Physics 1251 The Science and Technology of Musical Sound

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Physics 1251The Science and Technology of
Musical Sound

• Unit 3
• Session 33 MWF
• Percussion Instruments

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Physics 1251 Unit 3 Session 33 Percussion

• In the video (Amadeus), why did the soprano smile
  so broadly when she sang the high notes?

She was deliberately raising the pitch of the
formant in order to match the pitch of the notes
she was singing by changing the shape of her
vocal tract.
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Physics 1251 Unit 3 Session 33 Percussion

• QA
• Throat Singers?
• Dogs barking?
• Smoking?
• Whistling?
• Hyoid bones?

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Physics 1251 Unit 3 Session 33 Percussion

• 1' Lecture
• Percussion instruments are instruments that are
  struck.
• The timbre of their sound is determined by their
  vibration recipe.
• Their vibration recipe is determined by the modes
  of oscillation that are excited.
• Often percussion instruments do not have pitch.

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Physics 1251 Unit 3 Session 33 Percussion

• The Rain Stick
• By Seamus Heaney
• Upend the rain stick and what happens next
• Is a music that you never would have known
• To listen for. In a cactus stalk
• Downpour, sluice-rush, spillage and backwash
• Come flowing through. You stand there like a
  pipe
• Being played by water, you shake it again lightly
• And diminuendo runs through all its scales
• Like a gutter stopping trickling. And now here
  comes
• A sprinkle of drops out of the freshened leaves,

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Physics 1251 Unit 3 Session 33 Percussion

• The Rain Stick
• By Seamus Heaney
• Then subtle little wets off grass and daises
• Then glitter-drizzle, almost-breaths of air.
• Upend the stick again. What happens next
• Is undiminished for having happened once,
• Twice, ten, a thousand times before.
• Who cares if all the music that transpires
• Is fall of grit or dry seeds through cactus?
• You are like a rich man entering heaven
• Through the ear of a raindrop. Listen now again.
• From The Spirit Level (New York Noonday Press,
  1996) p3.

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Physics 1251 Unit 3 Session 33 Percussion

• The Percussion Instruments

Membranes
Strings
Drums
Piano
Hammer dulcimer
Percussion striking
Blocks, bells, shells
Bars
Plates
Cymbals, Gongs, Pans
Others
Xylophones, chimes
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Physics 1251 Unit 3 Session 33 Percussion

• 80/20The timbre of an instruments sounds depends
  on its vibration recipe.

fn n f1
Pitched
Amplitude
f1
2f1
3f1
4f1
fn m xn m f1
Unpitched
Amplitude
f01
Frequency
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Physics 1251 Unit 3 Session 33 Percussion

• The Oscillation of a Clamped Membrane

Mode (0,1)
d
f0 1 v/? v v(S/ s) f0 1 x0 1 /(p d) ?
v(S/ s) x0 1 2.405
Surface density s mass/area s density ?
thickness
Surface Tension S force/length
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Physics 1251 Unit 3 Session 33 Percussion

• Clamped Membrane vs String

Tension T
L
d
Linear density µ
Linear density µ mass/length
Surface density s mass/area
Tension T force
Surface Tension S force/length
fn n /(2 L) ? v(T/ µ) n 1, 2, 3, 4,
5, 6, 7.
fn m xn m /(p d) ? v(S/ s) x0 1 2.405
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Physics 1251 Unit 3 Session 33 Percussion

• The Oscillation of a Clamped Membrane

Mode (0,1)
f0 1 v/? v v(S/ s) f0 1 x0 1 /(p d) ?
v(S/ s) x0 1 2.405
d
Example d 0.30 m m 58 gm T 474 N
C p d .94 m Area 0.73 m2
s 0.058 kg/0.073
m20.8 kg/m2
S T/C 503. N/m
f0 1 x0 1 /(p d) ? v(S/ s)
2.405/(0.94)v(503/0.8) 64 Hz
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Physics 1251 Unit 3 Session 33 Percussion

• The Modes of Oscillation
  of an (Ideal) Clamped Membrane

Mode (0,1)
f0 1 x0 1 /(p d) ? v(S/ s) x0 1 2.405
Mode (1,1)
Mode (2,1)
f1 1 (x1 1 / x0 1) f0 1 x1 1 / x0 1 1.594
f2 1 (x2 1 / x0 1) f0 1 x2 1 / x0 1 2.136
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Physics 1251 Unit 3 Session 33 Percussion

• The Modes of Oscillation
  of a Clamped Membrane

Mode (0,1) xn m / x0 1 1
(1,1)1.594
(2,1)2.136
(0,2)2.296
(3,1)2.653
(1,2)2.918
(4,1)3.156
(2,2)3.501
(0,3)3.600
(5,1)3.652
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Physics 1251 Unit 3 Session 33 Percussion

• 80/20Membrane Acoustics
• The overtones of a circular membrane clamped at
  the edge are not harmonic and, therefore, they
  have no pitch.
• fn m (xn m /x01)f01
• The frequencies fnm of a membrane are (1)
  proportional to the square root of the ratio of
  surface tension of the head to the surface
  density ?v(S / s) and (2) inversely proportional
  to its diameter ?1/d.

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Physics 1251 Unit 3 Session 33 Percussion

• Demonstration
• Normal Modes of a Oscillation of a Clamped
  Membrane

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Physics 1251 Unit 3 Session 33 Percussion

• Ideal vs Real Membranes
• 80/20Real membranes have a lower frequencies than
  predicted for ideal membranes because of air
  loading the lowest frequencies are lowered the
  most.

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Physics 1251 Unit 3 Session 33 Percussion

• Mode Excitation
• 80/20Only those frequencies for which the modes
  were excited will appear in the vibration recipe.
• 80/20The highest frequency that can be excited by
  a mallet that is in contact with the surface for
  a period of Tcontact is
• f max 2/Tcontact

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Physics 1251 Unit 3 Session 33 Percussion

• Mode Excitation
• 80/20The highest frequency that can be excited by
  a mallet that is in contact with the surface for
  a period of Tcontact is
• f max 2/Tcontact

Tcontact ½ Tperiod 1/(2fmax )
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Physics 1251 Unit 3 Session 33 Percussion

• Demonstration
• Longitudinal Modes vs Transverse Modes for a Rod

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Physics 1251 Unit 3 Session 33 Percussion

• Longitudinal Wave (Sound Wave)

vL
h thickness
? density
E Youngs Modulus
Density ? mass/volume
vL vE/ ?
Youngs Modulus E stress/elongation
stiffness
fn n vL/(2L)
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Physics 1251 Unit 3 Session 33 Percussion

• Bending Wave in a Plate

vbend
h thickness
? density
E Youngs Modulus
Density ? mass/volume
vL vE/(.91 ?) vbend v1.8 f h vL

Youngs Modulus E stress/elongation
stiffness
fnm 0.0459 h vL( ynm /d)2
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Physics 1251 Unit 3 Session 33 Percussion

• The Modes of Oscillation
  of a Flat Cymbal

Mode (2,0) fn m / f0 1 1
(0,1)1.730
(3,0)2.328
(1,1)3.910
(4,0)4.110
(5,0)6.30
(2,1)6.71
(0,2)3.600
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Physics 1251 Unit 3 Session 33 Percussion

• 80/20 Plate Acoustics
• The overtones of a circular plate clamped in the
  center are not harmonic and, therefore, have no
  pitch.
• fn m (yn m /y20)2 f20
• The frequencies fnm of a circular plate are (1)
  proportional to the thickness ?h and (2) to the
  square root of the ratio of the stiffness and the
  density ?vE/? and (3) inversely proportional to
  the square of the diameter ?1/d2 .

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Physics 1251 Unit 3 Session 33 Percussion

• Summary
• Percussion instruments are instruments that are
  struck.
• Their vibration recipe is often not harmonic and,
  therefore, they do not have a definite pitch.
• For ideal circular edge-clamped membranes
  fnm ?(xnm /d)v(S/s).
• For circular plates free at the edge
  fnm ?h ? (ynm /d) 2 v(E/?).
• The maximum frequency excited by a mallet is f
  max 2/Tcontact.