ELEN E4896 MUSIC SIGNAL PROCESSING
Lecture 2:
Acoustics
1. Acoustics, Sound & the Wave Equation
2. Musical Oscillations
3. The Digital Waveguide
Dan Ellis
Dept. Electrical Engineering, Columbia University
dpwe@ee.columbia.edu
E4896 Music Signal Processing (Dan Ellis)
http://www.ee.columbia.edu/~dpwe/e4896/
2013-01-28 - 1 /16
1. Acoustics & Sound
• Acoustics is the study of physical waves
• Waves transfer energy without permanent
displacement of matter
• Common math for different media
gas, liquid, solid, EM
• Intuition: Pulse going down a rope
E4896 Music Signal Processing (Dan Ellis)
2013-01-28 - 2 /16
The Wave Equation
• For 1-D medium with displacement y(x, t):
2
c
curvature
2
y
=
x2
2
y
t2
acceleration
simple to derive from freshman physics...
y
y(0,t) = m(t)
y+(x,t)
E4896 Music Signal Processing (Dan Ellis)
x
y(L,t) = 0
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The Wave Equation
2
c
• Solution:
2
y
=
2
x
y(x, t) = y + (x
2
y
t2
ct) + y (x + ct)
+
y and rightward-moving y
sum of leftward-moving
traveling waves
shape does not change (set by initial conditions)
y+
c
y
y-
c
y(x,t)
= y+ + yx
E4896 Music Signal Processing (Dan Ellis)
2013-01-28 - 4 /16
Terminations & Reflections
• Boundary conditions include fixed points
e.g. held ends of string
• Superposition of
traveling waves
must match
constraints
y+
y(x,t)
= y+ + y–
hence reflections
• Any impedance change
y–
results in some reflection
x=L
energy loss...
http://www.youtube.com/watch?v=YOi8suJTwx8#t=1m35s
E4896 Music Signal Processing (Dan Ellis)
2013-01-28 - 5 /16
1-D Waveguides
• Plucked/struck string
tension S
mass/length ε
guitar, piano ...
L
• Acoustic tube
π2 S
2
ω = 2
L ε
pressure
e.g. clarinet or trumpet
vocal tract
• Solid bar
x
xylophone, ...
E4896 Music Signal Processing (Dan Ellis)
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2. Musical Oscillations
• (Pseudo) periodic oscillation
is central to musical pitch
Frequency
4000
3000
2000
1000
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Time
• Musical instruments create pitch
in different ways...
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Simple Harmonic Motion
• Basic 2nd order mechanical oscillation
ẍ =
2
x
x = A cos( t + )
resonance
• Spring + mass + damper
e.g. tuning fork
F = kx
m
ω2 =
ζ
x
k
m
• Not great as a musical instrument
only fundamental
E4896 Music Signal Processing (Dan Ellis)
low amplitude
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Relaxation Oscillator
• Alternating states
‘closed state’ collects energy from steady source
‘open state’ releases energy, then reverts to closed
• e.g.Vocal folds
p
u
• Oscillation period depends on tension
easy to adjust
hard to keep stable
E4896 Music Signal Processing (Dan Ellis)
2013-01-28 - 9 /16
Strings
• Canonical wave equation example
tension S
mass/length ε
L
π2 S
2
ω = 2
L ε
guitar (plucked)
piano (struck)
violin (bowed ...)
• Control of period
vary length L (frets)
vary tension S and/or length L (piano)
• Initial conditions = excitation
E4896 Music Signal Processing (Dan Ellis)
2013-01-28 - 10/16
Air Column
• Wave equation in air
U0 ejωt
kx = π
x=λ/2
pressure = 0 (node)
vol.veloc. = max
(antinode)
pressure waves traveling in tube
resonance of tube depends on length
coupled energy input
• Clarinet, oboe, organ, flute
finger holes disrupt waveguide (scattering)
first reflection determines oscillation period
E4896 Music Signal Processing (Dan Ellis)
2013-01-28 - 11/16
3. Digital Waveguides
• 1-D waveguide is easily discretized
spatial sampling
Julius Smith, 1992...
time sampling
nonlinear
element
scattering junction
(tonehole)
energy
acoustic
waveguide
ω=
πc
2L
(quarter wavelength)
waveguide ➝ delay line
scattering ➝ low-pass feedback
final load ➝ instrument body resonances
nonlinear function for energy input
https://ccrma.stanford.edu/~jos/wg.html
E4896 Music Signal Processing (Dan Ellis)
2013-01-28 - 12/16
Digital Waveguides
• Direct physical model + simplifications
Nut
-1
Initialize with pluck shape
-1
y+(0,t)
= –y-(0,t)
KarplusStrong
Bridge dispersion
+ radiation load
String Waveguide
y+(x,t)
Delay Lines
y-(x,t)
hreflec
y-(L,t)
= hreflec(t) * y+(L,t)
Delay z-L
Initialize with random values
s(t)
s(t)
L = SR/f0
hLP
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Waveguide Simulation
• Karplus-Strong Plucked String algorithm
Initialize with random values
Delay z-L
L = SR/f0
Lowpass
• Direct implementation of traveling waves
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2013-01-28 - 14/16
Karplus-Strong in Pd
• Pd’s delwrite~ / delread~ implement the
delay line
http://blog.loomer.co.uk/2010/02/karplus-strong-guitar-string-synthesis.html
E4896 Music Signal Processing (Dan Ellis)
2013-01-28 - 15/16
Summary
• Wave equation:
Simple physics leads to oscillations
• Musical acoustics:
Different ways to control steady tones
• Digital waveguides:
Natural, efficient simulation of tones
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