PHY2464 - The Physical Basis of Music
PHY
-2464
PHY-2464
Physical Basis of Music
Presentation
Presentation 20
20
Percussion
Percussion Instruments
Instruments -- II
Adapted
Adapted from
from Sam
Sam Matteson’s
Matteson’s
Unit
Unit 33 Session
Session 33
33
Sam
Sam Trickey
Trickey
Mar.
Mar. 18,
18, 2005
2005
PHY-2464
PHY
PHY-2464
Pres. 20 Percussion Instruments - I
Percussion = striking
We have already seen struck strings.
Now we turn to struck membranes,
bars, and other rather stiff objects
such as bells
&date
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 20 Percussion Instruments - I
The Percussion Instruments (*already discussed)
Strings*
Membranes
Drums
Piano*
Hammer dulcimer*
Cymbals, Gongs, Pans
PHYPHY-2464
Blocks, bells,
shells
Bars
Plates
Xylophones, chimes
Others
Pres. 20 Percussion Instruments - I
Amplitude
Amplitude
Recall! The timbre of an instrument’s sounds depends
on its spectrum (“vibration recipe”). An irregular
spectrum lacks characteristic pitch.
fn = n f1
f1
3f1
2f1
fn m = xn m f1
f01
Frequency
&date
Pitched
4f1
Unpitched !
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Oscillation of a Clamped Membrane
Surface density σ
Mode: (0,1)
d
f0 1 = v/λ
v/λ; v = √(S/ σ)
Surface Tension S
f0 1 = x0 1 /(π
/(π d) x √(S/ σ)
x0 1 = 2.405
Surface density σ= mass/area
σ= density x thickness
Surface Tension S= force/length
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Clamped Membrane compared with String
Surface density σ
d
Tension T
L
Linear density µ
Surface Tension S
• Surface density
σ= mass/area
• Surface Tension
S=
force/length
fn m = xn m /(π
/(π d) x √(S/ σ)
x0 1 = 2.405
&date
• Linear density
µ= mass/length
• Tension
T= force
fn = n /(2
/(2 L) x √(T/ µ)
n = 1, 2, 3, 4, 5, 6, 7…
7….
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Oscillation of a Clamped Membrane –
numerical example
Surface density σ
Mode: (0,1)
d
Surface Tension S
f0 1 = v/λ
v/λ; v = √(S/ σ)
f0 1 = x0 1 /(π
/(π d) x √(S/ σ)
x0 1 = 2.405
Example: d = 0.30 m; m = 58 gm; T = 474 N
• C = π d = .94 m • Area= 0.73 m2
• σ= 0.058 kg/0.073 m2=0.8 kg/m2
• S = T/C = 503. N/m
f0 1 = x0 1 /(π
/(π d) x √(S/ σ) = 2.405/(0.94)√(503/0.8)=
(503/0.8)= 64 Hz
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Oscillation Modes of an (Ideal) Clamped Membrane
Surface density σ
Mode: (0,1)
f0 1 = x0 1 /(π
/(π d) x √(S/ σ)
Surface Tension S
Mode: (1,1)
f1 1 = (x1 1 / x0 1) f0 1
x1 1 / x0 1 = 1.594
&date
x0 1 = 2.405
Mode: (2,1)
f2 1 = (x2 1 / x0 1) f0 1
x2 1 / x0 1 = 2.136
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Modes of Oscillation of a Clamped Membrane
Fx,y: x = # radial nodes, y = # circular nodes
Mode: (0,1)
(1,1)
(2,1)
(0,2)
(3,1)
xn m / x0 1 : 1
1.594
2.136
2.296
2.653
(1,2)
(4,1)
(2,2)
(0,3)
(5,1)
2.918
3.156
3.501
3.600
3.652
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Key FACTs about Membrane Acoustics:
&date
•
The overtones of a circular membrane clamped at
the edge are not harmonic and, therefore, they
have no well-defined 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
[√(S / σ)] and (2) inversely proportional to the
diameter [1/d].
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Demonstration:
Normal Modes of a Oscillation of a Clamped
Membrane
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Ideal vs. Real Membranes:
Key FACTs: Real membranes have lower
frequencies than predicted for ideal membranes
because of air loading. The lowest frequencies are
lowered the most.
&date
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Mode Excitation:
An observation of the obvious: only those
frequencies for which the modes were excited
will appear in the spectrum (“vibration recipe”).
Key FACT: The 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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Mode Excitation by mallet contact:
f max= 2/Tcontact
Tcontact = ½ Tperiod= 1/(2fmax )
&date
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Demonstration:
Longitudinal Modes vs Transverse Modes for a Rod
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Longitudinal Wave (Sound Wave)
vL
h: thickness
ρ: density
Y: Young’s Modulus
• Density
ρ= mass/volume
• Young’
Young’s Modulus
Y = stress/elongation
= stiffness
&date
• vL = √Y/ ρ
fn= n vL/(2L)
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Bending Wave in a Plate
vbend
h: thickness
ρ: density
Y: Young’s Modulus
• Density
ρ= mass/volume
• vL = √Y/(.91 ρ)
• Young’
Young’s Modulus
Y= stress/elongation
=stiffness
PHYPHY-2464
vbend = √[1.8 f h vL ]
fnm = 0.0459 h vL( ynm /d)2
Pres. 20 Percussion Instruments - I
Modes of Oscillation of a Flat Cymbal
&date
Mode: (2,0)
(0,1)
(3,0)
(1,1)
(4,0)
fn m / f0 1 : 1
1.730
2.328
3.910
4.110
(5,0)
(2,1)
(0,2)
6.30
6.71
3.600
PHY2464 - The Physical Basis of Music
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Key FACTs about Plate Acoustics:
•
The overtones of a circular plate clamped in the
center are not harmonic and, therefore, have no
well-defined pitch.
fn m = (yn m /y20)2 f20
•
The frequencies fnm of a circular plate are (1)
proportional to the thickness [h], (2) proportional
to the square root of the ratio of the stiffness and
the density [√E/ρ ] ,and (3) inversely proportional
to the square of the diameter [1/d2 ].
PHYPHY-2464
Pres. 20 Percussion Instruments - I
Summary:
•
•
•
•
•
&date
Percussion instruments are those whose
primary means of sound production is by being
struck.
Their spectru is often not harmonic and,
therefore, they do not have well-defined pitch.
For ideal circular edge-clamped membranes:
fnm = (xnm /d)√(S/σ).
For circular plates free at the edge:
fnm = h x (ynm /d) 2 √(E/ρ).
The maximum frequency excited by a mallet is
f max= 2/Tcontact.