Physics of Music Lecture 14 Percussion Acoustics

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Physics of MusicLecture 14Percussion Acoustics

• November 21, 2002
• Old Dominion University, Department of Physics
• Prof. Charles E. Hyde-Wright
• Ref Berg Stork Chapter 14

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Vibrations of Bars

• Vibrations in solids have many possible states of
  polarization.
• Sound waves can vibrate longitudinally
• Hit a bar on its end
• Sound waves can vibrate transverse
• Hit a bar with a transverse blow.
• Small amplitude transverse vibrations can be
  analyzed as superposition of vibrations along two
  axis perpendicular to direction of travel of
  wave.
• The direction of polarization of the vibration
  can change as the wave propagates
• Torsional (twisting) vibrations
• Hit a bar with an off axis transverse blow.

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Longitudinal Vibrations of Bars

• Longitudinal vibration of sound in bar propagates
  just like sound wave of air in tube.
• fn n vS / (2 L)
• Stiffness is provided by Youngs Modulus E,
  intrinsic to material (not geometry)
• E Pressure/(Fractional change in length)
• vS ? (E/r) Speed of sound in bar
• r Density of bar
• Metal is much stiffer than air, but also much
  more massive. Stiffness wins (by factor of 10 in
  vS)
• SteelE 21011 Pascal, r 8000kg/m3, vS 5000
  m/s
• Too high in frequency, not very useful musically

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Marimba
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Transverse Vibration of Bars

• Speed of sound strongly affected by geometry
• Compression on inside of curvature
• Stretch on outside of curvature
• fn n2 p vS K/ (8 L2), n 3.011, 5, 7,
• vS ? (E/r)
• K t / ? 12, bar of thickness t, free ends.
• K(1/2) ? (a2b2) a,b inner, outer radius
  (chimes)
• Harmonic ratios are 1.0 2.76 5.4 8.9
• K/L ltlt 1/10, lowers pitch back to musical range

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Transverse modes of bars(supported near ends)

• Non-Harmonic series

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Vibrations of Drumhead
Drumhead acts like many parallel strings, each of
different length. Vibrations on a string are
described by sine cosine. Vibrations on
membrane described by Bessel functions Geometry
dictates that harmonics are non-rational ratios
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Modes of Vibration of Membrane,
http//mathworld.wolfram.com/BesselFunctionoftheFi
rstKind.html

• Vibration described by 2 integers
• n number of nodal lines
• m number of nodal circles
• f(1/diameter) knm ?(F/s)
• F Tension on drumhead
• s areal density of drumhead
• knm zeros of Bessel function
• k01 1.0, k02 2.30, k03 3.60
• k11 1.59, k21 2.14, k31 2.65

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Bessel Functions Waves on a circular
membranehttp//en.wikipedia.org/wiki/Bessel_funct
ionApplicationshttp//physics.usask.ca/hirose/e
p225/animation/drum/anim-drum.htm
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Drumhead modes and relative frequency

• Hitting in middle only excites (0m) modes
• Hitting near edge excites (1m, 2m) modes
• Drum220.wav

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Tympani

• Tympani is tuned by coupling drumhead vibration
  to vibration of enclosed air.

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Tympani modes

• 01 mode decays very rapidly
• 11 mode couples strongly to enclosed air
• Volume of air increases effective mass, lowers
  frequency ?(stiffness/mass)
• Column of air (50cm/0.02cm) 2500 thicker
• Mass of air is 1/1000 of membrane
• Fractional mass increase 3.5 (exagerated)
• Shape of tympani gives different shifts to
  different modes 11, 21, 31,
• Lowest mode (11) is shifted down the most
• Relative Harmonic series 2345