Topic 3
Physical Acoustics Review
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WARNING!!!!
• This is a REVIEW.
• Mastery is ESSENTIAL to understand the
material later in the course.
• If you found this material particularly
challenging in previous courses it is,
spend the time on it NOW.
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Learning Objectives
• Outline the physical processes underlying
simple harmonic motion using the massspring model
• Describe the molecular basis of sound
wave propagation
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Spring Mass Model
• Mass (inertia)
• Elasticity
• Friction
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What is sound?
• It may be defined as the propagation of a
pressure wave in space and time.
• propagates through a medium
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Sound-conducting media
• Medium is composed of
molecules
• Molecules have “wiggle
room”
• Molecules exhibit random
motion
• Molecules can exert pressure
A
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Model of air molecule vibration
(Time 1)
Air molecules sitting side by side
Rest positions
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Model of air molecule vibration (Time 2)
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Model of air molecule vibration (Time 3)
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Model of air molecule vibration (Time 4)
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Model of air molecule vibration (Time 5)
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Model of air molecule vibration
a
Time
b
c
d
1
2
3
4
5
Distance
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Wave action of molecular
motion
Time
1
2
3
4
5
Distance
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Amplitude waveform
Position
Time
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Amplitude waveform
Question: How long will this last?
Amplitude
Time
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Model of air molecule vibration
Time
1
2
3
4
5
Questions:
Where is a region of compression?
Where is a region of rarefaction?
Pressure measuring device
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Pressure
For example…
Time
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Learning Objectives
• Define the key characteristics of sinusoidal
motion (amplitude, frequency/period and
phase)
• Outline the relationship between the
frequency and wavelength of a sound
wave
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Pressure vs. time (pressure waveform)
Amplitude
Phase (deg)
Phase: when a period
begins
Pressure
Period (T)
Frequency (F):
rate that waveform
repeats itself (1/T)
Time
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Phase
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Initiating a sound waves that
differ only in phase
A force is applied to molecule at frequency f and time t
same force applied at frequency f at time t+a where a < the period of vibration
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Features of a pressure
waveform
• Amplitude
–
–
–
–
Measured in pressure units
peak amplitude
peak-to-peak amplitude
Instantaneous amplitude
• Period and Frequency
– Period measured in time (basic quantity)
– Frequency is a rate measure (per unit time) expressed as Hertz
(s-1)
– May be expressed as octaves, semitones, etc
• Phase
– Measured in degrees (relative to period length)
– 0-360 degrees
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Spatial variation in pressure
wave
wavelength () is the
distance covering
adjacent high and low
pressure regions
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For example…
Pressure
Wavelength ()
Distance
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Relation between frequency and
wavelength
=c/F where
: wavelength
F: is the frequency
c: is sound speed in medium (35,000 cm/sec)
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Additional Concepts
• Propagation of waves
– Transmission
– Absorption
– Reflection
– Reverberation
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Learning Objectives
• Draw and describe time-domain and frequency-domain
representation of sound
• Distinguish between simple and complex sound sounds
with regard to physical characteristics and graphical
representations
• Distinguish between periodic and aperiodic sounds with
specific emphasis on terms such as fundamental
frequency/period, harmonics, and overtones
• Distinguish between continuous and transient sounds
• Describe how waves sum, define Fourier's theorem and
be able to describe the basics of Fourier analysis
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Graphic representation of sound
• Time domain
– Called a waveform
– Amplitude plotted as
a function of time
• Frequency domain
– Called a spectrum
– Amplitude spectrum
• amplitude vs. frequency
– Phase spectrum
• phase vs. frequency
– May be measured using a
variety of “window” sizes
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Same sound, different graphs
Time domain
Frequency domain
From Hillenbrand
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Classification of sounds
• Number of frequency components
– Simple
– Complex
• Relationship of frequency components
– Periodic
– Aperiodic
• Duration
– Continuous
– Transient
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Simple periodic sound
• Simple: one frequency component
• Periodic: repeating pattern
• Completely characterized by
– amplitude
– period (frequency)
– phase
• Other names: sinusoid, simple harmonic
motion, pure tone
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Simple periodic sound: Graphic
appearance
From Hillenbrand
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Complex periodic sounds
•
•
•
•
Complex: > one frequency component
Periodic: repeating pattern
Continuous
Frequencies components have a special relation
– Lowest frequency: fundamental frequency
• Symbol: fo
• Frequency component with longest period
– Higher frequency components: harmonics
• integer (whole number) multiples of the fo
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Complex periodic sounds: Graphic
appearance
• Time domain:
– repeating pattern of pressure change
– within the cycle, things look complex
• Frequency domain:
– spectral peaks at evenly spaced frequency
intervals
– “picket fence” appearance
• Auditory impression: sounds ‘musical’
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Complex periodic sounds: Graphic
appearance
From Hillenbrand
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Amplitude vs. Phase Spectrum
Amplitude spectrum:
different
Phase spectrum:
same
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Amplitude vs. Phase Spectrum
Amplitude spectrum:
same
Phase spectrum:
different
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(Complex) Aperiodic sounds
• Complex: > one frequency component
• Aperiodic: Does not repeat itself
• Frequency components are not
systematically related
• May be
– Continuous
– Transient
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Aperiodic sounds: Graphic
appearance
• Time domain:
– no repeating pattern of pressure change
• Frequency domain:
– the spectrum is dense
– No “picket fence”
• Auditory impression: sounds ‘noisy’
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Aperiodic sounds: Graphic
appearance
From Hillenbrand
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Analysis of complex waves
• Waves can be summed
• Complex waves are the sum of simple waves
• Fourier: French Mathematician:
– Any complex waveform may be formed by summing sinusoids of
various frequency, amplitude and phase
• Fourier Analysis
– Provides a unique (only one) solution for a given sound signal
– Is reflected in the amplitude and phase spectrum of the signal
– Reveals the building blocks of complex waves, which are
sinusoids
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Learning Objectives
• Draw and differentiate the waveform and
the waveform envelope
• Draw and differentiate the amplitude
spectrum, the phase spectrum and the
spectrum envelope
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The “envelope” of a sound wave
• Waveform envelope:
– imaginary smooth line that follows the peak of
the amplitude of a sound pressure waveform
• Spectrum envelope:
– Imaginary smooth line drawn on top of the
amplitude spectrum
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Waveform envelope
Time
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Spectrum envelope
From Hillenbrand
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Thought Question
Can an aperiodic and complex
periodic sound have identical
spectrum envelopes?
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Amplitude Spectrum: Window
Size
• “short-term” vs. “long-term average”
amplitude spectrum
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“Instantaneous” Amplitude
Spectra
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(Long Term) Average Amplitude Spectrum
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Learning Objectives
• Define an acoustic filter
• Draw and label a frequency response curve
• Draw and differentiate different types of acoustic
filters
• Define terms such as cutoff frequency, center
frequency, roll off rate, gain, and bandwidth
• Define and draw a basic filter system and relate
that to the source-filter theory of speech
production
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What is an “Acoustic” Filter
• holds back (attenuates) certain sounds and lets
other sounds through - selective.
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Why might we be interested in
filters?
• Human vocal tract acts like a frequency selective
acoustic filter
• Human auditory system behaves as a frequency
selective filter
• helps us understand how speech is produced
and perceived.
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Frequency Response Curve (FRC)
Center frequency
+
3 dB
Gain
passband
lower cutoff
frequency
upper cutoff
frequency
low
high
Frequency
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Operation of a filter on a signal
NOTE:
Amplitude spectrum describes a sound
Frequency response curve describes a filter
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Kinds of frequency selective
filters
Low-pass filters
– Lets low frequencies “pass through” and attenuates
high frequencies
High-pass filters
– Lets high frequencies “pass through” and attenuates
low frequencies
Band-pass filters
– Lets a particular frequency range “pass through” and
attenuates other frequencies
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Low Pass Filters
Gain
+
low
Frequency
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High Pass Filters
Gain
+
low
Frequency
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Band Pass Filter
Gain
+
low
Frequency
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Learning Objectives
• Define resonance, free and forced
vibration
• Outline how acoustic resonators behave
like acoustic filters
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Free vibration
• objects tend to vibrate at a characteristic
or resonant frequency (RF)
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Forced vibration
• A vibrating system can force a nearby
system into vibration
• The efficiency with which this is
accomplished is related to the similarity in
the resonant frequency (RF) of the two
systems
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Forced vibration
• If the RF of the two systems are the same,
the amplitude of forced vibration will be
large
• If the RF of the two systems are quite
different, the amplitude of forced vibration
will be small or nonexistent
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Resonance refers to
• Natural vibrating frequency of a system
• The ability of a vibrating system to force
another system into vibration
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Resonance
Acoustic (Cavity) Resonators
• Transmit sound frequencies with more
or less efficiency, depending upon the
physical characteristics
• Therefore, they act as filters, passing
through (and even amplifying) some
frequencies and attentuating others.
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Resonance
•
•
•
•
Acoustic (Cavity) Resonators
And since they act as filters, they have most
of the same features of a filter, even though
we might use different names.
Center frequency is often termed the
resonant frequency.
Frequency response curve often termed the
resonance curve.
Resonators may be sharply or broadly
“tuned” which refers to the roll-off frequency
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Resonator Features
Sharply tuned
Broadly tuned
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Resonator Features
Gain
Frequency
An example of the resonance characteristics
of the human vocal tract
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Learning Objectives
• Explain what the decibel is and why it is a
preferred way to quantify amplitude
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Signal amplitude vs. Signal loudness
• The bigger the signal – the louder the
signal
• Loudness is our perception of signal
amplitude
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What units do we use to measure
signal amplitude?
Up to this point, we’ve used pressure
• pressure = force/area
• cgs units = 1 dyne/cm2 = 1 barye = 0.1
pascal
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What units do we use to measure
signal amplitude?
Size may also be represented using
intensity
• Intensity = Power/area
– Power=Work/time
– Work=Force*distance
• Units: watts/m2 – not cgs
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Pressure-Intensity Relation
• Intensity is proportionate to Pressure2
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What is the decibel scale?
• We use the decibel scale to represent
signal amplitude
• We are used to using measurement scales
that are absolute and linear
• The decibel scale is relative and
logarithmic
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Linear vs. logarithmic
• Linear scale: 1,2,3…
• For example, the difference between 2
and 4 is the same as the difference
between 8 and 10.
• We say these are additive
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Linear vs. logarithmic
• Logarithmic scales are multiplicative
• Recall from high school math and hearing
science
10 = 101 = 10 x 1
100 = 102 = 10 x 10
1000= 103 = 10 x 10 x 10
0.1 = 10-1 = 1/10 x 1
Logarithmic scales use the exponents for the
number scale
log1010 = 1
log10100 = 2
log 101000=3
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Logarithmic Scale
• base doesn’t have to be 10
• In the natural sciences, the base is often
2.7… or e
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Logarithmic Scale
• Why use such a complicated scale?
– logarithmic scale squeezes a very wide range
of magnitudes into a relatively compact scale
– this is roughly how our hearing works in that a
logarithmic scales matches our perception of
loudness change
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For example,
linear
1
2
3
log
10
100
1000
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Absolute vs. relative measurement
• Relative measures are a ratio of a measure
to some reference
• Relative scales can be referenced to
anything you want.
• decibel scale doesn’t measure amplitude
(intensity or pressure) absolutely, but as a
ratio of some reference value.
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Typical reference values
• Intensity
– 10-12 watts/m2
– Threshold for normal hearing at 1000 Hz
• Sound Pressure Level (SPL)
– 20 micropascals
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However…
• You can reference intensity/pressure to
anything you want
For example,
• Post therapy to pre therapy
• Sick people to healthy people
• Sound A to sound B
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Now, let us combine the idea of
logarithmic and relative…
bel= log 10(Im/ Ir)
Im –measured intensity
Ir – reference intensity
A bel is pretty big, so we tend to use decibel
where deci is 1/10. So 10 decibels makes
one bel
dBIL = 10log 10(Im/ Ir)
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Intensity vs. Pressure
• Intensity is difficult to measure.
• Pressure is easy to measure – a
microphone is a pressure measuring
device.
• Intensity is proportionate to Pressure2
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Extending the formula to
pressure
Using some logrithmic tricks, this translates
our equation for the decibel to
dBSPL= (2)(10)log 10(Pm/ Pr) = 20log 10(Pm/ Pr)
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