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Standing WavesStanding WavesTime to read Chapter 3 of Time to read Chapter 3 of
Berg & StorkBerg & Stork
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String with ends fixedString with ends fixed
String is stretched = tensionString is stretched = tension
string wants to return to string wants to return to normal length …normal length …
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String with ends fixedString with ends fixed
String is stretched = tensionString is stretched = tension
… … but it overshoots and but it overshoots and keeps oscillatingkeeps oscillating
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fundamentalfundamental
22ndnd harmonic harmonic
33rdrd harmonic harmonic
44thth harmonic harmonic
Different vibration modesDifferent vibration modes
Animation courtesy of Dr. Dan Russell, Kettering UniversityAnimation courtesy of Dr. Dan Russell, Kettering University
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Standing waves are a superposition of Standing waves are a superposition of two counter moving wavestwo counter moving waves
Animation courtesy of Dr. Dan Russell, Kettering UniversityAnimation courtesy of Dr. Dan Russell, Kettering University
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vv vv
T/2 = T/2 = /(2v)/(2v)
f = v/f = v/
speed of the wave on speed of the wave on the string, NOT the the string, NOT the
speed of soundspeed of sound
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11 = 2 L = 2 L
ff11 = v/ = v/ = v/(2L) = v/(2L)
22 = L = L
ff22 = v/ = v/ = v/L=2 f = v/L=2 f11……
LL
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If the initial position of the string is one the the If the initial position of the string is one the the vibration modes, only that mode will be vibration modes, only that mode will be
“excited”“excited”
In general, the initial shape of the string will be a In general, the initial shape of the string will be a superposition of many modes. Each one will be superposition of many modes. Each one will be excited and evolve in time separately with their excited and evolve in time separately with their
own frequency.own frequency.
Different initial conditions will produce a Different initial conditions will produce a different timbre.different timbre.
http://www.falstad.com/loadedstring/http://www.falstad.com/loadedstring/
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Mersenne’s lawsMersenne’s laws
1
1
2 2
v Ff
L W L
fundamental fundamental frequencyfrequency
tensiontension mass per mass per lengthlength
lengthlength
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In other words …In other words …
1.1. Frequency is inversely proportional to lengthFrequency is inversely proportional to length
2.2. Frequency is proportional to square root of Frequency is proportional to square root of tensiontension
3.3. Frequency is inversely proportional to square Frequency is inversely proportional to square root of the string densityroot of the string density
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Vibration modes of membranesVibration modes of membranes
two two integersintegers
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You can also watch it on YouTube
http://www.youtube.com/watch?v=Zkox6niJ1Wc
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For a circular membraneFor a circular membrane
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Great visualization (with sound !) of Great visualization (with sound !) of membranes vibration modesmembranes vibration modes
http://www.falstad.com/membrane/j2/http://www.falstad.com/membrane/j2/
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Vibration modes of a bottle of beerVibration modes of a bottle of beer
fundamental modefundamental mode
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http://www.kettering.edu/~drussell/Demos.htmlhttp://www.kettering.edu/~drussell/Demos.html
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Fourier amplitudes of an empty beer bottle struck at the neckFourier amplitudes of an empty beer bottle struck at the neck
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ResonanceResonance
forceforce
Pushes at the natural frequency of the swing Pushes at the natural frequency of the swing increase the oscillation amplitudeincrease the oscillation amplitude
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For a resonance to occur the driving For a resonance to occur the driving force needs to have a frequency very force needs to have a frequency very
close to one of the natural close to one of the natural frequencies of the resonating object.frequencies of the resonating object.
It also helps if that mode has little It also helps if that mode has little damping.damping.
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Sound can play the role of a periodic force that can Sound can play the role of a periodic force that can excite a particular vibration mode excite a particular vibration mode if the frequencies if the frequencies
matchmatch
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Playing one note on the piano (C,E,F,G) Playing one note on the piano (C,E,F,G) makes the C3 “sing”makes the C3 “sing”
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Sympathetic string is Sympathetic string is not touched by the not touched by the
player but it resonates player but it resonates with the other stringswith the other strings
hardingfelehardingfele
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Resonance curveResonance curve
responseresponseat a given at a given frequencyfrequency
violinviolin loudspeakerloudspeaker
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Typical loudspeaker response in a roomTypical loudspeaker response in a room
valleys and valleys and peaks resulting peaks resulting from interaction from interaction
with walls, with walls, furniture, …furniture, …
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Examples of resonance:Examples of resonance:
• radio receiver (selects one frequency out of many radio receiver (selects one frequency out of many through resonant circuit)through resonant circuit)
• buildings and earthquakes, bridges and wind flutterbuildings and earthquakes, bridges and wind flutter
• child on a swingchild on a swing
• voice and musical instruments (formants)voice and musical instruments (formants)
• many phenomena in the emission and absorption of many phenomena in the emission and absorption of lightlight
• … …
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Resonant interaction with Saturn’s moons Resonant interaction with Saturn’s moons destabilizes some of the orbits in the ringdestabilizes some of the orbits in the ring
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This is not a This is not a string now, string now,
it’s the graph it’s the graph of the pressure of the pressure
x distancex distance
Standing sound waves in air tubesStanding sound waves in air tubes
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vvstring string v vsoundsound
nodes at nodes at the endsthe ends
nodes or nodes or antinodes at antinodes at
the endsthe ends
air tubes x stringsair tubes x strings
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closed endclosed end open endopen end
pressurepressure
displacementdisplacement
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/4/4
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1.5 7( )4
4 1.50.86
7
344 /400
0.86
L m
mm
v f
v m sf Hz
m
Example: closed-open tube, N=7Example: closed-open tube, N=7