Physics 1251 The Science and Technology of Musical Sound

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Physics 1251The Science and Technology of
Musical Sound

• Unit 1
• Session 6
• Helmholtz Resonators
• and Vibration Modes

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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• Foolscap Quiz
• What is the frequency of a simple harmonic
  oscillator that has a spring constant of k 50.0
  N/m and a mass m of 1.00 kg?

Frequency f 1/(2p)v(K/m) f
0.1592v(50.0/1.00) f 0.1592v(50.0) 1.13 Hz P
1/f 1/1.13 Hz 0.89 sec
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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• Put seat number on the Foolscap.
• Do you wish to sit here permanently?

Joe College 1/14/02 Session
1
Seat 123
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Physics 1251 Unit 1 Unit 1 Session 6
Helmholtz Resonators and Vibration Modes

• 1' Lecture
• A Helmholtz resonator is a simple harmonic
  oscillator where the mass is provided by the air
  in a narrow neck while the spring is provided by
  a volume of trapped air.
• The natural frequency of a Helmholtz Resonator is
  given by the formula
• f v/(2p)vA/ (V L)
• A area of neck v velocity of sound in air
• V volume of Bottle L length of neck

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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• 1' Lecture (contd.)
• When an object has n masses and n springs, there
  are n degrees of freedom and n modes of
  oscillation. Often each mode has a different
  frequency occasionally some frequencies are the
  same.

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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• Does Air have mass and weight?

How much?
Density ? mass/volume
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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes
Density of Air

• Density ? Mass/Volume

• ? 1. 2 kg/ m3

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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• The Bulk Modulus B is the springiness of a
  gas.
• B is equal to the change in pressure (in Pa) for
  a fractional change in volume.
• B ?p / (?V/V)

What is the increase in pressure if I decrease
the volume of trapped gas by 50? B 1.41 x 105
Pa. ?p (?V/V) B 0.50 (1.41 x105 ) 70 kPa .
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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes
Air has Springiness
F A B ( ?V/V) - (A2 B/V) x
0
0.33
0.50
20. N
30. N
0
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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes
Lowest Frequency
Highest Frequency
Largest Volume
Smallest Volume
k ? 1/V so f ? 1/vV
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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes
Turbulence
?
?
?
?
?
?
Simple Harmonic Motion of Air
?
Air mass ?
?
?
Oscillation of air mass
Air spring ?
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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• Two 500 ml Flasks
• Same Volume
• Same Length of neck
• Different diameter
• Same frequency?

?Smaller diameter
f 1/(2p)vk/m f 1/(2p)v(A2B/V) / (AL?)v
v B/? f v/(2p)vA/ (V L)
Larger ? diameter
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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• Helmholtz Resonator
• Ocarina

Open holes increase area of neck.

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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• Application of Helmholtz Resonator
• Ported Speaker Cabinet

Air Spring
Air mass
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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• Normal or Natural Modes of Oscillation

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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes
Two Masses on Two Coupled Springs
Spring ?
Mass ?
Spring ?
Mass ?
Mode 1
Mode 2
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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• 80/20A Simple Harmonic Oscillator has only one
  Normal or Natural Mode of Oscillation and only
  one frequency of oscillation.

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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• 80/20The number of Normal or Natural Modes of
  Oscillation is equal to the number of simple
  harmonic oscillators that are coupled together.

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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Modes

• 80/20Two Normal or Natural Modes of Oscillation
  are called degenerate if they have the same
  frequency.

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Physics 1251 Unit 1 Session 6 Helmholtz
Resonators and Vibration Mode

• Summary
• A Helmholtz Oscillator is a SHO comprised of an
  enclosed air volume and a narrow neck and has a
  single frequency.
• A normal or natural mode of vibration or
  oscillation is one of the fundamental ways that a
  device can move.
• The number of modes is equal to the number of
  simple harmonic oscillators in the system.
• Degeneracy means two or more normal modes have
  the same frequency.