Standing WavesStanding WavesTime to read Chapter 3 of Time to read Chapter 3 of

Berg & StorkBerg & Stork

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 …

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

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

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

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

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

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/

Mersenne’s lawsMersenne’s laws

1

1

2 2

v Ff

L W L

fundamental fundamental frequencyfrequency

tensiontension mass per mass per lengthlength

lengthlength

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

Vibration modes of membranesVibration modes of membranes

two two integersintegers

You can also watch it on YouTube

http://www.youtube.com/watch?v=Zkox6niJ1Wc

For a circular membraneFor a circular membrane

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/

Vibration modes of a bottle of beerVibration modes of a bottle of beer

fundamental modefundamental mode

http://www.kettering.edu/~drussell/Demos.htmlhttp://www.kettering.edu/~drussell/Demos.html

Fourier amplitudes of an empty beer bottle struck at the neckFourier amplitudes of an empty beer bottle struck at the neck

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

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.

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

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”

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

Resonance curveResonance curve

responseresponseat a given at a given frequencyfrequency

violinviolin loudspeakerloudspeaker

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, …

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

• … …

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

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

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

closed endclosed end open endopen end

pressurepressure

displacementdisplacement

/4/4

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