For an alternative self-study resource see
Sound Waves (University of Salford)
1. OSCILLATIONS -
VIBRATIONS - WAVES
Oscillation is a broad
term describing the variation of a physical or other property of
a system around a reference value.
The reference value is not necessarily fixed.
It is
defined --and may be re-defined-- in the context of the question
being investigated.
Consider the example
illustrated to the right: _ relative to the Earth's
center, a point on the surface moves in a circle; _ relative to the
Sun's center, the same point follows
a circular spiral;
_ relative to our Galaxy's center, the point follows a helical spiral, and so on.
Vibration is the back-and-forth motion
of a mass around a reference point, called its point of rest or equilibrium.
Vibration is a mechanical phenomenon and a specific type of oscillation.
Equilibrium is the condition
in which all forces acting on a system are balanced.
Unstable equilibrium: A state of equilibrium that breaks down
when the system is
disturbed (see below left). Stable equilibrium:
A state of equilibrium that persists despite moderate disturbances (see
below right).
All musical instruments --and, more generally, all vibrating objects
that produce sound-- operate around a stable
equilibrium.
At equilibrium there is no vibration and
no potential for
sound.
When a performer plucks a guitar string, blows into a trumpet,
or strikes a drum, the applied force disturbs the system's equilibrium
and
initiates vibrations that can produce sound. Once the excitation
stops, friction and other forms of resistance dissipate the energy
that sustained the vibration, which eventually dies out and the string / air
column / drumhead returns to its equilibrium state.
Vibrations and Waves
In most cases, the characteristics we observe in a
vibrating sound source have
corresponding features in the sound waves it produces.
Conversely, the observable characteristics of sound waves
can generally be traced back to the vibrations that
generated them.
Exceptions to this correspondence
are due to the properties of the medium through which the
energy of a vibration propagates in the form of a wave.
While not numerous, they are important and were identified
formally by Lord Rayleigh (John W. Strutt) in the late 1800s.
We'll discuss some of these exceptions in the second half of the module.
Simple harmonic motion is
the simplest type of vibration (e.g., a freely oscillating
pendulum - see the top animation to the right).
In simple harmonic motion, the restoring force --the force
that pulls a vibrating mass back towards its equilibrium
position -- is proportional to its displacement. In other words, the further the mass moves from
equilibrium, the stronger the restoring force.
Simple harmonic motion can be represented graphically by a
sinusoidal curve, commonly called a
sine signal. Such curves/signals are described by the
following equation, which we will only approach qualitatively:
y(t) = Asin(2πft+φ) - where:
* y: Displacement, represented on the vertical
axis (y). * t: Time, represented
on the horizontal axis (x). *
A: Amplitude, or maximum displacement
from equilibrium.
For sine signals the amplitude is constant and is
represented by a single
value on the vertical axis (y).
* f: Frequency, measured in
cycles per second; implied but not directly
represented on
the signal. * φ (or θ):Initial phase, indicating the signal's starting
position within its cycle;
measured in degrees or radians.
One complete cycle corresponds to 3600 or 2π
radians, where π ≈3.14...
The bottom animation to the right illustrates the relationship between steady circular motion and
sinusoidal motion.
Optional: The video
below, explains in detail the meaning and derivation of the sinusoidal equation
(from Khan Academy's high-school AP physics resources).
In a complex vibration the restoring force is
related to displacement in a complex way.
Complex
vibrations are described graphically by complex curves and can be understood
as combinations of multiple simple harmonic motions. Correspondingly,
complex signals can be represented as the sum of an appropriate set of sine signals. See, for
example, the animation to the right. We will return to this concept in
our discussion of spectrum.
Virtually all sounds we encounter or produce correspond to complex waves,
which originate in complex vibrations and are represented by complex
signals. A complex signal can, in turn, can be analyzed into a set of sine
signals each with its own amplitude, frequency, and starting phase.
A signal is a two-dimensional graphical representation of
a vibration or wave.
It plots: _ displacement from equilibrium of some quantity
(e.g., mass, velocity,
pressure, or voltage) on the vertical axis (y), over
_ time on the horizontal axis (x).
A signal
is therefore a time-domain representation showing how some variable
changes with time.
Signals representing waves are also commonly called waveforms,
term that is only appropriate in the case of transverse standing waves (more
later in the course).
A periodic vibration/wave is one
that repeats itself at regular time intervals.
By definition, all sinusoidal signals/waves are periodic.
Periodicity and regularity have important perceptual
consequences.
In general,
regularity supports prediction. In hearing, periodic sound waves give
rise to well-defined sensations of pitch.
3. PERIOD; FREQUENCY; AMPLITUDE; PHASE;
SIGNAL ENVELOPE
Period (T): The time
required for a vibrating system to complete one full vibration, or cycle.
In a signal, the period is measured along the horizontal time axis (x).
Period T is measured in number of seconds per cycle and is the inverse of
frequency f: T = 1/f .
Important: Time (t) and period (T) are
represented by different symbols because they are distinct concepts,
measured in different units: _ Time identifies a particular moment relative to a reference start
time, and is measured in seconds. _ Period describes the duration of one complete vibration cycle, and is
measured in seconds per cycle.
Frequency (f): Number of repetitions,
or cycles, per unit
of time.
Changes in frequency correspond primarily to changes in pitch. Frequency
f is measured in number of cycles per second or Hertz (Hz),
named after German physicist Heinrich Hertz. 1 Hz = 1 cycle per second - Frequency is the inverse of period
T: f = 1/T
Frequency is not represented
directly on the horizontal axis of a time-domain signal. Instead, it can be
inferred from the period (f=1/T): the shorter the period
the higher the frequency.
Frequency
is represented directly along the horizontal axis of a spectrum, to be
examined later in the module.
As illustrated under Simple
Harmonic Motion, steady rotation at a fixed angular velocity is
mathematically related to sinusoidal vibration at a fixed frequency.
Amplitude (A): Maximum displacement
from equilibrium. Changes in
sound-wave amplitude correspond primarily to changes in loudness.
For sine signals, amplitude is
constant, represented by a single value on the vertical
axis (y). Recall that the vertical axis of a time-domain
signal represents a quantity that changes over time.
Consequently, displacement takes different values at
different moments in time, of which amplitude is the maximum value.
For complex signals, amplitude varies over time. It is represented by a subset of points on the vertical
axis
(y) and is tracked by the signal envelope
(see further below).
The actual displacements of air molecules in
sound waves are extremely small. For speech,
displacement amplitude is in the order of millionths of an inch.
Phase (φ): A signal's position
within its vibration cycle
at a given moment in time. Phase can be measured in degrees (00
to 3600) or in radians , (0 to 2π radians). E.g.: 1/4 cycle = 900 = π/2
radians -- 1/2 cycle = 1800
= π radians -- 1 cycle =
3600 = 2π radians.
Perceptual Effects of Phase Changes
I. Changes in the absolute phase values of
otherwise identical non-simultaneous
signals are not perceptible. For example, the signals
shown below would sound identical when played in succession,
even though there is a phase-shift (time-lag) among
them.
II. Changes in the absolute phase values of simultaneous
signals with very different frequencies are not perceptible. For
example, changes in the phase relationship among the
spectral components of any given harmonic complex signal have no
perceptual effect, under ordinary listening
conditions. We will examine examples and
qualifications to this principle in our
discussion of spectrum.
III. In contrast, changes in phase relationships among multiple simultaneous
signals with identical or similar frequencies are rather salient.
Depending on the signals and listening conditions, these changes
can influence perceived loudness, pitch, timbre, and
sound-source location. We will examine these effects later in
the course.
Click on the image below, for an example of how a phase shift
can influence timbre (timbre: perceived quality or
character of a sound).
The signal shown above,
with its regular, sinusoidal peaks and valleys, represents a
simple, pure sine wave corresponding to the musical note A3
in equal-tempered tuning.
The peaks occur at regular intervals of ~4.54 ms, or ~0.00454 s. This interval is the period of the signal
T ~ 0.00454 ~ 1/220 s/cycle. The signal therefore repeats once every
~1/220 s or
~220 times per
second,
corresponding to a
frequency f
= 1/T ~220 Hz.
The signal shown above also represents
a sine wave, this time corresponding to the musical note A4.
Period b>T ~1/440 s/cycle, half that
of A3;
Frequency f~ 440 Hz, twice that
of A3.
The
vibration it represents produces a sound wave that travels
through the air
at approximately
the same speed as the wave
corresponding to A3.
However, the air molecules at each point
of the wave path oscillate
at twice the frequency.
The
amplitude of both signals is represented
by the vertical distance between the central horizontal line,
which
represents equilibrium, and either the top or bottom peak.
Signal Envelope
Signal Envelope is a boundary curve that traces the signal's amplitude
over time, capturing how the signal's energy changes with time. The
envelope follows the signal's successive maxima and minima, outlining
its overall amplitude pattern. Note that it includes points that do not belong to the signal.
In the illustrations to the left, the signal is shown in blue
and its envelope in
red.
For sinusoidal
signals, whose amplitude is constant, the envelope is a horizontal straight line, parallel to the
time
axis.
For all other signals, the envelope has a different shape, often consisting
of distinct time stages.
The terminology used to
describe these stages can be confusing.
We will adopt the
terminology illustrated on the image below left, commonly used in digital filter and synthesizer
design. The image below right, showing the signal of a trombone tone, illustrates the
envelope terminology more commonly used by musicians.
Signal envelopes and spectral envelopes
(discussed in the section on spectrum)
are the two features most frequently manipulated by audio engineers. We will
devote substantial attention to their physical and perceptual
correlates.
Signal Envelope Stages
Attack: describes how a
signal's amplitude/energy rises from its initial level toward its
maximum.
The shape of the attack, and its associated sonic characteristics depend on:
a) The inertia
characteristics of the sound source, which are related to its material, mass, shape,
tension, and other physical properties. b) The type of excitation
--that is, whether the source is plucked, bowed,
blown, struck, and so on.
The attack is perceptually salient: it often contributes strongly
to sound and sound source recognition. This is particularly the case
for
impulse signals (discussed below), in which the attack contains much of the signal's energy.
Decay: describes how vibration
energy in a system settles, following initial excitation, even if excitation
continues.
For mechanical/natural (as opposed to electronic or digital) sound sources,
this is the shortest envelope stage and is often omitted from
signal descriptions.
Sustain: describes how a sound source
responds during continuous excitation.
Sustain is generally the most regular and periodic stage of the
envelope. In continuous signals, it typically contains many of the
sound's characteristic features. Note: in acoustics, the "sustain" stage of a signal
envelope is also called the "steady
state." This term is approximate: neither the amplitude envelope nor
the spectral envelope
remains perfectly constant over time.
Release: describes how vibration energy in a
system naturally dies out --due to friction and other forms of resistance-- after the energy supply ends. [
Traditionally
referred to as Decay ]
In most
cases, the release stage lasts longer than the attack. The source's natural
resonant frequencies may continue to vibrate after the excitation stops,
with some frequencies persisting longer than others. The broader the
resonant response of a source the richer and shorter the release stage. We
will examine this relationship further in the next module, under
resonance.
Impulse and Continuous Signals
Impulse signals result from a very brief
excitation of a sound source, which is then left to vibrate on its own
until the energy dissipates through some form of friction/resistance. Impulse signals
typically contain an attack followed by a release (decay), without a
sustained stage.
Continuous signals result from
sustained excitation of a sound
source.
Their envelopes include all four stages in
the following order: Attack; Decay; Sustain; Release.
The energy content of a sound signal can be described using
three
measures of amplitude:
0-to-peak amplitude: The maximum positive displacement from
equilibrium. This is the standard
amplitude measure used in equations for
simple harmonic motion and sinusoidal signals.
Peak-to-peak amplitude: The difference between positive and negative
maximum displacements from equilibrium. This measure is useful for
describing signals that are not symmetrical around the time axis,
as is the case with most signals generated by musical
instruments.
Root-mean-square (RMS) amplitude:
A measure of the effective amplitude of a continuously varying
signal, linked to its perceived loudness.
The root mean square
pressure, PRMS, is mathematically defined
as the square root of the average of the squared pressure
variations over a given duration, usually one period.
Graphically it is represented by the area enclosed in a
signal
[Figure (a) below ]:
PRMS = <P2>0.5
To calculate the RMS value of any set of numbers:
1. Square each number. 2. Calculate the average
of the squared values. 3.
Take the
square root of that average. This procedure is necessary for quantities
such as sound pressure that oscillate between positive and negative
values. Simply calculating the average pressure displacement over one or more
complete cycles of a sine signal would produce zero.
For all sinusoidal signals,
peak and RMS amplitudes are interchangeable because their relationship is
fixed: RMS Amplitude = 0.707 * Peak Amplitude [Figure
(b) below ]
For complex signals, this relationship is not fixed
[Figure (c) below ]. Two complex signals with the
same peak amplitude may have different RMS amplitudes and vice versa. A
complex signal with higher RMS amplitude (e.g.,
bottom figure to the right) will generally sound
louder.
The difference between
peak RMS levels of complex signals is called the crest factor. A
punchy, uncompressed snare drum has a high crest factor (high
peaks, low RMS). A heavily compressed, distorted guitar chord has
a low crest factor (the RMS is almost as high as the peaks).
RMS amplitude maps more reliably to loudness than peak
amplitude. Human
hearing averages out sound over a window of ~100-300
milliseconds. RMS mathematically models this
response, better capturing the loudness of complex signals.
(a) RMS amplitude is a measure of the area outlined by the signal (highlighted) (b) 0-to-peak,
peak-to-peak, and RMS amplitudes of a sine signal. (c) 0-to-peak and RMS amplitudes of a
complex signal.
Application
Example: The Loudness of Advertisements
FCC rules require that the loudness of
television and Internet advertisements not exceed that of
surrounding programming.
Initially, compliance was enforced using peak amplitude.
Advertisers exploited this
limitation by using dynamic compression to increase RMS amplitude
and sound louder than regular programming, while keeping peak amplitude
within the permitted limits. The resulting runaway effect
contributed to the so-called loudness war.
In 2013,
the
FCC
updated its rules for more sophisticated loudness
assessment. One relevant measure is LUFS (Loudness Units relative to
Full Scale), which accounts for:
● The relationship between peak and RMS amplitude (i.e.,
the crest factor).
● The significant dependence of loudness on frequency.
The LUFS scale is also used by music streaming services
to normalize loudness levels across songs, reducing the need for
listeners to make substantial volume adjustments as they
move through a
playlist.
These issues will be explored
further in the Loudness module and are
particularly relevant to anyone planning to work in the sound-for-picture
industry.
The
principle of linear superposition
states that multiple sinusoidal waves can
coexist on a vibrating system independently. At any given moment, the
total displacement of the superimposed waves is equal to the
algebraic sum (i.e., a sum that takes into consideration signs: +,
-) of their original
displacements. It is the waves, rather than the vibrations themselves, that coexist
independently on a vibrating system, like a string. The resulting vibration at any point is
the combined effect of the superimposed waves.
The principle emerged from the
study of vibrating strings, treated as one-dimensional systems.
Galileo Galilei
introduced related ideas in the early 17th century.
Daniel Bernoulli
formalized the principle for vibrating systems in 1753..
The principle of interference
(see the animation below) follows from linear superposition. When two or more
vibrations/waves interact,
their combined amplitude may be larger (constructive interference) or smaller (destructive interference) than the amplitude of the individual vibrations/waves), depending on their phase relationship.
When two waves of different frequencies
(f1 and f2) and
amplitudes (A1 and A2)
combine, their phase relationship changes periodically. This
produces alternating constructive and destructive interference,
resulting in amplitude fluctuation:
the combined amplitude varies gradually and periodically between a
maximum of A1 + A2 and
a minimum of |A1 - A2|.
The fluctuation rate is equal to the absolute difference
between the two frequencies, |f1 -
f2|.
For example, in the graphs below, two sinusoidal signals A
and B have the same amplitude,
A, and the same initial phase, but different frequencies,
f1 and f2.
Adding the two signals together produces a complex signal (A + B)
whose amplitude
fluctuates between a maximum of 2A and a minimum of 0.
Because the signals have different frequencies, their
relative phase gradually shifts from perfectly in phase to
perfectly out of phase and back again.
This produces alternating constructive and destructive
interference and amplitude fluctuation at a rate equal to
|f1 - f2|. Depending on
fluctuation rate,
amplitude fluctuations can be perceived as beating(a sound with slowly
fluctuating level), roughness(a rattling sound), or combination tones(additional tones beyond the original ones). These
phenomena will be explored later in the course.
An extreme example of destructive interference
occurs when
two vibrations (waves) have the same frequency, amplitude, and
source location but a phase difference of 1800 (π
radians). Their displacements cancel at every moment,
resulting in no net vibration.
Complete or nearly complete destructive interference is used in
active noise-
and feedback-reduction systems. According
to a short
article in the December 10, 2004 issue of The New York Times, two 2005 Honda
car models were among the first cars to feature active noise-reduction systems based on this
principle. Versions of this technology are now commonplace in cars and, of course, headphones.
Total Constructive Interference
Total Destructive Interference
Two sine signals, A and B, have the same frequency and amplitude and
are in phase (00 phase difference). Their addition
produces total constructive interference.
The combined signal's amplitude equals the sum of the individual
amplitudes.
Two sine signals, A and B, have the same frequency and amplitude but a 1800 phase difference. Their addition
produces
total destructive interference. Because the signals are
perfectly out of phase, their displacements
have equal magnitudes but opposite signs at every moment. The
combined signal's displacement equals the algebraic sum of the
individual displacements: zero.
The absolute units for power (joules
per second or watts), intensity
(watts per square meter), and pressure (newtons per square
meter or pascals) are expressed on linear scales. However, sound power, intensity, and pressure levels are usually
expressed on a base-10 logarithmic (log) scale.
a)
Logarithmic scales better approximate how we perceive sound
According to Fechner's psychophysical law, named after 19th
century German physicist and philosopher Gustav Fechner,
perceived sensation
relates approximately logarithmically to the magnitude of a physical stimulus: multiplying a physical stimulus by a given factor corresponds
approximately to adding a fixed amount to the perceived sensation.
Therefore, logarithmic scales provide a more useful model of the relationship
between the the physical properties of sound waves and our perception of sound
than linear scales.
b) Logarithmic scales compress a very wide
range of values into a manageable range.
The audible range of sound power, intensity, and pressure is enormous in linear
terms. Expressed in logarithmic terms, the audible range
is compressed and becomes
more manageable, while remaining fine enough to match the resolution of
perception.
For example, the linear range between the
smallest (barely perceptible at 1,000 Hz) and largest (harmful to the ear
at all frequencies)
audible intensities amounts to approximately 1 trillion steps.
In contrast the same range in log math amounts to ~120 steps.
_
Linear range: | 10-12 to 1 W/m2 ] ~ 1012
steps.
_
Log range:
| log1010-12
- log101 | = | -12-0 | = 12 steps called
bels (after 19th-20th
century Canadian-American inventor, scientist, and engineer, A.G.
Bell).
According to perceptual experiments, hearing's average resolution for
intensity is 1/10th of a bel or 1 decibel (deci- from the Latin decimus
for tenth). Consequently, the range used is 12 * 10 = 120 dB
Standard versus Log Sound Levels
We express power, intensity, and pressure levels
of a sound logarithmically, in dB, because we are ultimately
interested in the sensation of sound rather than simply in physical
waves. The decibel formula expresses logarithmically the
ratio between absolute power/intensity/pressure values
and their corresponding
reference (i.e., smallest perceivable) values.
_ Sound Power Level: SWL = 10 * log
W / Wref _ Sound Intensity Level: SIL =
10 * log I / Iref _ Sound Pressure Level: SPL = 20 * log
P / Pref
(multiplier is 20 because power and intensity are
proportional to P2)
The reference sound level is
expressed as 0dB, and
corresponds to: Reference power: Wref = 10-12W Reference intensity: Iref
= 10-12W/m2 Reference pressure
change: Pref = 2*10-5Pa
(<1 billionth of change in atmospheric pressure:
~10.24 Pa) For example:
If I = Irefthen SIL =
10*log (Iref/ Iref) = 10 * log
1 = 10 * 0 = 0dB
The threshold of pain
(largest safe sound level) is expressed as 120dB, and
corresponds to:
Maximum (safe) power = 1 W Maximum (safe)
intensity = 1 W/m2 Maximum
(safe) pressure
change = 20 Pa
(<1 thousandth of change in atmospheric
pressure: ~10.24Pa) For example:
If I = 1 W/m2 and Iref
= 10-12W/m2then SIL =
10 * log (1 / 10-12 ) =
= 10 * log (100 / 10-12 )
= 10 * log (100-(-12) ) = 10 * log (1012 ) = 10 * 12
= 120dB
Key takeaways
Log sound levels in dB are not absolute but
relative to some reference level corresponding to barely audible;
Sound levels in dB derive
from ratios of physical quantities, express their level relationship, and are therefore unit-less;
Math operations --such as addition, subtraction, multiplication,
or division-- with dB numbers (log values) follow different rules than with
standard numbers (linear values);
Power is proportional to the square of
amplitude (A2) and to the square of frequency (f2). For sound waves in gases and liquids, power is also
proportional to the square of sound pressure (P2). Thus, producing higher
amplitudes/frequencies requires (emits) exponentially higher amounts of power and total
energy, and corresponds to exponentially higher amounts of intensity.
Logarithmic facts to memorize
(bookmark
this
online
dB calculator for future use)
Sound level facts (more in the Loudness
module)
Doubling (halving) the sound
level corresponds to +3dB (-3dB). For example: a) two guitars at 65dB each combine to produce
68dB; b) cutting in half the number of
instruments in an orchestra that produces 85dB,
will result in an orchestra that produces 82dB.
10-fold level increase (decrease) corresponds to +10dB (-10dB)
and to "twice
(half) as loud"; 100-fold level increase (decrease) corresponds to
+20dB (-20dB)
and to "4
times
(1/4th) as loud"; 1000-fold level increase (decrease) corresponds to
+30dB (-30dB)
and to "8
times
(1/8th) as loud"; and so on. For example: a) increasing an orchestra's average level by
20dB amounts to adding another
99 identical
orchestras, for a total of 100; b) reducing the sound entering/escaping a
recording studio by 30dB amounts to
blocking 99.9% of
the original sound energy, only
allowing 1/1000th through.
Frequency facts (more in the
Pitch module)
Successive doublings of a
frequency produce tones that sound progressively higher in
pitch but, at some level, are also identical and perfectly
blend when heard simultaneously (e.g., all notes
represented by the same letter name, which are 1 or more octaves apart).
Integer multiples of a
frequency produce tones that blend into a single tone
perception when heard simultaneously.
Examples of Sound Levels and Human Response
From the Noise Pollution Clearinghouse (http://www.nonoise.org)
Wave: The transfer of vibrational energy
through a
medium (e.g., air or a string). [ Defining waves is a complex task.
See this
extended definition ]
Simple (sine) wave: The propagation of simple harmonic
motion through a medium. Complex wave: The
propagation of complex vibration through a medium.
Waves originate from vibrations but are not equivalent to them.
Vibrations
describe motion at a source, whereas waves describe how that motion
and its energy propagate through a medium.
Consequently, waves depend on the
properties of the
propagation medium, while vibrations do not.
Sound waves in air consist of alternating compressions
--regions where pressure is above atmospheric pressure-- and
rarefactions/expansions --regions where pressure is below atmospheric pressure.
These pressure fluctuations spread away from the vibrating source
as a longitudinal wave, in which the air particles vibrate
parallel to the direction of wave propagation.
The video below visualizes sound waves generated by a vibrating string.
Temporal and Spatial Representations of Sound Waves
In sound wave signals,
the horizontal axis (x) can represent either of
two quantities:
a) Time: A temporal signal
of a
wave describes how sound pressure varies over time at a single point in space (top graph
to the
right). The horizontal axis represents time and can be used
to identify the wave's period. Temporal signals plot
the vibration of air particles at a single location, as a sound
wave passes through it, with the phase varying at different
moments in time.
b) Distance: A spatial signal
describes how sound pressure varies through space at a
particular moment in time (bottom graph to the right).
The horizontal axis represents distance and can be used to
identify the wave's wavelength (i.e., the distance
the wave will travel during a period, that is, during the time it
takes to complete one full cycle). Spatial signals plot the position of adjacent air
particles at a single moment in time, essentially freezing the
movement and distribution of wave energy through them. The phase
varies across adjacent points in space because the vibration
energy propagating through the air in the form of a wave reaches
those points at different times.
Wavelength (λ): The distance a wave travels during
the time it takes to complete one vibration cycle.
Unit: meters/cycle.
Wavelength λ depends on
● the period T
(and, consequently, the frequency f) of the source vibration
and
● the speed of wave propagation
c in the medium:
λ= c*T = c/f.
For a given propagation speed, the higher the frequency the shorter the period and the shorter the
wavelength.
The speed of sound, c, depends on the stiffness and
density of the propagation medium. For a longitudinal wave in a uniform
medium, such as air under ideal conditions, it can be expressed in terms of
the medium's elastic modulus E (measured in N/m2)
and density ρ is in (measured in Kg/m3):
c =
(E /ρ)0.5
For a wave travelling along a stretched string, the corresponding
equation is c = (T /μ)0.5, where T is
the string tension (linear elastic modulus, measured
in N/m) and μ is its linear density
(mass per unit length, measured in Kg/m). [ Note: both versions of the
equation lead to the same unit for c (m/s), as they should ].
Therefore, speed of sound is proportional to the
square root of stiffness and inversely proportional to the
square root of density:
The stiffer or more highly tensioned the
medium, the higher the wave speed.
Examples include hotter vs. cooler constrained air; looser vs. tighter string or drumhead; or metal vs.
wood.
The denser (more compact) the medium
the lower the
wave speed.
Examples include cold vs. hot unconstrained air or water vs.
wet balsa wood.
See to the right for an illustration of how stiffness affects the speed
at which sound waves propagate through a medium.
Speed of Sound in Air
At approximately 210C (700F),
the speed of sound in air is:
c ≈ 345 m/s
or c ≈ 1130 ft/s (rounded from
~343.8 m/s
and ~1128.2 ft/s to simplify
calculations)
As
the temperature rises, the air expands and: a) density drops, if the air is unconstrained and free to expand (e.g.,
outdoors), or b) stiffness rises, if the air is constrained and cannot expand (e.g.
indoors).
In both cases, the
speed of sound increases with temperature by ~0.6 m/s for each 10C (or ~1.1 ft/s
for each 10F) increase in temperature:
c≈ 345 +
(C-21)*0.6 m/s - c = 1130 +
(F-70)*1.1 ft/s
NOTES
In this animation, the distance between successive molecules represents
the strength of their bonds (i.e., stiffness), not the density of
each medium. Even though density increases from gases, to liquids,
to solids (reducing the speed of sound) stiffness increases
far more drastically (raising the speed of sound),
outweighing the effect of density and resulting in a net
increase in the speed of sound.
The stiffer of two media is not necessarily the denser. For example, diamond
vs. metals; bamboo vs. ironwood; carbon fiber composite vs.
steel; magnesium alloy vs. aluminum or steel.
For constant temperature and gas composition, speed of sound does
not depend on equilibrium (e.g. atmospheric) pressure.
Unit conversions:
1 foot = 0.3048 meters;
1 meter = 3.281 feet; C = (5/9)*(F-32); F = (9/5)*C+32
Transverse and Longitudinal Waves
Transverse waves:
waves that propagate perpendicularly to the motion of the
source vibration.
Examples include waves
traveling along a string, waves in the front and back plates of a
violin or waves across a drumhead.
Longitudinal waves:
waves that propagate parallel to the motion of the
source vibration.
Examples include
sound waves travelling through air or another gas,
medical ultrasound waves, or seismic P-waves.
The Inverse Square Law states that the intensity of a wave
spreading uniformly from a point source is inversely proportional to the
square of the distance from its source.
As the distance from the source increases, the intensity decrease in
proportion to the square of that distance.
For example:
Doubling the distance decreases
the intensity by a factor of 22 = 4, corresponding to a loss of ~6dB
(3dB loss for each halving of
intensity).
Increasing the distance tenfold decreases the intensity by a factor of
1010 = 100, corresponding to a loss of 20dB
(10dB loss for each tenfold reduction
in
intensity).
Reducing the distance to 1/100 its original value
increases the intensity
by a
factor of 100100 = 10,000, corresponding to a gain of 40dB
(10dB gain for each
tenfold increase in intensity).
The underlying assumption is that the energy radiated by the
source spreads uniformly in all directions, forming expanding
spherical wavefronts.
This assumption applies::
Theoretically, at all frequencies, assuming an idealized
point source, where wavelengths will always be much larger
than the source dimensions.
Practically, more so at lower the frequencies, where
wavelengths are more likely to be much larger than the
dimensions of the source.
(See Diffraction, in the next
module.)
Deriving the Inverse Square Law
I = W / A and, for a sphere,
A = 4πr2
=>
The intensity ratio at distances r1 and r2 from the center of the sphere
will be I1 / I2 = (r2 / r1)2 [ can you derive this from the equationI = W/A ? ]
Standing
Waves
are vibration patterns (on strings, air columns, listening
environments, etc.)
that form when energy propagating as a wave is trapped within two or more appropriately
positioned reflective boundaries.
Standing waves are a manifestation of the linear superposition and
interference principles. They arise from the combination of constructive and destructive
interference between a wave and its reflections. Standing
waves can also be understood as a form of resonance,
occurring at the natural frequencies of an oscillating system
(see Resonance in the next module).
Reminders: The
linear superposition principle states that,
at any given moment, the
total displacement of two superimposed waves is equal to the algebraic sum of their
individual displacements. The
interference principle extends the superposition principle: when two or more waves
combine, the resulting amplitude may be greater (constructive interference) or smaller (destructive interference) than the amplitudes of the individual waves, depending on their
relative phases.
Standing wave
frequencies
Standing waves form only
at frequencies whose wavelength satisfies a specific
relationship with the distance d between
reflective boundaries.
Assuming appropriately
positioned reflective boundaries:
_ the boundary distance d
must be an integer multiple of half the wavelength λ/2:
d = n * λ/2
(n: integer)
or
_ the wavelength λ must be an integer divisor of twice the
boundary distance 2d:
λ = 2d / n (n: integer)
Example:
Assume speed of
sound c = 360 m/s and two parallel reflecting boundaries,
separated by
d =
6m
The longest wavelength
that can form a standing wave corresponds to n =
1. So: wavelength λ = 2d/ 1
= 12 m/cycle and frequency f = c/λ= 360/12 = 30 Hz Additional standing waves will
occur at integer multiples of this frequency
(30*2=60Hz; 30*3=90Hz; 30*4=120Hz; etc.),
with correspondingly shorter wavelengths (12/2=6m;
12/3=4m; 12/4=3m; etc.).
Standing waves are
characterized by fixed points along the propagation medium where the
amplitude is maximum (antinodes) & minimum (nodes
- nodes have a zero amplitude at the extreme case of no
energy absorption at the reflective boundaries).
Standing waves are desirable in musical instruments
and other sound-producing systems, except on loudspeaker cones,
where they can interfere with the intended motion.
In contrast, they are undesirable in listening environments
because they produce spectrally unevensound levels across space.
Transverse Standing Waves
Transverse standing waves
form when energy propagating as a transverse wave is
trapped within two or more appropriately positioned
reflective boundaries.
Such waves are common in string and percussion
musical instruments.
For example, transverse standing waves on a violin's stings and back plate are responsible for the majority of
sonic energy generated by the instrument (which of the two,
in your opinion, contributes most of the energy?).
Transverse standing waves also form on drumheads,
cymbals, xylophone and vibraphone bars, and other materials used in musical instrument construction.
The animation to the right illustrates complex transverse standing-wave patterns,
known as Chladni patterns or Chladni figures,
named after
the German physicist and musician Ernst Chladni
(1756-1827).
The patterns were originally demonstrated on
metal plates excited at different
frequencies, with sand granules accumulating at the
nodes, where the plate's vibration amplitude is minimal.
The animations below-right simulate standing waves on a
string. Multiple standing-wave modes can occur simultaneously, combining
to produce a single, more complex signal. This will be
explored further in the discussion of Spectrum
and in subsequent modules.
Longitudinal standing
waves form when energy propagating as a longitudinal wave is
trapped within two or more appropriately positioned
reflective boundaries.
They can form inside
any air cavity:
Inside musical
instruments: Standing waves in the air cavities of musical instruments
help amplify and color (spectrally shape) the
sound produced. They are essential to all acoustic
instruments. In enclosed listening
environments: Standing waves in enclosed listening
environments are
undesirable because they contribute to uneven sound levels and
spectra throughout the space. Inside resonant traps: Standing waves inside
appropriately designed air cavities --placed within
enclosed listening spaces-- can function as sound
energy absorbers, particularly for low frequencies
(more during the Resonance module).
Interference between Sound Waves from
Spatially Separated Sources
A phenomenon analogous to standing waves on strings
and in air columns
occurs when two interfering identical waves originate from sources
separated by some distance.
In this case, the interference pattern extends through
space, producing regions of maximum and minimum sound
levels. These regions are analogous to a standing wave's
antinodes and nodes, respectively. Unlike a conventional
standing wave confined to a string or air column, however,
this spatial interference pattern spreads throughout an open
listening space.
In the animation below:
_ regions/sectors of contrasting colors represent areas of higher
sound levels, with a maximum near the center of each
sector.
_ sectors where the color changes very little represent
regions of lower sound levels, with a minimum near the
center of each sector. This phenomenon can occur whenever two or more loudspeakers
produce identical, coherent sound waves. Examples include mono
playback or reinforcement through a stereo system,
particularly when loudspeaker placement and signal
timing are not carefully managed.
Interference patterns and the resulting spatial
variations in sound pressure level (SPL) can be especially
pronounced during outdoor performances. In the absence of nearby
surfaces, there are fewer opportunities to diffuse the interference
pattern and improve SPL uniformity (more on Diffusion in
subsequent modules).
Experiment with the interactive animation below to explore how
the distance between sound sources and the wavelength of the sound
affect interference patterns.
Wavelength depends on both frequency and the speed of sound, which
in turn depends on properties of the propagation medium, including
its temperature.
Fourier's mathematical
principle --named after the 18th-19th-century French mathematician and physicist Joseph
Fourier-- states that a complex signals can be represented as
the sum of sinusoidal components. These components are also called Fourier components, spectral components, or
partials, and make up the signal’s spectrum.
Fourier analysis is the process of
decomposing a complex signal into its sinusoidal components. Fourier synthesis is the reverse process:
constructing a complex signal by adding sinusoidal components together.
The figure to the right illustrates the principle graphically and a mathematically (we will not
perform calculations using the
equation).
Periodic and Non-Periodic Signals
Periodic signals, also called
harmonic signals, include the signals of most
sounds produced by musical instruments playing individual
notes. Their sinusoidal components have frequencies that are
(close to) integer multiples of the lowest
component frequency, called the fundamental frequency
(ƒ0) and constituting the first harmonic
component.
The higher frequency
components are called upper i) harmonics,
ii) components, or iii) partials.
If the fundamental frequency is ƒ0, then the harmonic frequencies are: ƒ0, 2ƒ0, 3ƒ0, 4ƒ0 ... called the first, second, third, and fourth harmonics, respectively.
The harmonic components
above the fundamental are occasionally also referred to
as overtones. For example, the second harmonic
is the first overtone. To avoid confusion, we
will refrain from using the term.
The pitch of periodic (harmonic) complex signals
is generally definite, corresponding to
the fundamental frequency. This remains true even when
the fundamental component is
absent from the signal’s spectrum, a phenomenon
discussed in the Pitch module.
Non-periodic signals, also called
inharmonic signals,
include all signals whose sinusoidal components do not
follow an integer multiple relationship.
The pitch of non-periodic
(inharmonic) complex signals is indefinite, ambiguous, or entirely
absent, depending on _ the degree of inharmonicity (i.e., how far their
component frequencies depart from an integer-multiple
relationship); _ their spectral envelope; and _ the absolute and
relative duration of their spectral components. These relationships
will be discussed in the modules on Pitch, Loudness, and
Timbre.
_ Fourier-based Synthesizer used in class (a custom application
for Windows only).
To install it, save the ZIP
file, extract its content, and run SETUP.EXE.
Spectrum
A spectrum, or
frequency spectrum, is a
two-dimensional graphic representation of the frequencies and
(displacement/pressure/power) amplitudes of a
signal's sinusoidal components.
Frequency is plotted on the horizontal (x)
axis and amplitude/level is plotted on the vertical (y)
axis.
A spectrum provides a
recipe for a complex signal, indicating
the type (frequency) and amount (amplitude) of each ingredient (sinusoidal component)
contained in or required to
construct it. Spectra are frequency-domain
representations of sound, while signals are time-domain
representations.
Spectral Envelope
Analogous to the signal envelope
introduced earlier, the spectral envelope
is a boundary curve that traces the peaks of the spectrum,
showing how energy is distributed across the frequency range.
In the figure to the right, the individual spectral components are
shown in blue,
and the spectral envelope is shown in black.
A spectrum
describes the frequency content of a signal averaged
over a particular time interval. A sequence of spectra
taken over successive time intervals reveals how the
signal's frequency distribution changes with time.
For
musical signals, spectral distribution generally changes with
time: the amplitudes and, in many cases,
frequencies of individual spectral components vary throughout a
signal's duration. Differences in the amplitude envelopes
of individual components correspond to changes in
sound quality, which will be explored in the Timbre module.
A waterfall plot
(see to the right) is one of the possible graphical
representations of time-variant spectral distributions.
Waterfall plots are three-dimensional representations of
spectra that include frequency
(x axis), time (y
axis), and amplitude (z axis). _ Spectral envelopes for successive time intervals are displayed
along the frequency axis. _ Amplitude envelopes for individual
frequency components are
displayed along the time axis.
Consider a complex signal constructed
by adding two sinusoidal components with frequencies:
f1 = 250Hz and
f2
= 500Hz. 500Hz is twice 250Hz so the two components form
a harmonic signal. Its fundamental frequency is 250Hz, and its
perceived pitch generally corresponds to that frequency.
The graph below-left is the complex
signal produced by adding the two sinusoidal signals (shown in the graph
to the left).
The graph below-right shows the corresponding spectrum,
indicating the frequency and amplitude of each sinusoidal component.
Relationship between harmonic frequency components
and musical notation
Animation of the first 31 spectral components of a harmonic signal
with a fundamental frequency of 110Hz, corresponding to the musical
note A2). The components
are played in sequence (clip by Mladen Milicevic)
(a):
The musical pitch A2 in musical notation (fundamental frequency
of 110Hz). (b): The frequencies and approximate musical notation of
the first six harmonic components
of that note. (Adapted from Campbell and Greated
(1987), Musician's Guide to Acoustics).
Ideal Signal-Forms (also called 'waveforms') and
Spectra,
Key: A: amplitude of the first
component; n: component number
(a) Sawtooth signal
An ideal sawtooth signal
is formed by summing an
infinite series of harmonic components, with amplitudes proportional to A/n.
The odd- and even-numbered components have a 1800 phase
difference (image to the right).
In the
spectrum, the amplitudes of the even-number components are shown
on the opposite side of the reference axis from those of the
odd-numbered components to illustrate this phase difference.
(b) Triangular signal
An
ideal triangular signal is formed by summing only the odd-numbered harmonics,
with amplitudes proportional to A/n2. Successive
odd-numbered components have a
1800 phase difference (image to the right).
(c) Square signal
An ideal square signal is formed by summing
only the odd-numbered harmonics,
with amplitudes proportional to A/n. and in phase with one another
(image to the right).
Spectra
obtained through Fourier analysis
are rarely as clean and precise as the idealized examples shown
above. In practical analyses, the resulting frequency
components are often represented by frequency
bands (i.e. ranges) rather than exact, single-frequency
values (compare the left and right
figures below).
This
limitation reflects a fundamental
time-frequency trade-off. Improving
frequency resolution comes at the
expense of time resolution and vice
versa:
The more precisely we want to
track changes in a sound's energy
over time,
the shorter the analyzed time
interval must be. Short intervals,
however, produce broader frequency
bands and less precise frequency
estimates.
The more precisely we want to
determine the frequencies of the
components present in a sound, the
longer the analyzed time interval
must be. Longer intervals, however,
average out activity that occurs
within their duration, erasing any
changes.
This trade-off is related
conceptually to
Heisenberg's
uncertainty principle, named after 20th century German physicist Werner Heisenberg.
In quantum mechanics, the principle
states that certain pairs of
physical properties, such as
position and momentum, cannot both
be determined with increasing
precision.
In
addition, standard Fourier analysis
methods involve several assumptions
and practical limitations. For
example, the analysis: (a)
treats the analyzed signal as though
it extends indefinitely in time. (b)
assumes that all the energy, within
a frequency band returned in the analysis, lays at the band's upper end.
When these assumptions do not
adequately describe the signal or
the analysis conditions, the
resulting spectrum may exhibit spectral smearing:
apparent spectral energy spread
around the actual components,
introducing components that are
artifacts of the analysis process (see
below-right).
Frequency and amplitude values of the spectral components
used to synthesize the complex signal to the right.
Complex signal synthesized by adding
the two sinusoidal signals
described by
the spectrum
to
the left.
Spectrum returned from standard Fourier analysis of
17ms-long
portions
of same signal.
The short analysis interval provides relatively good temporal resolution
(17ms or 0.017s) but limited frequency resolution
(1/0.017 = ~59Hz). The two components consequently appear as broadened spectral bands rather than perfectly sharp spectral lines,
surrounded by spectral artifacts.sp;
Amplitude and Frequency
Modulation
In broadcast technology,
Amplitude
Modulation (AM) and Frequency Modulation (FM) are signal-processing techniques used
for the wireless transmission
of audio signals.
Audio signals can be
transmitted over large distances more effectively and with
less interference when they are carried by higher frequency signals.
This is because higher-frequency signals:
a) have more energy ( energy is proportional
f2 ), and are more directional, supporting more
efficient transmission and radiation of
electromagnetic energy
and b)
correspond to shorter wavelengths than
lower-frequency signals, requiring
smaller transmitting and receiving
antennas.
Consequently, audio signals are not
broadcast directly but in the form of
modulations (i.e. modifications) of high-frequency
"carrier" signals. A common application
is
AM and FM Radio.
High-frequency sinusoidal carrier (not shown) modified by a
low-frequency signal (black)
through
a) amplitude
modulation (producing the signal in red) or
b) frequency modulation (producing the signal in blue).
In
their simplest
forms:
i) Sinusoidal
Amplitude Modulation
(AM) involves
varying/modulating the amplitude
of a sinusoidal high-frequency carrier
signal according to a sinusoidal
low-frequency modulating signal. The
rate of amplitude variation equals the
modulating frequency, while the extent
of variation (i.e., the
modulation depth) depends on
the modulating amplitude.
The analogous musical effect is
tremolo,
perceived as periodic fluctuations in
loudness, similar to
those produced by interference and beating.
ii) Sinusoidal
Frequency Modulation
(FM) involves varying the
frequency of a sinusoidal high-frequency carrier
signal according to a sinusoidal
low-frequency modulating signal. The
rate of frequency variation equals the
modulating frequency, while the extent
of frequency variation (i.e., frequency
deviation) depends on the modulating
amplitude and defines the modulation
index [
optional details ] The
analogous musical effect is
vibrato,
perceived as periodic fluctuation in
pitch.
(a) Amplitude Modulation: A low-frequency
sinusoid (top) modulates the amplitude of a
high-frequency carrier (not shown), producing the AM
signal (bottom).
(b) Frequency Modulation: A low-frequency
sinusoid (black) modulates the frequency of a
high-frequency carrier (blue), producing the FM
signal (red, bottom).
Signal and Spectral Representations of
Amplitude-Modulation Rate and Depth
Based on the above graphs, can you deduce how modulation rate and depth
are represented in the spectrum and signalform of an
amplitude-modulated signal?
(answer below)
Sinusoidal amplitude modulation
of a sine signal with frequency fand amplitude
A), using a modulation rate fmod
(in Hz) and modulation depth m
(in %), produces:
a time-domain
signal whose amplitude fluctuates above and below
the original amplitude A, at a
rate equal to fmod
and by an amount determined nonlinearly by the
modulation depth m, and
a
frequency-domain spectrum containing three
sinusoidal components: the original carrier (frequency f, amplitude
A) and two
additional components, called sidebands,
whose frequencies and amplitudes depend on
the modulation rate and depth: _
Lower sideband:
Frequency flow
= f - fmod
Amplitude Alow
= 1/2mA _ Higher sideband:
Frequency fhigh =
f + fmod
-
AmplitudeAhigh = 1/2mA
Thus, the two sidebands have equal
amplitudes and are located symmetrically
above and below the carrier frequency.
In other words, modulation
rate determines the frequency separation between
the carrier and its sidebands, while modulation
depth determines the amplitude (and therefore
energy) of the sidebands relative to the
carrier.
For example:
Consider a
sinusoidal signal with frequency f
= 2000Hz and amplitude A = 4, amplitude-modulated at a
rate fmod = 10Hz and depth
m = 60% = 0.6.
Time-domain
representation
The resulting AM signal retains its carrier frequency
of 2000Hz, but its
amplitude fluctuates above and below A=4,
completing 10
amplitude fluctuations per second.
Frequency-domain representation
The resulting spectrum contains three
components: Carrier: Frequency f
= 2000Hz; Amplitude A = 4 Lower sideband: Frequency flow
= f - fmod =
2000-10 = 1990Hz; Amplitude
Alow = 1/2mA
= 0.5*0.6*4 = 1.2 Upper sideband: Frequency fhigh =
f + fmod
= 2000+10 = 2010Hz; Amplitude
Ahigh = 1/2mA
= 0.5*0.6*4 = 1.2
The resulting AM spectrum therefore consists of a 2000 Hz carrier with amplitude 4, accompanied by two equal-amplitude sidebands at 1990 Hz and 2010 Hz, each with amplitude 1.2.
Loyola Marymount University - School of Film &
Television
