Fundamentals of Acoustics
The Nature of a Sound Event
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Sound consists of vibrations of air molecules
Air molecules are analogous to tiny superballs
Sound occurs when air molecules are disturbed
and made to ricochet off of each other
The Nature of a Sound Event

The ricochets cause the density of the air
molecules to oscillate
Rarefied
Normal
Compressed
The Nature of a Sound Event

The ricochets cause the density of the air
molecules to oscillate back and forth
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Wave Types
Sound consists of longitudinal waves
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propagation
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oscillation
The wave’s
oscillation is in
the same
direction as its
propagation
Water waves are transverse waves
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propagation
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oscillation
The wave’s
oscillation is
perpendicular to
the direction of its
propagation
Sound
Propagation
Sound waves
propagate in a sphere
from the sound
source (try to imagine
a spherical slinky).
Note that the
molecules themselves
are not travelling.
What spreads is the
energy of the wave.
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Sound Perception

Speed of sound (in air):
1128 ft./sec (344 m/sec)

When sound waves reach the eardrum, they
are transduced into mechanical energy in
the middle ear

The mechanical motion is transduced into
electrical current in the inner ear. The
auditory nerves interpret the current as
sound
Sound Wave Plots

Sound waves are typically represented with
molecular density as a function of time
compressed
normal
time
rarefied
molecular density
Music vs. Noise
Musical sounds are typically periodic – the wave repeats regularly
repeats
Sine wave
Though they don’t exist in
nature, sine waves are often
useful for demonstrating
properties of sounds
Noise is aperiodic – there is no repeating pattern
Noise
Properties of a Musical Event
A musical event can be described by four properties.
Each can be described subjectively, or objectively (in
terms of measured properties)
Subjective
Pitch
Objective
Frequency
Volume
Amplitude/Power/Intensity
Timbre
Overtone content
Duration in beats
Duration in time
Frequency/Pitch
Frequency is measured in cycles per second, or Hertz (Hz)
f = 2 Hz
Wavelength (l), the
distance between
corresponding points
on the wave, is the
inverse of frequency.
l =
c
f
=
1000 ft./sec.
2 cyc./sec.
one second
l
=
500 ft./cyc.
Frequency/Pitch
Middle A = 440 Hz
l = 2.3 ft.
20 Hz
l = 50 ft.
<
frequencies
audible to
humans
< 20,000 Hz (20 kHz)
l = 0.05 ft.
Sound wavelengths are significantly larger than light wavelengths
Waves reflect from a surface if its
height/width is larger than the wavelength
Waves refract around surface if the
surface dimensions are smaller than
the wavelength
This explains why we can hear sound from around corners,
but cannot see around corners:
Light wavelengths are far too small to refract around any
visible surface
Our Pitch Perception is
Logarithmic
Equivalent pitch intervals are perceived according to an
equivalent change in exponent, not in absolute frequency
For example, we hear an equivalent pitch class with every
doubling of frequency (the interval of an octave)
Frequencies of successive octaves of concert A
55
110
220
440
880
1760
3520
55 x 2 0
55 x 2 1
55 x 2 2
55 x 2 3
55 x 2 4
55 x 2 5
55 x 2 6
Our Pitch System is Based on
Equal Division of the Octave
12 Tone Equal Temperament –
the octave is divided into twelve equal increments
We can describe an octave by:
• choosing a starting frequency
n/12
• multiply it by 2
for n = 0 to 11
A
220
0
x 212
220
A#
220
1
x 212
233
B
220
2
x 212
247
C
220
3
x 212
261.6
C#
220
4
x 212
277
D
220
5
x 212
293.6
D#
220
6
x 212
311
E
220
7
x 212
329.6
F
220
8
x 212
349.2
F#
220
9
x 212
370
G
220
10
x 212
392
Higher octaves may be created by doubling each frequency
Lower octaves may be created by halving each frequency
G#
220
11
x 212
415.3
Phase
Phase = “the position of a wave at a certain time”
If two waveforms at the same frequency do
not have simultaneous zero-crossings, we say
they are “out of phase”
Wave 1 + Wave 2
Wave 1
Wave 2
Two waves at the
same frequency but
different phase
In terms of sound perception, phase can be critical or imperceptible,
as we’ll see...
Loudness
Loudness is related to three measurements:
• Power
• Pressure
• Intensity
All three are related to changes in sound
pressure level (molecular density)
Molecular Motion is Stationary




As sound travels, molecules are not traveling with
the sound wave
What is traveling is an expanding sphere of energy
that displaces molecules as it passes over them
How strong is the force behind this energy wave?
The more force is contained in a sound wave, the
greater its perceived loudness.
Power
Power = the amount of time it takes to do work
(exert force, move something)
Power is measured in watts, W
There are two difficulties in measuring sound power levels.
The range of human hearing encompasses many millions of watts.
Sound power level is also relative, not absolute. Air
molecules are never completely motionless.
Given these two difficulties, sound power levels are
measured on a scale that is comparative and logarithmic,
the decibel scale.
Logarithmic Scale
Logarithm = exponent
(an exponent is typically an integer, a logarithm not necessarily)
102 = 100
103 = 1000
log10100 = 2
log101000 = 3
102.698 = 500
log10500 = 2.698
102.875 = 750
log10750 = 2.875
Logarithms allow us to use a small range of
numbers to describe a large range of numbers
The Decibel Scale

The decibel scale is a comparison of a
sound’s power level with a threshold level
(the lowest audible power level of a sine
tone at 1 kHz).
Threshold (W0):
W0 = 10-12 watts
Power level of a given sound in watts, LW(dB):
L W (dB) = 10*log10 (W/W0 )
Decibels
Typical power levels:
Soft rustling leaves
Normal conversation
Construction site
Threshold of pain
10 dB
60 dB
110 dB
125 dB
Halving or doubling sound power level results in a change of
3 dB.
For example, a doubling of the threshold level may be
calculated:


 
 410 12 
LW(dB) = 10log10 
 10log10  2  3.01 dB
12 
 
 210

Thus, a power level of 13 dB is twice that of 10 dB. A
power level of 60 dB is half that of 63 dB, and so on.
Pressure changes
The degree of fluctuation present in a vibrating object
Peak pressure level:
Maximum change in sound pressure level
(more generally: in a vibrating system, the
maximum displacement from equilibrium position)
The amplitude level fluctuates with the wave’s oscillation.
Thus, power is the cause, pressure change is the result
Pressure changes
Also may be described as changes in sound pressure level
(molecular density).
Pressure level is measured in Newtons per square meter (N/m 2 )
-5
Threshold: 2 x 10 N/m2 (p0 )
There is a direct relationship between pressure and power
levels:
For any propagating wave (mechanical, electric, acoustic, etc.)
the energy contained in the wave is proportional to the square
of its pressure change.
Pressure changes are also expressed in decibels, but in a way
that describes an equivalent change in power level:
L W (dB) = 10*log10(W/W0) = 10*log10(p/p0)2 = 20*log10(p/p0)
logmn = nlogm
This is how pressure is
measured
Pressure changes
In audio parlance, “amplitude” (the degree of
pressure change) is often equated with “loudness.”
The reason is that modifications to volume are made by
adjusting the amplitude of electrical current sent to an
amplifier.
But perceived loudness is actually based on power level plus
the distance of the listener from the source.
Intensity
Power corresponds to the sphere of energy expanding
outward from the sound source
The power remains constant, spread evenly over the
surface of the sphere
Perceived loudness depends primarily on the sound
power level and the distance from the sound event
Power combined with distance is intensity, I,
measured in watts per square meter (W/m2 ).
Intensity is also measured in decibels:
L I (dB) = 10*log 10(I/I 0 )
-12
I0 = 10
W/m 2
Timbre
The perceived difference in sound quality when two
different instruments play at the same pitch and loudness
Sine waves are useful as demonstrations because they are a
wave with one frequency only, thus they are often termed
pure tones
Natural sounds are composed of multiple frequencies
To understand how a wave can be composed of multiple
frequencies, we can consider the behavior of a wave in a
bounded medium, such as a string secured at both ends (or
air vibrating within a pipe)
Timbre
When we pluck a string, we initiate wave motion
The wavelength is twice the length of the string
The perceived pitch is the fundamental, the speed
of sound divided by the wavelength
Timbre
This curved shape represents the
string’s maximum deviation
It’s more accurate to think of it as a
series of suspended masses (kind of like
popcorn strung together to hang on a
Christmas tree).
Timbre
Each suspended mass can vibrate independently.
Thus, many simultaneous vibrations/frequencies
occur along a string.
When a string is first plucked, it produces a
potentially infinite number of frequencies.
Timbre
Eventually, the bounded nature of the string confines wave
propagation and the frequencies it can support
Only frequencies that remain in phase after one propagation
back and forth can be maintained; all other frequencies are
cancelled out
Only frequencies based on integer subdivisions of the string’s
length, corresponding to integer multiples of the fundamental,
can continue to propagate
Timbre
These frequencies are called harmonics
NOTE:
These frequencies are
equally spaced
Therefore, they do
not all produce the
same pitch as the
fundamental
Therefore, other
frequencies are
introduced
…etc.
Timbre

Harmonics are well known to many
instrumentalists
–
–
Strings
Brass
Timbre

The first six harmonics are often the
strongest:
220
440
660
880
1100
1320
Fundamental
Octave
Perfect fifth
Octave
Major third
Perfect fifth

People can learn to “hear out” harmonics
Timbre
Instruments and natural sounds usually
contain many frequencies above the
fundamental
 These additional frequencies, as part of the
total sound, are termed partials
 The first partial is the fundamental

Timbre
The first partial is the fundamental
 Other terms are also used
 Overtones are partials above the
fundamental (the first overtone is the
second partial)
 Harmonics are partials that are integer
multiples of the fundamental

The Spectrum

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


Jean Baptiste Fourier (1768-1830) discovered a
fundamental tenet of wave theory
All periodic waves are composed of a series of
sinusoidal waves
These waves are harmonics of the fundamental
Each harmonic has its own amplitude and phase
The decomposition of a complex wave into its
harmonic components, its spectrum, is known as a
Fourier analysis
The Spectrum
It is often more useful to represent complex
waveforms with a spectral plot as opposed to a time
domain plot
=
time domain
spectral domain
amplitude as a function of time
amplitude as a function of frequency
Sound in Time

Our perception of sound and music events is
determined by the behavior of frequency
and loudness over time
Sound in Time

All instruments can be characterized by
changes in amplitude over time (the
envelope)
loudness
trumpet
bowed violin
harp
Changes in amplitude often correspond with changes in
frequency content...
time
Sound in Time
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

Most instrument’s sound begins with an initial
transient, or attack, portion
The transient is characterized by many high
frequencies and noise
Example: the scraping of a bow or the chiff of
breath
An instrument’s distinctiveness is determined
primarily by the transient portion of its sound
Sound in Time
Following the transient, instruments usually
produce a steady-state, or sustained, sound
 The steady state is characterized by

–
–
Periodicity
Harmonic spectrum
The Spectrogram
Most natural sounds (and musical instruments) do not
have a stable spectrum.
Rather, their frequency content changes with time.
The spectrogram is a three-dimensional plot:
Vibraphone note at 293 Hz (middle D)
2) frequency
3) power of a given frequency (darkness level)
1) time
The instrument’s sound is characterized by the fundamental at 293 Hz and the fourth harmonic at 1173 Hz.
The attack also contains noise below 2 kHz, the tenth harmonic at 2933 Hz and the seventeenth harmonic at 4986 Hz.
Once the steady state portion sets in, the highest harmonic fades first, followed by a fading of the fundamental.
Localization



The auditory system localizes events through
interaural time delay – the sound wave reaches the
nearer ear a few milliseconds before it reaches the
farther ear
For stereo systems, using delay for localization is
impractical because it requires people to listen
from a “sweet spot”
Localization effects are simulated through
differences in loudness
Localization



In a multi-speaker system, a sound emanating
from one speaker will be localized at that speaker
A sound produced at equal volume from two
speakers will be perceived as a “phantom image”
placed in space between them
Changing the volume balance between two
speakers will cause the phantom image to “drift”
towards the louder speaker
Measurement and Perception
Our perception of auditory events is based
on all these measurements in combination
 And more
 An auditory event may be more than the
sum of its parts

Measurement and Perception
Phase

Changing the phase of components in a steadystate tone produces no perceptible change in
sound, although the shape of the wave may change
noticeably
Measurement and Perception
Phase




The behavior of components in the attack segment is likely
to be far more complex than in the steady state segment
Changing the phase of attack components can change the
character of the attack
Solo performance sounds different from group
performance because no two players can ever sound at
exactly the same time; thus the attack is blurred
Since an instrument’s characteristics are defined primarily
by the attack, the phase of attack components is critical
Measurement and Perception
Timbre





We have discussed timbre as the result of overtone content
It is also judged by the sound’s envelope
Research in sound synthesis has shown the envelope shape
to be more definitive than an exact match of overtone
content
The attack portion is critical—a faster attack can be
confused with “brightness” (more high frequency overtones)
Considerable research has gone into the creation of “timbre
space,” a multi-dimensional plot in which timbres are
classified according to overtone content, envelope and
attack time
Measurement and Perception
Loudness
While intensity is the measurement most closely correlated to
loudness, the perception of volume is based on a number of
factors, not all of them entirely measurable.
Measurement and perception
Loudness
Perceived loudness is frequency-dependent
Perceived equal
loudness of sine
tones
This is why
many receivers
have a
Loudness knob
Equal loudness curves (Fletcher, Munson, 1930s).
Measurement and perception
Loudness
Perceived loudness is frequency-dependent
Within close frequency ranges, perceived loudness is
proportional to the cube root of intensity
Two violins playing the same pitch will generate twice the
intensity of one violin, but will not sound twice as loud
To achieve twice the volume, eight violins are required
Measurement and perception
Loudness
Perceived loudness is bandwidth-dependent
Increasing the bandwidth (component frequency
content) of a sound makes it sound louder, even if the
intensity remains constant
Despite many efforts, no one has suceeded in
creating a definitive perceptual scaling system for
loudness
Measurement and Perception
Loudness
Some have argued that estimation of loudness is not
automatic (measurable), but depends on a number of
higher-level estimations of distance, import, context, etc.
Hermann Helmholtz, On the Sensations of Tone (1885):
…we are exceedingly well trained in finding out by our
sensations the objective nature of the objects around us,
but we are completely unskilled in observing these
sensations per se; and the practice of associating them
with things outside of us actually prevents us from being
distinctly conscious of the pure sensations.
Measurement and Perception
Conclusion
Objective measurements can tell us more about sound
events
By the same token, they give us insight into what we
don’t know
This course will examine music in technical terms
This examination will give us some new insights
It will also give us an idea of where music crosses the
barrier from the objective (acoustics) to the subjective
(magic?)