Physics of Music Lecture 2: Waves and Sound

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Physics of MusicLecture 2 Waves and Sound

• Phys 332W
• Prof. Charles E. Hyde-Wright

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Demonstrations

• 9/7/06
• Mass on spring
• (available in 215), PASCO interface
• Needed Motion sensor (bi-directional pulley
  interface) Rod stand, spring, mass, string.
• 9/14/06
• Torsion Wave machine
• Speaker w/ and w/out baffle

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Good Waves and Bad Waves
BergStork, 2nd edition

• Fig 1-22 Bad cartoon (Infinite velocity)

• Fig 1-23 Good wave (finite maximum velocity).

Velocity rate of change of position slope of
position vs time graph
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Why do things wiggle?--Inertia

• Mass on a spring, child on a swing.
• At the equilibrium point, the net force is
  zeroyet the motion continues (in fact the
  equilibrium point is the point of maximum
  velocity).
• Force equals mass times acceleration
• Acceleration is the rate of change of velocity
• When the Force goes to zero, the velocity doesnt
  go to zero, the velocity stops changingthe
  object keeps going at the same speed.
• See graphs with PASCO

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Why do things wiggle? Stiffness

• As a spring stretches or compresses, there is a
  force striving to restore the equilibrium
  condition.
• The same is true for
• a stretched string as it is plucked, bowed, or
  hit
• A reed as it is bent from its equilibrium
• The bar of of xylophone or marimba
• For simple systems, the restoring force is
  proportional to the amount of stretch.

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What determines frequency of wiggle?

• Stiffness divided by inertia
• Simple Hookes Law spring
• F - k x
• But force mass times acceleration Fm a
• k/mass F/(x mass)
• k/m a/x units of 1/time2
• Frequency is proportional to square root of
  stiffness divided by inertia.
• f 2 p ?k/m 2 p w

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Mass on a Spring
Position vs Time
Velocity vs Time
Acceleration vs Time
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Lissajous figures of mass on a spring
Acceleration vs. Position
Velocity vs. Position
Velocity and position are 90 out of phase
a/x F/(m x) -k/m w2
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Basic Wave Phenomena

• Inverse Square Law
• Polarization
• Superposition
• Constructive and Destructive Interference

• Huygens Principle
• Reflection
• Refraction
• Diffraction

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Inverse Square Law

• The energy density stored in a wave is
  proportional to the square of the wave amplitude
  (Intensity).
• As a wave pulse travels outwards from a source,
  the total energy is conserved (ignoring
  dissipation to heat).
• Since surface area grows as the square of
  distance, the amplitude of a FREELY expanding
  wave decreases linearly with distance, the
  intensity decreases with the square of distance.
• If the wave is confined to a tube, the surface
  area remains constant with distance, the wave
  intensity remains constant

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Inverse Square Law

• Surface of Area expands as square of distance
  from source
• Intensity decreases as one over radius squared.

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Polarization

• Sound is longitudinally polarized.
• Surface water waves combine longitudinal and
  transverse polarization
• Light is a transverse wave.
• Electric and magnetic fields are the capability
  of exerting electromagnetic forces on charges in
  the path of the wave.
• Matter waves electron, atom, are polarized with
  respect to the spin of the particle

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Superposition (Physics)

• Sound
• You can hear and understand my voice, even if
  music is playing in the background.
• The total sound wave reaching your ear is just
  the simple sum of voice music
• Light
• Your ability to see me is unaffected by the
  presence of other light waves bouncing around the
  room.

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Superposition (Perception)

• Two or more tones (especially in harmonic ratio)
  will blend into a single sound, retaining the
  pitch of the fundamental
• Two light waves of different color (e.g. yellow
  and blue) will blend to form a new color, (green)
  even if no green light is present. Red and blue
  combine to form a color that does not correspond
  to any unique wavelength of light.
• Constructive Interference

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Superposition (Mathematics)
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Huygens Principle

• Each point on a wave acts as a source of a wave
  traveling outwards (in 3-dimensions, with 1/r2
  intensity).
• The initial phase of the outward wave equals the
  phase of the source point.
• The total wave amplitude at any point is equal to
  the sum of all waves coming from all the source
  points on the earlier wave.
• Waves diffract around obstacles

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Constructive and Destructive Interference

• Two identical sources, in phase, combined to
  produce a sound intensity of 4 times either
  source by itself
• Two identical sources, 180 degrees out of phase,
  combine to exactly cancel each other.
• In this case, removing one source increases the
  sound level.

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Refraction
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Thermal Inversion Refraction

• Thermal inversion (e.g. night time). Air is
  cooler near ground

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Thermal Gradient Refraction

• Usually, the air nearest the ground is hottest.
  This creates an upward refraction of sound

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Wind Shear Refraction

• Wind shear creates a focussing of sound downwind,
  a defocussing upwind

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Beats

• Combination of two tones.
• File http//www.physics.odu.edu/hyde/Teaching/Fall
  04/Lectures/Beats.wav contains two pitches in
  ratios 1001/1000, 1002/1000, 1004/1000,
  1016/1000, 1032/1000, 1064/1000, 1128/1000.
• A musical half-step is a ratio 1059/1000
• This ratio is the twelve root of 2
• 12 equal ratio half-step intervals in an octave

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Beats www.physics.odu.edu/hyde/Teaching/Fall04/Le
ctures/beats.wav
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Beats

• The sound wave on the previous slides was
  constructed by adding two waves of close, but
  different frequencies.
• What your ear hears is the product of two waves
  A pitch at the average frequency modulated by a
  amplitude at the difference of the two
  frequencies.
• Your ear/brain/mind is doing Trigonometry
• sin(wd)tsin(w-d)t 2 sin(wt) cos(dt)
• Sum of two pitches product of average and
  difference
• Try this in Excel, or on your calculator!

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Trigonometry oscillations, not triangles.

• A Sin(wt)
• A amplitude of wave
• (wt) is the argument, or phase, of the wave.
• The sine wave starts at zero when its phase is
  zero, and repeats everytime the phase increases
  by 2p.
• Period time to repeat 2p/w 1/f
• A cos(wt)
• Cosine wave is same as sine wave, but shifted ¼
  oscillation.
• Cosine wave starts at its maximum1 when phase0

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Doppler (Phys332W 2000 project,
www.physics.odu.edu/hyde/chw.htm)
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Ultrasound

• Frequency gt 20 KHz
• Medical imaging uses simultaneous phase and
  intensity measurement with an array of
  piezo-electric transducers.