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Standing Waves Time to read Chapter 3 of Berg & Stork

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String with ends fixed String is stretched = tension string wants to return to normal length …

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String with ends fixed String is stretched = tension … but it overshoots and keeps oscillating

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fundamental 2 nd harmonic 3 rd harmonic 4 th harmonic Different vibration modes Animation courtesy of Dr. Dan Russell, Kettering University

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Standing waves are a superposition of two counter moving waves Animation courtesy of Dr. Dan Russell, Kettering University

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l v v T/2 = l /(2v) f = v/ l speed of the wave on the string, NOT the speed of sound

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l 1 = 2 L f 1 = v/ l 1 = v/(2L) l 2 = L f 2 = v/ l 2 = v/L=2 f 1 … L

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If the initial position of the string is one the the vibration modes, only that mode will be “excited” In general, the initial shape of the string will be a superposition of many modes. Each one will be excited and evolve in time separately with their own frequency. Different initial conditions will produce a different timbre. http://www.falstad.com/loadedstring/

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Mersenne’s laws fundamental frequency tension mass per length length

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In other words … 1. Frequency is inversely proportional to length 2. Frequency is proportional to square root of tension 3. Frequency is inversely proportional to square root of the string density

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Vibration modes of membranes two integers

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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 membrane

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Great visualization (with sound !) of membranes vibration modes http://www.falstad.com/membrane/j2/

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Vibration modes of a bottle of beer fundamental mode

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

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Fourier amplitudes of an empty beer bottle struck at the neck

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Resonance force Pushes at the natural frequency of the swing increase the oscillation amplitude

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For a resonance to occur the driving force needs to have a frequency very close to one of the natural frequencies of the resonating object. It also helps if that mode has little damping.

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Sound can play the role of a periodic force that can excite a particular vibration mode if the frequencies match

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Playing one note on the piano (C,E,F,G) makes the C3 “sing”

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Sympathetic string is not touched by the player but it resonates with the other strings hardingfele

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Resonance curve response at a given frequency violin loudspeaker

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Typical loudspeaker response in a room valleys and peaks resulting from interaction with walls, furniture, …

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Examples of resonance: radio receiver (selects one frequency out of many through resonant circuit) radio receiver (selects one frequency out of many through resonant circuit) buildings and earthquakes, bridges and wind flutter buildings and earthquakes, bridges and wind flutter child on a swing child on a swing voice and musical instruments (formants) voice and musical instruments (formants) many phenomena in the emission and absorption of light many phenomena in the emission and absorption of light … …

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Resonant interaction with Saturn’s moons destabilizes some of the orbits in the ring

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This is not a string now, it’s the graph of the pressure x distance Standing sound waves in air tubes

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v string v sound nodes at the ends nodes or antinodes at the ends air tubes x strings

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closed end open end pressure displacement

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l /4

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Example: closed-open tube, N=7

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