PHY2464 - The Physical Basis of Music
PHY
-2464
PHY-2464
Physical Basis of Music
Presentation
Presentation 12
12
Helmholtz
Helmholtz Resonators
Resonators
Adapted
Adapted from
from Sam
Sam Matteson’s
Matteson’s
Unit
1
Session
Unit 1 Session 66
Sam
Sam Trickey
Trickey
Feb.
Feb. 22,
22, 2005
2005
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Pres. 12 Helmholtz Resonators
Outlook •
Start with a reminder about analyzing
components of musical instrument (cultural
artifact explored “post facto”)
•
Reminder about plucked, struck, string, and
bowed string instruments: the hollow body is a
common element.
•
The hollow body has key roles in both the
“frequency filter” subsystem and the “antenna”
subsystem.
PHY2464 - The Physical Basis of Music
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Pres. 12 Helmholtz Resonators
Reminder about the four primary subsystems that
comprise a musical instrument:
(1) a mechanical energy source;
(2) a frequency generator;
(3) a frequency filter system;
(4) an antenna.
♩♪♫
~
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f1 f2 f3 f4
~
fn
Pres. 12 Helmholtz Resonators
Fact: The “frequency generator” component is
basically the same for all stringed instruments
(though details of exciting it differ widely).
Frequency of nth harmonic on a string under
tension T with linear density µ :
fn = [n/(2L)] × √(T/ µ)
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 12 Helmholtz Resonators
Fact: Coupling of the strings to the top and bottom
plates accentuates or attenuates various
frequencies.
♩♪♫
~
f1 f2 f3 f4
fn
~
Top & bottom plates plus cavity
acoustics filter harmonics
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How can we model the response of the body and air
cavity in its filtering of the harmonics?
Recall – guitar
cross-section
Neck
Truss rod
Sound hole
Bridge
Body
Brace
PHY2464 - The Physical Basis of Music
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Violin cross section (bass bar missing on this side)
Finger board
Scroll & Tuning
Body
pegs
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f-hole
Sound Post
Pres. 12 Helmholtz Resonators
Vibration is transmitted to the body and air cavity by
the action of the Bridge. What are the basic
properties of the air cavity?
Rocking
motion
Helmholtz
Resonance
Sound post
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 12 Helmholtz Resonators
Jogging the memory:
What are the frequency f and period P of a
simple harmonic oscillator (SHO) that has a
spring constant of k = 50.0 N/m and a mass m of
4.00 kg?
Frequency = f = 1/(2π)√(K/m)
f = 0.1592√(50.0/4.00)
f = 0.1592√(12.5) = 0.563 Hz
P =1/f = 1/0.563 Hz = 1.777 sec
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Does Air have mass and weight?
How much?
Density = ρ = mass/volume
PHY2464 - The Physical Basis of Music
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Density of Air
• Density = ρ =
Mass/Volume
• ρair = 1. 2 kg/ m3
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The “Bulk Modulus” B is a measure of 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)
Bair = 1.41 x 10 5 Pa
What is the increase in pressure if I decrease the volume of
trapped gas by 50%?
∆p = (∆V/V) B = 0.50 (1.41 x10 5 ) = 70 kPa .
PHY2464 - The Physical Basis of Music
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Springiness of air
F = A B ( ∆V/V) = - (A2 B/V) y
∆V
Fair
V
∆V/V:
Force:
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0
0
∆V
Fair
0.33
20. N
0.50
30. N
Pres. 12 Helmholtz Resonators
Lowest Frequency
Highest
Frequency
k ∝ 1/ V
k ∝ 1/V
Smallest Volume
Largest kVolume
∝ 1/ V
k ∝ 1/ V
so
f ∝ 1/ V
PHY2464 - The Physical Basis of Music
Pres. 12 Helmholtz Resonators
Simple Harmonic
Motion of Air (Dr.
Sam Matteson perperforming)
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↕
⃕
⃔
⃔ ⃕
⃕
⃕
Turbulence
⃔
⃔
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Air “mass” →
OscillationAir
of “spring”
air
→
mass
Pres. 12 Helmholtz Resonators
Because the “Bulk
Modulus”
Modulus” B indicates
the springiness of a gas,
B also is related to the
speed of wave
propagation in the gas
vsound = B / ρ
PHY2464 - The Physical Basis of Music
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Two 500 ml Flasks
• Same Volume V
• Same Length of neck L
• Different diameter
←Smaller
Larger →
diameter
diameter
Same frequency?
f = 1/(2π
1/(2π)√[k/m]
f = 1/(2π
(ALρ)]
1/(2π)√[(A2B/V) / (ALρ
v= √ B/ρ
B/ρ
f = v/(2π)√[A/ (V L)]
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Helmholtz Resonator
•
Ocarina
Open holes increase area of “neck.”
PHY2464 - The Physical Basis of Music
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Application of Helmholtz Resonator:
Ported Speaker Cabinet
Air “Spring”
Air “mass”
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A String has more than one mode of vibration
(recall harmonics)
Does a Helmholtz Resonator also?
If the spring analogy works, yes.
Normal or Natural Modes of Oscillation
PHY2464 - The Physical Basis of Music
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Two Masses on Two Coupled Springs
Spring ———→
Mass ————→
Spring ————→
Mass ————→
Mode 1
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Mode 2
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A Simple Harmonic Oscillator has only one
Normal or Natural Mode of Oscillation and only
one frequency of oscillation.
The number of Normal or Natural Modes of
Oscillation is equal to the number of simple
harmonic oscillators that are coupled together.
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 12 Helmholtz Resonators
An informative web site on the relationship between
mass-spring vibrations and
Helmholtz Resonators:
http://www.phys.unsw.edu.au/~jw/Helmholtz.html
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Pres. 12 Helmholtz Resonators
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.