Stationary Waves Stationary waves are produced by superposition of two progressive waves of equal...
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Transcript of Stationary Waves Stationary waves are produced by superposition of two progressive waves of equal...
Stationary Waves
Stationary waves are produced by superposition of two progressive waves of equal amplitude and frequency, travelling with the same speed in opposite directions.
http://www2.biglobe.ne.jp/~norimari/science/JavaEd/e-wave4.html
Production of Stationary Waves
A stationary wave would be set up by causing the string to oscillate rapidly at a particular frequency.If the signal frequency is increased further, overtone patterns appear.
Properties of a stationary wave (1)
Stationary waves have nodes where there is no displacement at any time.In between the nodes are positions called antinodes, where the displacement has maximum amplitude.
A vibrating loop
N A N A N
VibratorVibrator
Properties of a stationary wave (2)
The waveform in a stationary wave does not move through medium; energy is not carried away from the source.The amplitude of a stationary wave varies from zero at a node to maximum at an antinode, and depends on position along the wave.
Vibrations of particles in a stationary waveAt t = 0, all particles are at rest because the particles reach their maximum displacements.At t = ¼T, Particles a, e, and i are at rest because they are the nodes.Particles b, c and d are moving downward. They vibrate in phase but with different amplitude.Particles f, g and h are moving upward. They vibrate in phase but with different amplitude.
t = 0
t = ¼T
t = ¼T
t = ⅜T
t = ½T
a
bc
d
e
fg
hii
ab c d
ef g h
ii
Properties of a stationary wave (2)
All particles between two adjacent nodes (within one vibrating loop) are in phase.
Video
1. Stationary waves (string)
2. Stationary waves (sound)
Modes of vibration of strings
Picture of Standing Wave Name Structure
1st Harmonicor
Fundamental
1 Antinode2 Nodes
2nd Harmonicor
1st Overtone
2 Antinodes3 Nodes
3rd Harmonicor
2nd Overtone
3 Antinodes4 Nodes
4th Harmonicor
3rd Overtone
4 Antinodes5 Nodes
5th Harmonicor
4th Overtone
5 Antinodes6 Nodes
L = ½λ1
f1 = v/2L
L = λ2
f2 = v/L
L = 1½λ3
f3 = 3v/2L
L = 2λ4
f4 = 2v/L
L = 2½λ5
f5 = 5v/2L
http://id.mind.net/~zona/mstm/physics/waves/standingWaves/standingWaves1/StandingWaves1.html
L
Investigating stationary waves using sound waves and
microwavesMoving the detector along the line between the wave source and the reflector enables alternating points of high and low signal intensity to be found. These are the antinodes and nodes of the stationary waves.The distance between successive nodes or antinodes can be measured, and corresponds to half the wavelength λ.If the frequency f of the source is known, the speed of the two progressive waves which produce the stationary wave can be obtained.
Reflector
DetectorWave source
Resonant Frequencies of a Vibrating String
From the experiment, we find that There is a number of resonant frequencies
in a vibrating string, The lowest resonant frequency is called
the fundamental frequency (1st harmonic), The other frequencies are called overtones
(2nd harmonic, 3rd harmonic etc.), Each of the overtones has a frequency
which is a whole-number multiple of the frequency of the fundamental.
Factors that determine the fundamental frequency of a
vibrating string
The frequency of vibration depends on the mass per unit length of the string, the tension in the string and, the length of the string.
The fundamental frequency is given by
T
Lfo 2
1 where T = tension
= mass per unit length L = length of string
Vibrations in Air Column
When a loudspeaker producing sound is placed near the end of a hollow tube, the tube resonates with sound at certain frequencies.Stationary waves are set up inside the tube because of the superposition of the incident wave and the reflected wave travelling in opposite directions.
http://www.walter-fendt.de/ph11e/stlwaves.htm
Factors that determine the fundamental frequency of a
vibrating air column
The natural frequency of a wind instrument is dependent upon The type of the air column, The length of the air column of the
instrument.
Open tube Closed tube
Name
Modes of vibration for an open tube
Picture of Standing Wave Structure
1st Harmonicor
Fundamental
2 Antinodes1 Node
2nd Harmonicor
1st Overtone
3 Antinodes2 Nodes
3rd Harmonicor
2nd Overtone
4 Antinodes3 Nodes
4th Harmonicor
3rd Overtone
5 Antinodes4 Nodes
5th Harmonicor
4th Overtone
6 Antinodes5 Nodes
L = ½λ1
f1 = v/2L
L = λ2
f2 = v/L
L = 1½λ3
f3 = 3v/2L
L = 2λ4
f4 =2v/L
L = 2½λ5
f5 = 5v/2L
Modes of vibration for a closed tube
Picture of Standing Wave Name Structure
1st Harmonicor
Fundamental
1 Antinode1 Node
3rd Harmonicor
1st Overtone
2 Antinodes2 Nodes
5th Harmonicor
2nd Overtone
3 Antinodes3 Nodes
7th Harmonicor
3rd Overtone
4 Antinodes4 Nodes
9th Harmonicor
4th Overtone
5 Antinodes5 Nodes
L = ¼λ1
f1 = v/4L
L = ¾λ3
f3 =3v/4L
L = 1¼λ5
f5 =5v/4L
L = 1¾λ7
f7 = 7v/4L
L = 2¼λ9
f9 =9v/4L
The quality of sound (Timbre)
The quality of sound is determined by the following factors: The particular harmonics present in addition
to the fundamental vibration, The relative amplitude of each harmonic, The transient sounds produced when the
vibration is started.
1st overtone Fundamental
2nd overtone
3rd overtone
resultant
http://surendranath.tripod.com/Harmonics/Harmonics.html