Physics of Music Lecture 3: Standing Waves

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Physics of MusicLecture 3 Standing Waves

• Traveling Waves,
• Reflections
• Standing Waves
• Prof. Charles E. Hyde-Wright
• Autumn 2004

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Standing Electron Waves in an Atomic Corral

• www.almaden.ibm.com/vis/stm/corral.html

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Reflections

• Waves reflect when the medium changes
• String fixed at one (or both ends)
• Sound in a closed end pipe
• Sound in a open end pipe (less obvious)
• Light striking a mirror
• Electrons at a crystal boundary

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Waves on a string

• What happens when a wave pulse on a stretched
  string reaches the end (e.g. bridge of violin)?
• How can an arbitrary wave pulse always have zero
  amplitude at the end of the string?
• The physical force anchoring the string generates
  a wave pulse of opposite amplitude traveling in
  the opposite direction.

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Boundary value problem

• Wave amplitude 0 at each end of string
• Equations describing wave motion dont know about
  ends of string.
• String tied at one end acts like infinite long
  string with negative wave pulse traveling in
  opposite direction.

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Resonance Standing Waves

• An arbitrary wave on a string will slosh back and
  forth and slowly dissipate (if not continuously
  driven).
• At special Resonant Frequencies a string will
  vibrate with a Standing wave.
• The left traveling wave and the right traveling
  wave exactly add to create a stable wave.

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Radians and Circles

• 1 radian angle subtending an arc of length 1
  radius
• Circumference of circle 2pr
• 1 full revolution (360o) 2p radians.

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Sinewaves

• Simple oscillations with unique frequency f are
  sine or cosine functions
• Sin(x) is a periodic function that goes through
  one complete oscillation as its argument (x)
  increases by 2p.
• Sin(0)0,
• Sin(p/2)1 (max value)
• Sin(p)0
• Sin(3p/2)-1 (min value)
• Cos(x)Sin(xp/2) Cosine is just shifted sine.

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• For a snapshot of a wave at one instant in time
  x (2p z /l). Wave repeats when position z
  changes by wavelength l.
• For a time recording of a sound wave at one
  place x (2p f t). Wave repeats when time t
  increases by period T 1/f

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Standing Waves ln(5.0m)/n, n1, 2,
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Standing waves from Traveling waves
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Properties of simple standing waves in
1-dimensional volume of length L

• Wavelengths lnL/(2n), n1, 2,
• If traveling wave velocity is v, then wave
  repeats in time l/v.
• Frequencies fn v/ln.
• All anti-nodes of equal amplitude
• All nodes equally spaced ln/2

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Standing Waves From Superposition

• Standing Wave is product of oscillation in space
  (wavelength l) and oscillation in time (frequency
  f)
• Amplitude sin(2p x/l) sin(2p t f)
• The wave repeats when x increases by l.
• The wave repeats when t increases by 1/f.
• Remember discussion of beats
• The product of two sine waves is equal sum of two
  waves with the arguments added or subtracted
• (1/2)Amplitude cos(2p x/l-2p t f) cos(2p
  x/l2p t f)
• These are two traveling waves, with opposite
  direction and amplitude.

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Sound waves in a tube

• A sound wave is a longitudinal oscillation of
  density, driven by a longitudinal oscillation in
  pressure.
• The velocity of motion of the air is determined
  by the amplitude of the pressure oscillation, the
  viscosity of the air, and the geometry of the
  tube.
• Impedance (Amplitude of Driving Force) /
  (Amplitude of velocity response).
• Impedance of sound in tube is r v / S
• r Density of air approximately 1 kg /m3
• v speed of sound in air 340 m/s (room
  temperature)
• S cross sectional area of tube

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Acoustic Impedance of finite Tube

• Profile of air velocity in cross section of tube.
• At closed end, Air velocity0 (impedance is
  infinite)
• At open end, Impedance is (almost) zero
  (Effective area is infinite)

1/area0
Vair0
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Boundary conditions on Sound Waves in a Tube
(Length L, radius r)

• Closed End
• Pressure wave at maximum (antinode)
• Velocity wave at zero (node)
• Open End
• Pressure wave at zero
• Velocity wave at maximum
• End correction for open end
• Effective length L 0.61 r
• T.R. Rossing, The Science of Sound, section 4.5

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Standing Sound (Pressure) Waves. Tube open at
one end and closed at other.
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Standing Sound (Velocity) Waves. Tube open at
one end and closed at other.
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Resonance

• Each natural frequency of string, sound in tube,
  vibrating drum head, atoms in a trap (see Prof.
  Sukenik), electrons in an atom, quarks in a
  nucleus is a resonance.
• For narrow, isolated resonances, there is a
  characteristic behavior The reflecting waves
  return in phase with the driving wave A large,
  stable, amplitude builds up.
• What happens when we drive to drive a wave at the
  wrong frequency?
• The reflecting waves cancel as often as they add,
  the resulting vibration is chaotic and small
  amplitude.