Fundamentals of Acoustics

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Fundamentals of Acoustics
2
The Nature of a Sound Event

• 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

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The Nature of a Sound Event

• The ricochets cause the density of the air
  molecules to oscillate

Normal
Compressed
Rarefied
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The Nature of a Sound Event

• The ricochets cause the density of the air
  molecules to oscillate back and forth

5
Wave Types

• Sound consists of longitudinal waves

The waves oscillation is in the same direction
as its propagation
propagation
oscillation
Water waves are transverse waves
The waves oscillation is perpendicular to the
direction of its propagation
propagation
oscillation
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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

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Sound Wave Plots

• Sound waves are typically represented with
  molecular density as a function of time

compressed
normal
time
rarefied
molecular density
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Music vs. Noise

• Musical sounds are typically periodic the wave
  repeats regularly

repeats
Sine wave
Though they dont exist in nature, sine waves are
often useful for demonstrating properties of
sounds
Noise is aperiodic there is no repeating pattern
Noise
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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
Objective
Pitch
Frequency
Volume
Amplitude/Power/Intensity
Timbre
Overtone content
Duration in beats
Duration in time
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Frequency/Pitch
Frequency is measured in cycles per second, or
Hertz (Hz)
one second
f 2 Hz
Wavelength (l), the distance between
corresponding points on the wave, is the inverse
of frequency.
l
1000 ft./sec.



500 ft./cyc.
l
2 cyc./sec.
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Frequency/Pitch
Middle A 440 Hz
2.3 ft.
l
frequencies audible to humans
lt
20 Hz
lt
20,000 Hz (20 kHz)
l 0.05 ft.
l 50 ft.
Sound wavelengths are significantly larger than
light wavelengths
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Waves reflect from a surface if its height/width
is larger than the wavelength
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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
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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
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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

A
A
B
C
C
D
D
E
F
F
G
G
220
233
247
261.6
277
293.6
311
329.6
349.2
370
392
415.3
Higher octaves may be created by doubling each
frequency
Lower octaves may be created by halving each
frequency
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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
Two waves at the same frequency but different
phase
Wave 2
In terms of sound perception, phase can be
critical or imperceptible, as well see...
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Loudness
Loudness is related to three measurements

• Power

• Pressure

• Intensity

All three are related to changes in sound
pressure level (molecular density)
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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.

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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.
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Logarithmic Scale
Logarithm exponent
(an exponent is typically an integer, a logarithm
not necessarily)
102 100
log10100 2
103 1000
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
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The Decibel Scale

• The decibel scale is a comparison of a sounds
  power level with a threshold level (the lowest
  audible power level of a sine tone at 1 kHz).

Threshold (W0)
Power level of a given sound in watts, LW(dB)
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Decibels
Typical power levels
Soft rustling leaves
10 dB
Normal conversation
60 dB
Construction site
110 dB
Threshold of pain
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
LW(dB)
3.01 dB
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.
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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 waves
oscillation. Thus, power is the cause, pressure
change is the result
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Pressure changes
Also may be described as changes in sound
pressure level (molecular density).
Threshold
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 (dB) 10log10(W/W0)
10log10(p/p0)2
20log10(p/p0)
W
This is how pressure is measured
logmn nlogm
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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.
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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
Intensity is also measured in decibels
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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)
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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
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Timbre
This curved shape represents the strings maximum
deviation
Its more accurate to think of it as a series of
suspended masses (kind of like popcorn strung
together to hang on a Christmas tree).
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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.
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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 strings length, corresponding to integer
multiples of the fundamental, can continue to
propagate
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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.
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Timbre

• Harmonics are well known to many instrumentalists
• Strings
• Brass

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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

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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

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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

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The Spectrum

• 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

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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
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Sound in Time

• Our perception of sound and music events is
  determined by the behavior of frequency and
  loudness over time

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Sound in Time

• All instruments can be characterized by changes
  in amplitude over time (the envelope)

loudness
time
trumpet
bowed violin
harp
Changes in amplitude often correspond with
changes in frequency content...
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Sound in Time

• Most instruments 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 instruments distinctiveness is determined
  primarily by the transient portion of its sound

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Sound in Time

• Following the transient, instruments usually
  produce a steady-state, or sustained, sound
• The steady state is characterized by
• Periodicity
• Harmonic spectrum

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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 instruments 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.
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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

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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

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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

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Measurement and Perception
Phase

• Changing the phase of components in a
  steady-state tone produces no perceptible change
  in sound, although the shape of the wave may
  change noticeably

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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 instruments characteristics are defined
  primarily by the attack, the phase of attack
  components is critical

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Measurement and Perception
Timbre

• We have discussed timbre as the result of
  overtone content
• It is also judged by the sounds 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 criticala 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

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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.
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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).
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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
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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
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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.
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Measurement and Perception
Conclusion
Objective measurements can tell us more about
sound events
By the same token, they give us insight into what
we dont 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?)