Physical Acoustics
Hearing, auditory perception, or audition
is the ability to perceive sound by detecting
vibrations, changes in the pressure of the
surrounding medium through time, through
an organ such as the ear.
Hearing is the result of a complex interaction
of physics, physiology, perception and
cognition.
Sinusoids
The Basic Sound
Reading: Yost Ch. 2
Definitions of Sound
“If a tree falls in the woods, and no one’s around to hear it, does it
make a sound?”
In physics, sound is a vibration that propagates
as a mechanical wave of pressure and
displacement, through a medium such as air or
water. Any object that can vibrate can give rise
to sound. To vibrate, an object must have
inertia and elasticity (in practice, all do).
• Inertia: expresses the fact that force
must be applied to displace an object.
• Elasticity: opposing force that returns
object back towards its starting point.
In physiology and psychology, sound is the
reception of mechanical waves and their
perception by the brain. Thus, “sound” is
anything that we hear (audible).
Vibrations
Vibration is the movement of an object
from one point in space to another point
and usually back again to or through the
first point.
• Regular vibrations – yield a subjective
quality of “tonality” or “pitch”.
– Pure tone: sinusoidal wave at a
single frequency (fixed period).
– Complex tones: multiple frequencies,
may vary in time.
• Irregular – subjective quality of
“dissonance”.
• Random – “noisy”
Tone: 1000 Hz
Tone: 440 Hz
Harmonic complex
(200, 400, 600 Hz)
Field cricket
Whistle
White noise
Pink noise
Simple Harmonic Motion 1
An object needs inertia (mass) and elasticity
(spring-like quality) to vibrate.
In a simple mass-spring model, if the mass
is displaced a distance x from its resting
position, which requires an inertial force
equal to mass times acceleration (Newton’s
2nd law of motion):
F = m ∙ a (= m ∙ d2x/dt2)
then an equal and opposite restoring elastic
force proportional to the displacement is
exerted by the spring (where s is the spring
constant or stiffness):
x
F=-s.x
When the mass is released, it will move
back towards its resting position, continue
past it to –x because the mass has
momentum, and then attempt to return to its
resting position again; it will oscillate.
-x
Simple Harmonic Motion 2
For one-dimensional simple harmonic
motion, where the equation of motion is
given by:
m ∙ d2x/dt2 = - s ∙ x
general solutions are of the form:
x(t) = A sin(2πft + θ)
where A is the maximum displacement
from the resting position, f is the
frequency of vibration (equal to √s/m),
and θ is the starting phase.
Thus, the general motion for a simple,
regularly vibrating object is sinusoidal in
nature.
Properties of Sine Waves
A sinusoid describes a smoothly varying
relationship between object displacement
and time.
Three variables determine a sinusoid:
Amplitude: the magnitude of
displacement, i.e., how far the object
moves from rest.
Frequency: rate at which the vibration
goes through one full cycle of motion (in
cycles/sec. or Hertz, Hz).
Period is the time it takes to complete
one full cycle. Pr = 1/f, or f = 1/Pr
Phase: Starting point in the displacement
cycle at which vibration begins (degrees
of angle).
Unit circle with
radius = A
Fourier Analysis
• Joseph Fourier (1768 - 1830)
showed that all (periodic)
vibrations could be resolved
into a weighted sum of a series
of sinusoidal vibrations (called
a Fourier series).
• Each frequency of the series is
called a Fourier component.
• The inner ear decomposes
sound in a manner analogous
(but not identical) to Fourier
analysis.
Psychological Correlates of Sinusoids
Amplitude (A): Generally related to
loudness.
Frequency (f): Typically related to
the subjective quality of pitch.
Phase (θ): Only detectable under
certain conditions:
• A phase difference at the two
ears is perceived as a sound
laterally displaced from the
midline of the head.
• Phase changes in spectral
components of complex sounds
can be perceived (with difficulty).
Loudness
increases
Loudness
decreases
Pitch
decreases
Pitch
increases
Frequency
 Frequency of a sinusoid is the number of
cycles it completes per second.
 Inversely related to the period of vibration
(duration of one cycle): f = 1 / Pr
0
5
10
15
Time-ms
20
200 Hz simple vibration
200 Hz complex vibration
(200+400+600 Hz sinusoids)
Phase
Sine wave (θ = 0°)
 Defined in angular degrees: one
cycle of sinusoidal vibration
rotates through 360° (2π
radians).
 Starting phase called the phase
angle, expressed relative to the
zero degree condition.
Sine wave (θ = 90°)
= Cosine wave
Sine wave (θ = 180°)
Relative Phase
Sometimes, two sinusoids are said to be
out of phase with each other.
If the two sinusoids have the same
frequency and are out of phase, then
they must have different starting phases.
In such a case, one sinusoid leads (i.e.,
reaches its peak first) and the other
lags, and the phase difference is
constant with time.
If the two sinusoids have different
frequencies, then the phase difference
will change with time. The starting phase
difference is figured at time 0, the
instantaneous phase refers to the phase
difference figured at any other time.
Amplitude
Displacement of sine wave varies with time
[x(t) instantaneous amplitude]. How do we
specify amplitude, if it’s not stationary?
Non-time dependent
amplitude measures for
sinusoids include:
Peak amplitude =
maximum positive
displacement (A) of the
sine wave.
Peak-to-peak amplitude
= total displacement from
max. positive to max.
negative peaks (2A).
RMS Amplitude
For non-sinusoidal sounds (e.g. noise), peak
or peak-to-peak measures are inadequate.
Gaussian Random Noise
Take an average over some time period? But
the instantaneous amplitude of the waveform
would average to zero.
Squaring the instantaneous displacement
amplitude makes all negative values positive.
Square root of the mean (of) squared
amplitude values is called the root mean
squared (rms) amplitude of the signal.
• RMS amplitude is computed over (at
minimum) one cycle (Pr = T) with the
following equation:
Arms = √1/T ∫0T x2(t)dt
•
For sinusoids, rms amplitude is ~70% of
peak amplitude (0.707Apeak).
Probability
Damped Oscillations
 No vibration occurs in the absence
of resistive forces, i.e. friction.
 Friction attenuates amplitude of
oscillation, so that over time,
oscillation dies out unless it is
reinforced by adding energy.
 Rate of damping is such that ratio
of successive peaks is constant.
 Damping rate varies because
resistive forces depend on the
nature of the vibrating material:
pure materials (e.g. crystal glass)
tend to dampen more slowly than
complex materials (e.g., wooden
desk)
Rate of damping: In this example,
each peak is ½ that of previous
peak (i.e., A declines in powers of
2 over time)
Summary
• Objects have inertia and elasticity, therefore can
vibrate and (possibly) produce audible sound.
• Sound waveforms can be regular (simple or
complex), irregular, or random (indeterminate).
• Vibration waveform described by three
parameters: Amplitude, frequency, and starting
phase.
• All vibrations can be resolved as a weighted
sum of sinusoids (“Fourier series”)
• Amplitude of free vibration declines over time at
a constant ratio due to friction, and the rate of
decline is different for different materials.