin(kx - \omega t)\,$

and

$y_2\; =\; y_0\, \sin(kx +\omega t)\,$

where:

y₀ is the amplitude of the wave,
ω (called angular frequency, measured in radians per second) is 2π times the frequency (in hertz),
k (called the wave number and measured in radians per metre) is 2π divided by the wavelength λ (in metres), and
x and t are variables for longitudinal position and time, respectively.

So the resultant wave y equation will be the sum of y₁ and y₂:

$y\; =\; y_0\, \sin(kx - \omega t)\; +\; y_0\, \sin(kx + \omega t).\,$

Using the trigonometric sum-to-product identity for 'sin(u) + sin(v)' to simplify:

$y\; =\; 2\, y_0\, \cos(\omega t)\; \sin(kx).\,$

This describes a wave that oscillates in time, but has a spatial dependence that is stationary: sin(kx). At locations x = 0, λ/2, λ, 3λ/2, ... called the nodes the amplitude is always zero, whereas at locations x = λ/4, 3λ/4, 5λ/4, ... called the anti-nodes, the amplitude is maximum. The distance between two conjugative nodes or anti-nodes is λ/2.

Standing waves can also occur in two- or three-dimensional resonators. With standing waves on two dimensional membranes such as drumheads, illustrated in the animations above, the nodes become nodal lines, lines on the surface at which there is no movement, that separate regions vibrating with opposite phase. These nodal line patterns are called Chladni figures. In three-dimensional resonators, such as musical instrument sound boxes and microwave cavity resonators, there are nodal surfaces.

Examples[edit source | edit]

One easy to understand example of standing waves is two people shaking either end of a jump rope. If they shake in sync the rope can form a regular pattern of waves oscillating up and down, with stationary points along the rope where the rope is almost still (nodes) and points where the arc of the rope is maximum (antinodes)

Sound waves[edit source | edit]

The hexagonal cloud feature at the north pole of Saturn was initially thought to be standing Rossby waves.^[3] This explanation has recently been disputed though.^[4]

Standing waves are also observed in physical media such as strings and columns of air. Any waves traveling along the medium will reflect back when they reach the end. This effect is most noticeable in musical instruments where, at various multiples of a vibrating string or air column's natural frequency, a standing wave is created, allowing harmonics to be identified. Nodes occur at fixed ends and anti-nodes at open ends. If fixed at only one end, only odd-numbered harmonics are available. At the open end of a pipe the anti-node will not be exactly at the end as it is altered by its contact with the air and so end correction is used to place it exactly. The density of a string will affect the frequency at which harmonics will be produced; the greater the density the lower the frequency needs to be to produce a standing wave of the same harmonic.

Light[edit source | edit]

Standing waves are also observed in optical media such as optical wave guides, optical cavities, etc. Lasers use optical cavities in the form of a pair of facing mirrors. The gain medium in the cavity (such as a crystal) emits light coherently, exciting standing waves of light in the cavity. The wavelength of light is very short (in the range of nanometers, 10^-9 m) so the standing waves are microscopic in size. One use for standing light waves is to measure small distances, using optical flats.

Mechanical waves[edit source | edit]

Standing waves can be mechanically induced into solid medium using resonance. One easy to understand example is two people shaking either end of a jump rope. If they shake in sync the rope will form a regular pattern with nodes and antinodes and appear to be stationary, hence the name standing wave. Similarly a cantilever beam can have a standing wave imposed on it by applying a base excitation. In this case the free end moves the greatest distance laterally compared to any location along the beam. Such a device can be used as a sensor to track changes in frequency or phase of the resonance of the fiber. One application is as a measurement device for dimensional metrology.^[5]^[6]

References and notes[edit source | edit]

^ This article incorporates public domain material from the General Services Administration document "Federal Standard 1037C".
^ Blackstock, David T. (2000), Fundamentals of Physical Acoustics, Wiley–IEEE, ISBN 0-471-31979-1 , 568 pages. See page 141.
^ A Wave Dynamical Interpretation of Saturn's Polar Region, M. Allison, D. A. Godfrey, R. F. Beebe, Science vol. 247, pg. 1061 (1990)
^ A laboratory model of Saturn’s North Polar Hexagon, A. C. Barbosa Aguiar, P. L. Read, R. D. Wordsworth, T. Salter, Y. H. Yamazaki, Icarus, vol. 206 (2009)
^ M.B. Bauza, R.J Hocken, S.T Smith, S.C Woody, (2005), The development of a virtual probe tip with application to high aspect ratio microscale features, Rev. Sci Instrum, 76 (9) 095112 .
^ http://www.insitutec.com

External links[edit source | edit]

Java applet of standing waves on a vibrating string
Java applet of transverse standing wave
Java applet showing the production of standing wave on a string by adjusting frequency
Standing Wave Lessons - Includes helpful graphics, flash animations, YouTube videos

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English Journal

Acoustic field radiated into a transversely isotropic solid from a small aperture spherical surface.

Every AG, Wenke I, Aebi L, Dual J.SourceSchool of Physics, University of the Witwatersrand, Wits 2050, South Africa. arthur.every@wits.ac.za
Ultrasonics.Ultrasonics.2011 Oct;51(7):824-30. Epub 2011 Apr 5.
The acoustic field modelling reported in this paper finds application in the design of a scanning probe tip for measuring the near-surface elastic properties of solids and surface structures at high frequencies and with high spatial resolution. The underlying concept is for a longitudinally polarize
PMID 21514947

Instantaneous axial force of a high-order Bessel vortex beam of acoustic waves incident upon a rigid movable sphere.

Mitri FG, Fellah ZE.SourceLos Alamos National Laboratory, MPA-11, Sensors & Electrochemical Devices, Acoustics & Sensors Technology Team, MS D429, Los Alamos, NM 87545, USA. mitri@lanl.gov
Ultrasonics.Ultrasonics.2011 Aug;51(6):719-24. Epub 2011 Mar 1.
The present investigation examines the instantaneous force resulting from the interaction of an acoustical high-order Bessel vortex beam (HOBVB) with a rigid sphere. The rigid sphere case is important in fluid dynamics applications because it perfectly simulates the interaction of instantaneous soun
PMID 21450326

Japanese Journal

Global dynamics above the ground state energy for the focusing nonlinear Klein-Gordon equation

Nakanishi Kenji,Schlag W.
Journal of Differential Equations 250(5), 2299-2333, 2011-03-01
… This equation admits a unique positive stationary solution Q,called the ground state. … The analogue of thesolutions found by Duyckaerts, Merle [13], [14] for the energy critical wave andSchr¨odinger equations appear here as the unique one-dimensional stable/unstablemanifolds approaching ±Q exponentially as t → ∞ or t → ?∞, respectively. …
NAID 120002770950

Analysis Method of Ground Wave Propagation over Land-to-Sea Mixed-Path by Using Helmholtz-Kirchhoff Integral Theorem

KAWANO Toru,GOTO Keiji,ISHIHARA Toyohiko
IEICE Transactions on Electronics E94.C(1), 10-17, 2011
… In this paper, we have derived a novel integral representation for the ground wave propagation over land-to-sea mixed-paths by applying the Helmholtz-Kirchhoff integral theorem. … By using the method of stationary phase applicable uniformly as the stationary phase point approaches the endpoint of the integral, we have derived the asymptotic solution for the scattered fields consisting of the first-order and the second-order diffraction terms. …
NAID 130000454325