Transcript Physics 1251 The Science and Technology of Musical Sound

Physics 1251
The Science and Technology
of Musical Sound
Unit 1
Session 6
Helmholtz Resonators
and Vibration Modes
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/(2π)√(K/m)
f = 0.1592√(50.0/1.00)
f = 0.1592√(50.0) = 1.13 Hz
P =1/f = 1/1.13 Hz = 0.89 sec
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Put seat number on the Foolscap.
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Joe College
1/14/02
Seat #123
Session #1
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/(2π)]√[A/ (V L)]
A: area of neck
v: velocity of sound in air
V: volume of Bottle
L: length of neck
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
1′ Lecture (cont’d.):
•
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.
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Does Air have mass and weight?
How much?
Density = ρ = mass/volume
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Density of Air
• Density = ρ =
Mass/Volume
• ρ = 1. 2 kg/ m3
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 = Δp / (ΔV/V)
What is the increase in pressure if I decrease the
volume of trapped gas by 50%? B = 1.41 x 105 Pa.
Δp = (ΔV/V) B = 0.50 (1.41 x105 ) = 70 kPa .
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Air has “Springiness”
ΔV
V
ΔV/V:
Force:
F = A B ( ΔV/V) = - (A2 B/V) x
0
0
ΔV
0.33
20. N
0.50
30. N
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Lowest Frequency
Highest Frequency
Largest Volume
k ∝ 1/V
so f ∝ 1/√V
Smallest Volume
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Turbulence
↕
⃔
Simple Harmonic
Motion of Air
Air “mass” →
OscillationAir
of“spring”
air
→
mass
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Two 500 ml Flasks
• Same Volume
• Same Length of neck
• Different diameter
←Smaller
Larger →
diameter
diameter
Same frequency?
f = 1/(2π)√[k/m]
f = 1/(2π)√[(A2B/V) / (ALρ)]
v= √ B/ρ
f = v/(2π)√[A/ (V L)]
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Helmholtz Resonator
•
Ocarina
Open holes increase area of “neck.”
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Application of Helmholtz Resonator:
Ported Speaker Cabinet
Air “Spring”
Air “mass”
Physics 1251
Unit 1 Session 6
Helmholtz Resonators and Vibration Modes
Normal or Natural Modes of Oscillation
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
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.
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.
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.
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.