Fundamentals of Sound
MODULE 2A: BASIC ACOUSTICS
Vibrations - Signals - Spectra - Envelopes - Waves

 

 
Physical attributes of acoustic waves - Part I: Introduction
   Vibrations, Signals, Signal Envelopes, Frequency/Period, Amplitude, Phase
   Logarithmic Scales
   Root-Mean-Square Amplitude
   Linear Superposition & Interference
Physical attributes of acoustic waves - Part II
   Sound Waves - Inverse Square Law 
   Standing Waves
   Fourier Analysis/Synthesis - Spectrum & Spectral Envelope - Modulation

 

 

 

   

Physical attributes of acoustic waves - Part I: Introduction

For an alternative self-study resource see Sound Waves (University of Salford)
 

 
Oscillation: A broad term, describing the variation (back-and-forth) of any physical or other property of a system around a reference value.

The reference value is not fixed for each system. Rather, it is always defined (and re-defined) in the context of some question of interest. Consider the example to the right:
Relative to the earth's center (i.e. with the earth's center as the reference point and simplifying the earth as a sphere), a given point on the earth's surface moves in a circle.
Relative to the sun's center, the same point moves in a circular spiral. Relative to the center of our galaxy, the same point moves in a helical spiral, ... and so on.
Vibration: The back-and-forth motion of a mass around a point of reference, called point of rest or equilibrium. Vibration is a mechanical phenomenon, a special case of oscillation.

Equilibrium: the condition arising when all forces applied to a system are perfectly balanced.

Unstable equilibrium: equilibrium that breaks down if disturbed (first image, below). 
 
Stable equilibrium: equilibrium that is maintained in spite of disturbances (second image, below). 

All musical instruments (and, more generally, all vibrating objects) operate under stable equilibrium conditions. During equilibrium there is no vibration and no potential for sound.
When performers pluck a guitar string, blow into a trumpet, or strike a drum, they apply a force that disturbs the given equilibrium, initiating vibrations that can produce sound.
Some time after they stop applying any force, the vibrations stop and the string / air column / drum head returns to its equilibrium state, due to friction "consuming" the energy that initiated the vibration.

Vibrations vs. Waves: In most cases:
a) observations on the vibrations of a sound source have corresponding features in the resulting sound waves, and
b) observations on sound waves have more-or-less direct origins in vibrations. 
Exceptions to such correspondence are rare but important and were first identified by Lord Rayleigh (John W. Strutt) in the late 1800s. We'll discuss some of them in the second half of the module.
 


     
Simple Harmonic Motion: the simplest type of vibration; for example, a free pendular motion. See the animation to the right (top).

In a simple harmonic motion, the restoring force, F=m*a (i.e. force pulling the vibrating mass m back towards its point of rest at an acceleration (a)) is proportional to the displacement (distance away) from its resting point.

Simple Harmonic Motion can be described graphically by a sinusoidal curve, usually referred to as  sine curve / sine signal. Such curves/signals are described by the following equation:  
y(t) = Asin(2πft+φ). For our course, you only need to develop a qualitative understanding of this equation.

In a signal that represents a simple harmonic motion
y: displacement (y axis)     t: time (x axis)   
A: amplitude or maximum displacement - for sine signals this is constant, represented by a single point on the y axis,    
f: frequency (cycles/s)
φ (or θ): starting phase/angle in the cycle (1 cycle->2π->3600    π: 3.14...; In terms of degrees, π: 1800, or 1/2 cycle)
 
The animation to the right (bottom), 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).
 
  
 
Complex vibration/wave:  a vibration in which the restoring force is related to displacement in a complex way. 

Complex vibrations are described graphically by complex curves. All such vibrations can be seen as the result of the combination of multiple simple harmonic motions. By extension, the curves/signals that describe them are essentially the sum of an appropriate number of sine curves/signals (see, for example, the animation to the right).
We will return to this when we discuss "Spectrum".

Practically all sounds we encounter or create correspond to complex waves, which have their origin in complex vibrations and are represented by complex signals, which, in turn, can be derived from (analyzed into) some set of sine signals each with its own amplitude, frequency, and starting phase values.
 

 
(Acoustical) Signal:  A two-dimensional graphic representation of a vibration/wave, plotting
_ y or vertical axis: displacement (move away from equilibrium) of some mass or a number
   of other concepts/measures (e.g. velocity, pressure)
   over
_ x or horizontal axis: time.

A signal is therefore a "time-domain representation" showing how some variable changes with time.
Signals of waves are also (erroneously) referred to as waveforms.

Sinusoidal (sine) signals represent sinusoidal vibrations/waves and complex signals represent complex vibrations/waves.
 
The signal, above, with its regular, sinusoidal peaks and valleys, represents a simple/sine/pure wave, corresponding in frequency to the note A3 (in equally tempered tuning).

The peaks of the signal appear at regular intervals of 4.54 milliseconds, or 0.00454 seconds, or 1/220th of a second. So the Period of the signal is 1/220th of a second. This means that the signal (and the wave it represents) repeats itself 220 times per second or has a Frequency of 220 Hertz (Hz). Note that Frequency=1/Period & Period=1/Frequency.
 

The signal, above, also represents a sine wave. This one corresponding in frequency to the note A4.

This signal repeats twice as fast as the one to the left, at a frequency of 440 Hz. The vibration it represents would travel through air to our ear about as quickly as the one represented to the left, but would cause the air molecules to vibrate at a rate twice as fast.
The Amplitude of both signals is represented by the distance between the top (or bottom) peak and the central horizontal line, representing the point of rest (equilibrium).
 

 

  		The Signal Envelope is a boundary curve that traces the signal's amplitude through time and, consequently, captures how energy in the signal changes with time. It encloses the area outlined by all maxima of the motion represented by the two-dimensional signal. It includes points that do not belong to the signal (in the images: signal is in blue, envelope is in red). 

For sinusoidal signals, the envelope is a horizontal straight line, parallel to the x axis (see above). For all other signals, the envelope has a different shape, made out of separate time stages/portions.
 
The terms describing a signal envelope's stages can be confusing.
We will adopt the terminology illustrated below left, used in digital filter and synthesizer design. The image below right (signal of a trombone tone) illustrates the more common envelope stage terminology, employed by musicians.

Signal envelopes and spectral envelopes (described under "Spectrum") are the two features that audio engineers are most likely to manipulate. We will be devoting substantial time to the exploration of their physical and perceptual correlates.
The attack stage captures how energy rises in a signal from 0 to its maximum level.

The shape and corresponding sonic characteristics of the attack depend on
a) the inertia characteristics of the sound source (related to source material, mass, shape, tension, etc.) and
b) the type of excitation (i.e. plucked, bowed, blown, struck, etc.).
The attack portion of a signal is perceptually salient (i.e. noticeable/characteristic), particularly so for impulse signals (see just below), where the attack portion contains most of the signal's energy. 

The decay portion captures how vibration energy in a system settles, following initial excitation and assuming continuous excitation.

For mechanical/natural (as opposed to electronic or digital) sound sources, the decay stage is the shortest envelope stage.

The sustain stage captures how a sound source responds during continuous excitation.

It is the most regular/periodic of the envelope stages and, for continuous signals, contains a sound's most salient features.
Note: in acoustics, the "sustain" stage of a signal envelope is also called "steady state," even though neither it's amplitude nor its spectral envelopes remain truly steady with time.

The release stage captures how energy in a system naturally dies out following the end of energy supply.

In most cases, the release stage is longer than the attack stage, with the natural resonant frequencies of a sound source dying out last. The broader the resonant response of a source the richer and shorter the release stage (more during the next module, under "Resonance").

Impulse signals: signals resulting from the instantaneous (very short) excitation of a sound source, which is then left to vibrate on its own until the energy dies out (due to some form of friction/resistance).
Impulse signals only include the attack and release (decay) envelope portions.
 
Continuous signals: signals resulting from continuous excitation of a sound source. Continuous signals include all four signal envelope stages, occurring in this order: attack, decay, sustain, release.

  
 

Periodic vibration/wave: A vibration/wave that repeats itself at regular time intervals.

Periodicity and regularity are attributes with specific perceptual significance. In general, regularity encourages prediction. In terms of sound, periodic waves give rise to well-defined pitch sensations.
 
Period (T): The time it takes for a vibrating body to complete a single full vibration, called cycle, represented on the x axis of a signal.
Period is measured in seconds/cycle and is the inverse of frequency f. So, T = 1/f .

Note that time (t) and period (T) are related concepts but do not describe the same thing, which is why they are represented by different symbols.
Frequency (f): Number of repetitions per unit time. Changes in frequency correspond primarily to changes in pitch.
Frequency is measured in number of cycles per second or Hertz (Hz - 1 Hz = 1 cycle/second), after German physicist Heinrich Hertz.
Frequency is the inverse of period T. So, f = 1/T.

Frequency values are not directly represented in signals. They are represented indirectly by period values T = 1/f.
Frequency values are directly represented on the x axis of a spectrum (see the section on "Spectrum," further below).

The relationship between rotation at a steady angular velocity (i.e. steady rotation at fixed values in angle-degrees/s) and vibration at a steady frequency (i.e. sinusoidal motion at fixed values in cycles/s) has been illustrated under "simple harmonic motion," above.

Amplitude (A): Maximum displacement ( of mass, velocity, pressure, etc.) away from the point of rest. Changes in amplitude correspond primarily to changes in loudness.
For sine signals, amplitude is represented as a single point on the y axis.
Note: for sound waves, displacement amplitudes of air molecules are minute (~ 1 millionth of an inch for speech).

As already noted, the y axis represents displacement (move away) from equilibrium (point of rest) as time passes, which has different values at different moments in time.
For a given sine signal, the maximum such value is that signal's amplitude.

For complex signals, amplitude is represented by a subset of points on the y axis of , tracked by the signal envelope.
For spectra, amplitude is represented by the entire y axis, which describes the total energy per frequency or frequency band within a signal (more in the section on "Spectrum," further below).


 

Phase: Position on the vibration cycle at a given moment in time.
Phase is measured in degrees (00-3600) or as a proportion of a full cycle = 2π radians (1/4 cycle: π/2;   half cycle: π;  etc.).

Changes in the absolute phase values of non-simultaneous signals are not perceptible.
E.g. If the signals, below, were played in succession they would sound identical, in spite of the fact there is a phase-shift (time-lag) among them. The same is the case for signals played simultaneously but having very different frequencies.
So, changes in the phase relationship among the spectral components of any given harmonic complex signal (see "Spectra", below) have no perceptual effect (examples).

 
In contrast, changes in phase relationships among multiple simultaneous signals, that are identical or close in frequency, are rather salient and correspond to loudness, timbre, pitch, and perceived source-location changes, to be described later in the course. Click on the image, below, for an example of the influence of phase-shift on timbre (i.e. sound quality).
  

  

  
 

 

Logarithmic Scale for Power, Intensity, and Pressure

 

Intensity (I): A measure of the amount of energy a vibration (wave) produces (carries) per second, through a unit area.
                       Defined as Power over Area. Measured in Watts per square meter (W/m2).

Power is proportional to the square of amplitude (A2) to the square of frequency (f 2) and, for waves in gases & liquids, to the square of pressure (P2). So:
Producing higher amplitudes/frequencies requires (emits) exponentially higher amounts of power (i.e. energy per unit time) and total energy and corresponds to exponentially higher amounts of Intensity.

The absolute units for Power (Joules/s or Watts), Intensity (Watts/m2), and Pressure (N/m2 or Pacals) are all linear. However, when applied to sound, all these quantities are usually expressed on a 10-base logarithmic (log) scale.

 

Short review of logarithms and logarithmic operations
 
Logarithmic operations are mathematically complimentary to power operations.

Reminders: 103 = 10x10x10 = 1000;     24 = 2x2x2x2 = 16;      32 = 3x3 = 9
 
                     xa * xb = xa+b - e.g.: 105 x 103 = 105+3 = 108
                     xa / xb = xa-b   - e.g.: 105 / 103 = 105-3 = 102    -    103 / 105 = 103-5 = 10-2 = 1/102

Logarithm - General Definition: The log with base x of a number y is the power a to which we must raise the base, x, in order to get y.
In math notation: logxy = a  <=>  y = xa .  [by definition, logx1 = 0 -  logx0 = undefined for all non-zero bases and has infinite values for base 0].

For example:
   log1010 = 1  because 10 = 101
   log101000 = 3  because 1000 = 103
   log102 = 0.3  because 2 = 100.3
   log22 = 1  because 2 = 21
   log28 = 3  because 8 = 23
   log210 = 3.322  because 10 = 23.322

Addition in log math is equivalent to multiplication in linear math.
For example:
    log101000 + log10100 = log10103 + log10102 = 3 + 2 = 5 = log10105 = log10100,000 = log10(1000 * 100)
 
Similarly, subtraction in log math is equivalent to division in linear math.
For example:
    log101000 - log10100 = log10103 -  log10102 = 3 - 2 = 1 = log10101 = log1010 = log10(1000/100)

 

Two reasons we use log rather than linear scales to measure sound Power, Intensity, and Pressure:
 
a) As stated in Fechner's Psychophysical Law (after 19th century German physicist and philosopher, Gustav Fechner), perception relates (approximately) logarithmically to physical stimuli. That is, multiplication in the value of a physical stimulus corresponds to addition in the resulting sensation. 
Therefore, log math provide a better model to the relationship between physics (waves) and sensation (sound) than linear math.
 
b) Log math permit the compression of very large scales into smaller and easier manageable ranges, which are still fine enough to match the resolution of perception.
For example, the linear range between the smallest (barely perceptible at 1,000Hz) and largest (harmful to the ear) audible Intensities is |10-12 to 1 W/m2|,  amounting to 1,000,000,000,000 (1 trillion) steps.
In contrast the same range in log math is:
|log1010-12 - log101| = |-12-0| = 12 Bells (after 19th-20th century Canadian-American inventor, scientist, and engineer, A.G. Bell) or 120 decibels (dB) ("deci-": Latin for "one tenth").

Smallest perceivable, or REFERENCE sound level: 0dB (less than 1 billionth of change in atmospheric pressure)
     0dB corresponds to a) Power = Wref = 10-12 W,  b) Intensity = Iref = 10-12 W/m2, and  c) Pressure = Pref = 2*10-5 Pa (Pa: N/m2)
Largest safe sound level: 120dB
     120dB corresponds to a) Power = 1 W,   b) Intensity = 1 W/m2, and   c) Pressure = 20 Pa

Note: Sound level measurements assume 10-base log values and the base is usually omitted form the notation. So: log1010-12 = log10-12

We 
So, we express Power, Intensity, and Pressure levels of a sound logarithmically, in dB (i.e. 1/10 of a Bell) because we are ultimately interested in the sensation of sound rather than simply in physical waves. In using logarithms, we essentially compare absolute power/intensity/pressure values to the corresponding reference 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 
(Optional: why the constant for SPL is 20  but for SWL and SIL is 10? - Tip: Power and Intensity are proportional to P2)

Key takeaway messages:

a) log sound levels in dB are not absolute but relative to some reference level corresponding to "barely audible";
b) dB levels derive from ratios of physical quantities and are therefore unit-less;
c) one cannot perform math operations (addition/subtraction/multiplication/division) on dB the same way one would do with linear (regular) numbers.

Key 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).
    e.g.     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 Increase (decrease) in sound level corresponds to +10dB (-10dB) and to "twice (half) as loud";
    100-fold Increase (decrease) in sound level corresponds to +20dB (-20dB) and to "4 times (1/4th) as loud";
    1000-fold increase (decrease) in sound level corresponds to +30dB (-30dB) and to "8 times (1/8th) as loud"; and so on.
    e.g. a) to increase an orchestra's average level by 20dB you need to combine it with another 99 identical orchestras for a total of 100;
            b) reducing the sound entering/escaping a recording studio by 30dB corresponds to blocking 99.9% of the original sound energy
                and 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, also identical and perfectly blending when heard simultaneously (e.g. all notes with 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)
Common sounds Noise Level [dB] Musical Dynamics Effect

Rocket launching pad
(no ear protection)

180

  

Irreversible hearing loss

Carrier deck jet operation
Air raid siren

140

  

Painfully loud

Thunderclap

130

  

 

Jet takeoff (200 ft)
Auto horn (3 ft)

120

  

Maximum vocal effort

Pile driver
Rock concert

110

  

Extremely loud

Garbage truck
Firecrackers

100

fff

Very loud

Heavy truck (50 ft)
City traffic

90

ff

Very annoying
Hearing damage (8 Hrs)

Alarm clock (2 ft)
Hair dryer

80

f

Annoying

Noisy restaurant
Freeway traffic
Business office

70

mf

Telephone use difficult

Air conditioning unit
Conversational speech

60

mp

Intrusive

Light auto traffic (100 ft)

50

p

Quiet

Living room
Bedroom
Quiet office

40

pp

 

Library
Soft whisper (15 ft)

30

ppp

Very quiet

Broadcasting studio

20

  

 

 

10

  

Just audible

 

0

  

Hearing begins

 

See also an additional table of commonly encountered sound levels.
We will return to this topic during our discussion on loudness.

 

  

  

Root-Mean-Square (Pressure) Amplitude
The amount of energy in a signal can be measured in terms of

a) 0-to-peak amplitude (maximum positive displacement away from rest; standard amplitude measure in the equation for the simple harmonic motion and in sinusoidal signals),
b) peak-to-peak amplitude (difference between positive and negative maximum displacement away from rest; useful when describing energy content of signals that are not symmetrical around the time axis; e.g. most signals generated by musical instruments)
c) root-mean-square (RMS) amplitude (for sinusoidal signals, RMSAmplitude = 0.707*PeakAmplitude). 

Root mean square Pressure or PRMS is a measure of the area outlined by the signal (see Figure b, below) and is mathematically defined as the square root of the average of the square of the pressure of the sound signal over a given duration (usually one period):

PRMS = (P2)average0.5 

To calculate the RMS of any set of numbers:
     i) square each number in the set,
     ii) calculate the set's average and
     iii) calculate the square root of the average.
This process is necessary for all quantities that, like sound signals, oscillate between positive and negative values.
For example, just calculating the straight average amplitude over one or more full cycles of any sine signal would result in 0.

For sine signals, peak and RMS amplitudes are interchangeable since their relationship is fixed.
(RMSAmplitude = 0.707*PeakAmplitude)
 
In contrast, two complex signals with the same peak amplitude may have different RMS amplitudes and vice versa.
The complex signal with the higher RMS amplitude (e.g. the one to the right, bottom) will sound louder (more in the "Loudness" module).
Figure (a) 0-to-peak, peak-to-peak, and RMS amplitudes of a sine signal
Figure (b) RMS amplitude is a measure of the area outlined by the signal (highlighted).

 

Application Example:

FCC rules mandate that the loudness of TV or Internet advertisements do not exceed that of other adjacent programming. This mandate was originally enforced in terms of peak amplitude.
Advertisers used dynamic compression to raise RMS amplitude while staying within the peak amplitude limits, winning the "loudness war."

In 2013, FCC rules were updated to enforce more sophisticated compliance assessment measures (e.g. LUFS: Loudness Units Full Scale). LUFS units take into account the loudness implications of
     a) the difference between Peak and RMS amplitude and
     b) the dependence of loudness on frequency (more in the "Loudness" module).
The same scale is also used by music streaming services in order to normalize, as much as possible, loudness levels among songs and reduce the need for users to significantly change listening levels as they go through a playlist.

The above issues will be addressed during our "Loudness" module and are directly relevant to anyone planning to enter the sound-for-picture business. ;

 

  

 
Linear Superposition & Interference

 

The linear superposition principle states that, at any given moment in time, the total displacement of two superimposed vibrations (waves) is equal to the algebraic sum (i.e. a sum that takes into consideration signs: +, -) of the displacements of the original vibrations (waves). 
This principle was first introduced by Bernoulli (late 1700s) and came out of the study of strings, considered as one-dimensional systems.  It stated that many sinusoidal vibrations can co-exist on a string independently of one another and that the total effect at any point is the algebraic sum of the individual vibrations. 

To be more precise, if the compound vibration at any point of a string is the algebraic sum of the individual vibrations, it is the sinusoidal waves (and not the vibrations) that can co-exist on a string independently of one-another.

 

The interference principle (see the animation, below) is an extension of the linear superposition principle. It states that, 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 combining two or more vibrations (waves) with different frequencies (f1, A1 & f2, A2), their periodically changing phase relationship results in periodic alterations between constructive and destructive interference, giving rise to the phenomenon of Amplitude Fluctuation: amplitude that fluctuates periodically between A1+A2 and A1-A2, at a rate equal to the frequency difference of the original vibrations (waves) |f1-f2|. 
 


 

In the graphs, below, two sines A & B have the same amplitude A and the same starting phase, but different frequencies (f1 & f2).

Adding them together will produce a complex signal (A+B) whose amplitude fluctuates between a max (2A) and a min (0) value because their periodically changing phase relationship (gradually moving from perfectly in-phase to perfectly out-of-phase) results in a periodic and gradual move between constructive and destructive interference at a rate equal to |f1-f2|.
Depending on their rate, such fluctuations can be perceived as beating (sound with slowly fluctuating level), roughness (rattling sound), or combination tones (tones additional to the original tones), to be described later in the semester.
 


Interactive tool exploring interference between two sines with different frequencies
(alternative interactive tool)

 

As an extreme example of destructive interference, if two vibrations (waves) have the same frequency, amplitude, and source location but a 1800 phase difference, the two vibrations (waves) will cancel one another, resulting in no vibration (wave) at all.

Complete (or almost complete) destructive interference is often used in active noise and feedback reduction systems. According to a short story in the December 10, 2004 issue of New York Times, two 2005 Honda car models were the first to carry active noise reduction systems, based on this principle. Versions of this feature are by now commonplace in car manufacturing and, of course, headphones.
 

Constructive Interference
 

 

Sines A & B have the same frequency, amplitude, and phase (i.e. 00 phase difference). Their addition results in total constructive interference: the combined signal's amplitude is equal to the algebraic sum of the initial amplitudes (assuming both signals originate from the same point in space).

Sines A & B have the same frequency and amplitude but a 1800 phase difference. Their addition results in total destructive interference: the combined signal has amplitude 0, which is, again, the algebraic sum of the two amplitudes. Since the sines are perfectly out of phase, at every moment in time their displacements have equal values but opposite signs, resulting in a sum of 0 (assuming both signals originate from the same point in space).

 

 

 

 

Physical attributes of acoustic waves - Part II

 

Sound waves - Inverse Square Law

 

 
Wave: Transfer of vibration energy across a medium (e.g. air, string, etc.).

Simple/Sine Wave: Transfer of a simple harmonic motion across a medium.
Complex Wave: Transfer of complex vibration/motion across a medium.

Waves originate in but are not equivalent to vibrations. Waves depend on the properties of the propagation-medium while vibrations do not.
Sound waves in air are manifested as alternating condensations/compressions (with pressure values above atmospheric pressure) and rarefactions/expansions (with pressure values below atmospheric pressure) that spread away from the vibrating source longitudinally (i.e. parallel to the direction of vibration), in the form of pressure fluctuations.

The video, below, is a visualization of sound waves caused by a vibrating string. [Defining waves is a complex task. See this extended definition]

 

 
For signals of sound waves, the x axis may represent one of two measures:

a) time, outlining the wave's period; in this case, the signal is temporal and describes how wave energy varies with time, at a single point in space (graph the the right-top).
In temporal signals, the phase of a wave is different at different moments in time, reflecting the vibration of air particles at a given point in space.

b) distance, outlining the wave's wavelength (i.e. the distance the wave will travel during a period); in this case, the signal is spatial and describes how wave energy is spread in space at a given moment in time (graph the the right-bottom).
In spatial signals, the phase of a wave is different for adjacent points in space, because the vibration energy propagating through the air in the form of a wave reaches these 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 is a function of a vibration's period T (and, consequently, frequency f) and of the speed of wave propagation c within a given medium: λ= c*T (equivalent to λ= c/f).
In all cases, the higher the frequency the shorter the period and the shorter the wavelength.

 


(click here for a wavelength calculator)

 
Speed of sound (c): c = (E / ρ)0.5 where E is Young's modulus in N/m2 (measure of stiffness) and ρ is density in Kg/m3.
For example: Speed of sound on a string with tension T and linear density ρ (density: mass per unit length): c = (T / ρ)0.5.

Our key take-aways from this equation are:
_ the stiffer/tenser the medium (e.g. metal vs. wood vs. air; looser vs. tighter string or drumhead) the higher the speed of sound;
_ the denser/more compact the medium (e.g. cold vs. hot air / water vs. wood) the lower the speed of sound.

See to the right for a visualization of the impact of stiffness on the speed with which sound waves propagate through a medium.
     * In this animation, the distance among molecules per medium represents degree of
       bond strength (i.e. stiffness), not density. As noted, higher density corresponds to
       lower sound speed.

Speed of sound in air:    c = 345m/s at 210C     or    1130ft/s at 700F
[C: temperature in Celsius - F: temperature in Fahrenheit]

The speed of sound increases with air temperature because higher temperatures expand the air and
a) reduce its density, if the air is unconstrained and free to expand (e.g. outdoors), or
b) increase its stiffness, if the air is constrained and cannot expand (e.g. indoors).

The speed of sound increases by 0.6 m/s for each Celsius degree of increase in temperature.
More generally: cCelsius = 345 + (C-21)*0.6 m/s.
The speed of sound increases by 1.1 ft/s for each Fahrenheit degree of increase in temperature.
More generally: cFahrenheitt = 1130 + (F-70)*1.1 ft/s.

Unit conversions & formulas  
    1 foot = 0.3048 meters;    1 meter = 3.281 feet;    C = (5/9)*(F-32);    F = (9/5)*C+32

 

Transverse waves: waves propagating perpendicularly to the motion of the vibration
(e.g. waves on a string, waves on the front and back plates of a violin, waves on a drumhead, etc.)
 
 

 

Longitudinal waves: waves propagating parallel to the motion of the vibration
(e.g. sound waves in air or other gases).
 
           Images sources: https://waitbutwhy.com/2016/03/sound.html    -    https://www.acs.psu.edu/drussell/demos/waves/wavemotion.html

 

Source: Institute of Sound and Vibration Research

 
Experiment with this wave simulator

 

Inverse Square Law

 

The Inverse Square Law states that the intensity of a wave at a given point in space is inversely proportional to the square of the distance from its source.

For example:
_ doubling the distance from a source will reduce the Intensity by a factor of 2x2=4, corresponding to a 6dB loss (3dB loss for each halving of Intensity).
_ increasing the distance tenfold will reduce the Intensity by a factor of 10x10=100, corresponding to a 20dB loss (10dB loss for each tenfold reduction in Intensity).
_ decreasing the distance by a factor of 100 will increase the Intensity by a factor of 100x100=10,000, corresponding to a 40dB gain (10dB gain for each tenfold increase in Intensity).
_ etc.

The underlying assumption is that the energy from a vibrating source spreads away from the source in a spherical manner. This applies
a) theoretically to all frequencies, for the idealized case of a "point source" &
b) practically to low frequencies, which are more likely to have wavelengths that are much larger than the source's dimensions (see diffraction, next module).

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 equation I = W/A ? ]

 
                                                  


 

 

 

Standing Waves

 

Standing waves are vibration patterns (on strings, air columns, listening environments, etc.) arising 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 and arise from the combination of constructive and destructive interference between a sound wave and its reflection(s). Standing waves can also be understood as a form of resonance, occurring at an oscillating system's natural frequencies (see "Resonance" in the next module).
Experiment with this standing wave simulator.

Reminders:
The linear superposition principle states that, at any given time, the total displacement of two superimposed vibrations (waves) is equal to the algebraic sum (i.e. a sum that takes into consideration signs: +, -) of the displacements of the original vibrations (waves). 
The interference principle is an extension of the superposition principle. It states that, 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.

 

Trapped waves create standing waves only if their wavelength λ is an integer divisor of twice the distance d between boundaries. So: λ ϵ D(2d)
Conversely, trapped waves create standing waves only if the distance d between boundaries is an integer multiple of the waves' half-wavelength λ/2. So: d = n * λ/2 (n: integer).

Example:
Assuming speed of sound c = 360m/s and two reflecting boundaries d = 6m apart:
The longest-wavelength standing wave will be λ = 2d = 12m and will correspond to frequency f = c/λ = 360/12 = 30Hz
Additional standing waves will be created with integer devisor wavelengths (12/2=6m; 12/3=4m; 12/4=3m; etc.) and, consequently, integer multiple frequencies (30x2=60Hz; 30x3=90Hz; 30x4=120Hz; etc.)

Standing waves are characterized by fixed points along the wave 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 undesirable in listening environments.
 

 

Transverse Standing Waves arise when energy propagating as a transverse wave is trapped within two or more, appropriately positioned, reflective boundaries.

Their generation is quite 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 are also generated on drum skins, cymbals, xylophone/vibraphone bars, and other material used in musical instrument construction.

The animation, to the right, illustrates complex patterns of transverse standing waves (referred to as Chladni patterns/figures, after 18th-19th century German physicist and musician, Ernst Chladni) forming on a metal plate that is being excited at two different frequencies. Sand granules settle at the nodes.

The animations below-right, simulate standing waves on a string. All these waves occur simultaneously, combining into a single signal (more under "spectrum" and during the following modules).

(source)

 
Longitudinal Standing Waves arise when energy propagating as a longitudinal wave is trapped within two or more, appropriately positioned, reflective boundaries.

Longitudinal standing waves can be generated inside any air cavity.
_ Standing waves within the air cavities of musical instruments are useful and, in some cases necessary, amplifying and "coloring" the sound produced.
_ Standing waves in air cavities outside a musical instrument (e.g.t (> in any enclosed listening space) are undesirable because they contribute to uneven
   sound level and frequency distribution within the space.


(source)

Interactive tool exploring standing waves (The Physics Classroom)

   

Interference of sound waves from sources separated by some distance

A phenomenon analogous to standing waves on strings or air columns is observed if two identical, interfering waves originate from two sources that are separated by some distance. In this case, the observed interference pattern (i.e. gradual shifts between constructive and destructive interference) spreads away from the sources and outlines areas of minimum (nodes) and maximum (antinodes) sound levels in space (a form of moving "standing" wave!).

In the following example, the animation portions with contrasting alternating colors correspond to areas of higher levels (maximum level at each sector's center) and animation portions with minimal change in color correspond to areas of lower levels (minimum level at each narrow band's center).
This phenomenon arises, for example, whenever two or more loudspeakers produce exactly the same sound waves, as is the case during mono playback over a stereo system, or during many live sound reinforcement contexts that don't involve professional engineers.

 

 
Interference patterns and the associated SPL (sound pressure level) spatial non-uniformity are particularly pronounced in open-air performances, where there are no reflections to diffuse wave energy and, potentially, improve SPL uniformity.

Play around with the interactive animation, below, to qualitatively explore the relationship among sound source distance, sound wave frequency, and interference patterns.

 

[source]
For a slightly different visualization, click here.
 

 

 

Fourier Analysis/Synthesis - Spectrum & Spectral Envelope - Modulation

 

Fourier's mathematical law (after 18th-19th century French mathematician and physicist, Joseph Fourier) states that all complex signals can be reduced to the sum of a set of sines, called the Fourier/sine/spectral components or partials, which make up the complex signal’s spectrum. 
     _ Analysis of a complex signal into its sine components is called Fourier analysis.
     _ The reverse process of constructing a complex signal out of a set of sinusoids, is called Fourier synthesis.
The figure to the right provides a graphical and a mathematical expression of this statement (we will not be performing any operations with the equation). 

Periodic signals are also called harmonic signals (such as the signals corresponding to most solo musical sounds) and have sine components whose frequencies are integer multiples of the lowest frequency component, referred to as:
the 1st harmonic component or the "fundamental", usually designated as ƒ0.
All other components are called harmonic components or just harmonics.

If the fundamental component has frequency 1ƒ0, then the components above the fundamental have frequencies 2ƒ0, 3ƒ0, 4ƒ0 (referred to as the 2nd, 3rd, 4th harmonic) and so forth. 

Periodic (harmonic) complex signals have a rather definite pitch that matches in frequency the frequency of the fundamental component. This appears to be the case even if the fundamental component is not present in the signal’s spectrum (more during the module on "pitch"). 

All other types of signals are non-periodic and are called inharmonic signals. 
Non-periodic (inharmonic) signals have a rather indefinite pitch or no pitch at all, depending on the degree of inharmonicity (i.e. on how far away its frequency components are from an integer multiple relationship) and on the absolute and relative duration of their spectral components (more during the discussions on pitch, loudness, and timbre).

The term "overtones" is occasionally also used to refer to any components with frequencies above that of the lowest component. They are designated as harmonic overtones to refer to to the upper components/harmonics of harmonic signals or inharmonic overtones to refer to the upper components of inharmonic signals.

  
    
Experiment with this Fourier synthesis java applet.

 
Here is an additional Fourier Synthesis Simulation (you may need the Java console, available for download here).

You may also download the Fourier-based Synthesizer used in class (Custom application, Windows only).
To install, save the .zip zipped file, double-click on it to un-package it, and run the SETUP.EXE file.


 

Spectrum

Spectrum (or frequency spectrum) is a two-dimensional graphic representation that indicates the frequency (x axis) and amplitude/level//pressure/power (y axis) values of a signal's sinusoidal components.
It is a "recipe," indicating the type (frequency) and amount (amplitude) of each ingredient (sine component) contained in or required to make the corresponding complex signal. Spectra are frequency-based representations of sound, while signals are time-based representations.

Analogously to Signal Envelope, already described, the Spectral Envelope is a boundary curve that traces the peaks of the spectrum, capturing how energy is distributed across the frequency range (in the image to the right: spectral components are in blue, spectral envelope is in black). 

 

 
Spectra always illustrate a signal's "recipe" for a specific time-slice, with multiple successive such time-slices illustrating how the signal's frequency energy distribution changes with time.

For all musical signals, spectral distribution does change with time. Consequently, the amplitude and frequency values of each spectral component also change with time. Differences in amplitude envelopes among a spectrum's sinusoidal components correspond to sound quality changes that we will discuss during the "Timbre" module.

The "waterfall" visualization (to the right) is a 3-D spectrum that includes frequency (x axis), amplitude (z axis) and time (y axis).
Spectral envelopes per time slice are displayed along the x (frequency) axis.
Amplitude envelopes per frequency component are displayed along the y (time) axis.   

See and hear spectrograms of sample instruments.

 

2 sines

 

Graphical  Example of Synthesis

Complex signal made out of two sine components with frequencies
f1 = 250Hz and f2 = 500Hz.
The pitch of a complex tone corresponding to this complex signal would match the pitch of the fundamental component (250Hz).

The graph below-left is the complex signal resulting from the addition of the two sine signals in the graph to the left.
The graph below-right is the spectrum of the complex signal, indicating the frequency and amplitude values of the sine components making up the complex signal.


superposition (addition) of the 2 sines

 

the spectrum of the above signal

 

 

Relationship between harmonic frequency components
and musical notation

 

Animation of the first 31 spectral components of a harmonic signal beginning at 110Hz, (tone with pitch A2), played back in sequence (clip by Mladen Milicevic)

Harmonic components of the note A3

  

 

(a): A2 musical pitch (fundamental frequency: 110Hz), represented in musical notation.
(b): the frequencies and (approximate) notational representation of the first six harmonic components of the tone in (a)
(after Campbell & 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) An ideal sawtooth signal (far right) is formed by summing an infinite number of all harmonic components,
with amplitudes A/n and 1800 phase-shift between odd- and even-numbered components, illustrated on the spectrum (right).

The blue points marking the amplitudes of the even components are on the opposite side of the blue points marking the amplitude of the odd components, to illustrate the described phase shift.

(b) An ideal triangular signal (far right) is formed by summing an infinite number of only odd-numbered harmonic components,
with amplitudes A/n2 and 1800 phase-shift between successive odd components, illustrated on the spectrum (right).

(c) An ideal square signal (far right) is formed by summing an infinite number of only odd-numbered harmonic components,
with amplitudes A/n and in-phase with one another, illustrated on the spectrum (right).

 

Animated spectra; 4 ideal signal-forms
 

(Different animation angle)
 

First 6 harmonic components of
a sawtooth signal analogous to those generated by strings
 
Pitch/note values of each component of the signal to the left
(assuming fundamental = 256Hz) &
corresponding standing waves on a string 

  

Experiment, again, with this Fourier synthesis java applet
   [Optional: detailed discussion of spectra]

  

 

Fourier analysis drawbacks

Spectra arising from Fourier analysis are, unfortunately, never as clean and precise as the ones in the above figures. 
The frequency values returned from the analysis are frequency ranges/bands, not single frequency values. In the figures, below, compare the spectral lines (left) vs. bands (right).

There is a time/frequency trade-off in that:

a) the more precisely we want to know how the energy of a spectrum changes with time, the broader will be the frequency bands returned by the analysis and the less precisely the frequency values will be represented and
b) the more precisely we want to know the frequencies in a spectrum, the longer the signal chunks analyzed will have to be and the less precisely the level changes of a sound over time will be captured. 

In other words, we can only increase the frequency resolution of the analysis at the expense of the time resolution, and the converse.
This trade-off is related to Heisenberg's uncertainty principle, (after 20th century German physicist, Werner Heisenberg). Oversimplified, it states that the more precisely we determine the position of a particle the less precisely we may know its momentum (momentum = mass*velocity), and the converse.

In addition, there is a series of assumptions accompanying the standard mathematical implementation of Fourier analysis:
   (a) the signal analyzed has infinite duration and
   (b) all the energy within a frequency band, returned in the analysis, lays at the band's upper end.
Violation of these assumptions results in spectral "smearing." (i.e. spectral components that are artifacts of the analysis process, surrounding the actual spectral components of a signal - 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 resulting from Fourier analysis of 17ms-long portions
of the same signal.
Temporal resolution: 17ms or 0.017s
 Frequency resolution: 1/0.017 =~ 59Hz.

 

Amplitude and Frequency Modulation
 

In broadcast technology, the terms Amplitude Modulation (AM) and Frequency Modulation (FM) describe two specific signal processing techniques used in the wireless transmission of audio signals.

In simple terms, audio signals can be transmitted over large distances more robustly and with less contamination if they are "carried" by higher frequency signals, which
   a) have more energy (energy is proportional to the square of frequency),      and
   b) can be transmitted/received by smaller antennas than those required 
       for lower frequencies
       (the higher the frequency the shorter the wavelength).

So, rather than being transmitted directly, the intended signal is transmitted in the form of a modulation (i.e. modification) of a high-frequency carrier signal.
Most common application: 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 its simplest case:

i) Sinusoidal Amplitude Modulation refers to the sinusoidal variation of a carrier signal's amplitude, at a rate equal to the modulating signal's frequency and at a degree (or depth) equal to the modulating signal's amplitude.
The analogous musical term is "Tremolo," describing the sensation of slow loudness fluctuations, similar to the sensation associated with interference and "beating."
 
ii) Sinusoidal Frequency Modulation refers to the sinusoidal variation of a carrier signal's frequency, at a rate equal to the modulating signal's frequency and degree equal to some proportion of the modulating signal's frequency, depending on "modulation index." [  optional details  ]
The analogous musical term is "Vibrato," describing the sensation of slow pitch fluctuations. 

In (a), below, the bottom signal is an AM signal, resulting when the top sine signal modulates the amplitude of a high-frequency carrier sine signal (not shown).
In (b), a low-frequency sine (black) is modulating a high-frequency carrier sine (blue), resulting in the FM signal at the bottom (red).
 

 
(a)
 

 
(b)
 
Experiment with these AM & FM applets

 

 

Signal and spectral representations of amplitude-modulation rate and depth
 
   


Based on the above graphs, can you deduce the spectral correlates of the rate and depth parameters of amplitude modulation?
(answer just below)

Sinusoidal amplitude modulation [with modulation rate fmod (in Hz)  and modulation depth m (in %)] of a sine signal (with frequency f and amplitude A) results in a signal with

a) amplitude that fluctuates above/below A by an amount that depends on the modulation depth and at a rate equal to the modulation rate, and
 
b) spectrum that has three components; the original sine (f; A) and two sidebands, also determined by the modulation rate and depth:
    _ a low-frequency sideband,   flow  = f - fmod    -    Alow  = 1/2mA  
    _ a high- frequency sideband, fhigh = f + fmod   -    Ahigh = 1/2mA   (Ahigh = Alow)
 
In other words, modulation depth defines the energy shared by the two sidebands as a percentage of the energy of the original sine

For example:

If the amplitude of a sine signal (f = 2000Hz and A = 4) is modulated at a rate fmod = 10Hz and depth m = 60%  then
 
     a) the resulting AM signal will have the same frequency (fAM = 2000Hz) but amplitude that fluctuates above/below A=4
         at a rate equal to 10 times/second.
     b) the spectrum of the resulting AM signal will have three components: the original sine (f = 2000Hz and A = 4) and two sidebands:
     flow  = f - fmod  = 2000-10  = 1990Hz    -    Alow  = 1/2mA = 0.5*0.6*4 = 1.2
     fhigh = f + fmod = 2000+10 = 2010Hz    -    Ahigh  = 1/2mA = 0.5*0.6*4 = 1.2

 

  

Loyola Marymount University - School of Film & Television