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Chapter 3 Transverse standing waves Resonance and the Overtone Series Mersenne’s laws Longitudinal standing waves Other standing waves and applications http://hyperphysics.phy-astr.gsu.edu/Hbase/waves/string.html http://www.kw.igs.net/~jackord/bp/n2.html www.physics.umd.edu/lecdem/misc/phys102/PH102chap03.ppt
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Identical waves moving in opposite directions
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Motion of the spring
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Motion of point along the spring
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Stretched strings End effects
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Standing wave representation
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Impossible “standing waves”
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Standing Waves in a Stretched String
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Stretched String Frequencies and Wavelengths f = v / λ N = 1 λ = 2L f = v/2L = f 1 N = 2 λ = L f = v/L = 2f 1 N = 3 λ = 2L/3 f = 3v/2L = 3f 1 N = 4 λ = L/2 f = 2v/L = 4f 1 N = 5 λ = 2L/5 f = 5v/2L = 5f 1 N = 6 λ = L/3 f = 3v/L = 6f 1
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Notes of the Overtone Series
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Mersenne’s Laws f = fundamental frequency L = length of string F = tension in string W = mass per unit length of string
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Mersenne’s First Law f α 1/L F and W constant If L => 2 L then f => ?? If L => L / 2 then f => ??
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Mersenne’s First Law f α 1/L F and W constant If L => 2 L then f => f / 2 If L => L / 2 then f => 2 f
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Mersenne’s Second Law f α √F L and W constant If F => 9 F then f => ?? If F => F / 4 then f => ??
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Mersenne’s Second Law f α √F L and W constant If F => 9 F then f => 3 f If F => F / 4 then f => f / 2
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Mersenne’s Third Law f α 1/ √W F and L constant If W => 4 W then f => ?? If W => W / 9 then f => ??
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Mersenne’s Third Law f α 1/ √W F and L constant If W => 4 W then f => f / 2 If W => W / 9 then f => 3 f
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Rope Wave Example 110 grams => 3 loops (frequency f) What happens with 990 grams?
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Rope Wave Example 990 grams => 1 loop (frequency f and wavelength λ) What mass will produce Wavelength λ/2?
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Rope Wave Example 990 grams => 1 loop (frequency f and wavelength λ) Mass of 250 grams will produce wavelength λ/2.
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Nanoguitar Silicon strings: length 6 – 12 microns diameter 150 – 200 nanometers http://www.news.cornell.edu/releases/nov03/nemsguitar.ws.html
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Nanoguitar
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Longitudinal Standing Waves
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Kundt’s tube
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Air Columns End effects
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“End Effect” Experiment
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End at right “open” End at right “closed”
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End configurations for tubes 1. Phase change at open and closed ends
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End configurations for tubes 2. Node or antinode at open and closed ends
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Open Tubes
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“Open” Tube Frequencies and Wavelengths f = v / λ N = 1 λ = 2L o f = v/2L o = f o N = 2 λ = L o f = v/L o = 2f o N = 3 λ = 2L o /3 f = 3v/2L o = 3f o N = 4 λ = 2L o /4 f = 4v/2L o = 4f o N = 5 λ = 2L o /5 f = 5v/2L o = 5f o N = 6 λ = 2L o /6 f = 6v/2L o = 6f o
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Closed Tubes
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“Closed” Tube Frequencies and Wavelengths f = v / λ N = 1 λ = 4L c f = v/4L c = f c N = 2 does not exist. N = 3 λ = 4L c /3 f = 3v/4L c = 3f c N = 4 does not exist. N = 5 λ = 4L c /5 f = 5v/4L c = 5f c N = 6 does not exist.
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Open and Closed Tube Comparison
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Tube Quiz
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Fundamental Frequency Open and Closed Tubes If L o = L c what is f o : f c If L o = 2L c what is f o : f c
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Closed Tube Resonances
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Applications to Musical Instruments
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The Flute
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The Recorder
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Actual Recorder Finger Hole Positions
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The Clarinet
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The Saxophone
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The Trumpet
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Various Standing Waves Flame Tube Aluminum Rod Velocity of Sound in Aluminum Chladni Plates (photos)
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Chladni Plates
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Violin Body Vibrations
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Standing Waves in a Membrane Rectangular Membrane Applet Circular Membrane Applet Rectangular Membrane AppletCircular Membrane Applet
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Beaker Breaker using sound wave resonance Teacup Standing Waves
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Tacoma Narrows Bridge Collapse
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The End Any Questions?
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