Acoustic Waveforms Acoustic Waveforms Simple Harmonic Motion ...

Acoustic Waveforms 

Robert Mannell 

Macquarie University 

Simple Harmonic Motion (1) 

• A single cycle of a sine wave can be 

depicted as if it were a point on a circle 

moving anti-clockwise (they are 

mathematically equivalent). 

• As we move anti-clockwise from the 3 

o’clock position (0°), we pass through 90° 

(12 o'clock) 180° (9 o'clock), 270° (6 

o'clock) and 360° same as 0°. 


• Note that simple harmonic motion (SHM) 

is a characteristic of many physical 

systems, including a pendulum, a child’s 

swing or a tuning fork. 

• Ideal systems oscillate forever. 

• Natural systems displaying SHM gradually 

or quickly reduce in amplitude. This is 

known as “damping”. 

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Acoustic Waveforms 

In this topic we will examine:- 

1. Simple Harmonic Motion 

2. Continuous Waveforms and Damping 

3. Adding Waveforms and Phase 

4. Speech Waveforms 


• Note that as we rotate anticlockwise from 0° we rise to a 

maximum value, back to the starting value (zero) and 

then to the minimum value and back to zero again. 

• Note that the angle can also be expressed in radians, 

where 2π (2 x pi) radians equals 360°. 

• One whole rotation (2π or 360°) is equal to the period (T) 

Waveforms and Damping (1) 

• A sine wave is a waveform exhibiting ideal 

simple harmonic motion. Such a system 

loses no energy (or has its energy 

replenished from outside the system). 

• Such a waveform can also be called a 

continuous waveform as it continues 

forever without eventually reducing to zero 

intensity. 

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• A sine wave is a waveform generated by a 

system that is characterised by ideal 

simple harmonic motion. 

• A sound wave exhibiting these 

characteristics would be a pure tone. 


• This loss of energy in an oscillating system 

is know as damping. 

• A damped waveform is also know as a 

non-continuous waveform. 


• The terms weakly damped and strongly 

damped are relative terms. 

• When compared to speech sounds, a 

tuning fork is said to be weakly damped as 

it dies down very slowly when compared to 

each of the glottal pulses generated by the 

vocal folds during phonation (glottal pulses 

die out in a very small fraction of a 

second). 

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• Real systems are never ideal. All naturallyoccurring 

systems lose energy (eg. as heat due 

to friction). 

• A tuning fork lasts a while, but doesn't ring 

forever. Its intensity decreases over a relatively 

long period (say about half a minute). 

• A tuning fork looses energy as heat (both 

internally as a consequence of heat loss during 

physical deformation and externally as a 

consequence of friction with air). 


• A damped waveform can die out quickly or 

slowly. 

• A waveform that dies out quickly is said to 

be strongly damped, as it loses energy 

quickly. 

• A waveform that dies out slowly is said to 

be weakly damped, as it loses energy 

slowly. 


• Damping is also a characteristic of systems that 

produce sounds with very complex spectral 

patterns. 

• If you hit your fist on a wooden table it will 

produce a "bang" that dies out very rapidly. The 

"bang" possesses a very complex aperiodic or 

random spectrum that dies out in less than a 

second and so the sound and the system 

generating it are much more highly damped than 

a tuning fork and the sound it produces. 

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• Note that it is common to refer to both a 

sound and the system that generates it as 

possessing some relative degree of 

damping. 

Adding Waveforms & Phase (2) 

Here we can see the effect 

of adding two pure tones, 

one of 100 Hz and the other 

of 500 Hz. The 500 Hz tone 

has half the sound pressure 

level of the 100 Hz tone. In 

the bottom part of the 

diagram we can see the 

two pure tones as dashed 

lines. A simple addition of 

the dashed lines results in 

the unbroken line. The 

unbroken line clearly has a 

more complex pattern than 

either of the two pure tones. 


• In the next slide we add together three 

waveforms with frequencies of 100, 200 

and 300 Hz. (Highest common factor 100) 

• They differ in the positions of the start of 

each wave cycle. 

• In the left image they all start at 0°. 

• In the right image they start at 0°, 90° and 

180° (going from the lowest to highest 

frequency) 

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• So far, we have examined simple 

waveforms that resemble sine waves. 

When we plot a pure tone we can easily 

see its period (or frequency) and 

amplitude (sound pressure level). 

• The vast majority of natural sounds are not 

pure tones but are complex sounds that 

can be thought of as the combination of 

two or more pure tones. 


• Note, in the previous slide, that the complex 

pattern repeats with the same period as the 

100 Hz tone. 

• 100 Hz is the highest common integer factor of the 

frequencies of the two tones (100 and 500 can 

both be divided by 100 to give an integer result). 

• The frequency of a complex wave is always equal 

to the frequency of the highest common factor of 

the sine waves being added to produce it. 

• The repetition frequency of the complex pattern is 

called its fundamental frequency (F 0). 


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• As you have seen, differences in the 

relative starting point of the three waves 

results in very different shapes to the 

resulting complex waves. 

• These differences in starting point are 

known as phase differences and these 

differences can be expressed as a phase 

angle (in degrees). 


When we add together two waves that differ 

slightly in frequency (less than 20Hz), the 

resultant waveform has a very low F0 which can 

be heard as an intensity fluctuation, which is know 

as beating. 

Speech Waveforms (1) 

• A waveform is a two dimensional representation of 

a sound. The two dimensions in a waveform 

display are time and intensity. The vertical 

dimension is intensity and the horizontal 

dimension is time. 

• Waveforms are known as time domain 

representations of sound as they display changes 

in intensity over time. 

• The intensity dimension actually displays sound 

pressure. Sound pressure is a measure of the tiny 

variations in air pressure that we are able to 

perceive as sound. Intensity in these waveforms is 

a simple linear scaling of sound pressure (not dB). 

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Here we can see 2 

waves (dotted lines) 

added together with 

different phase angles. 

At 0° and 360° (the two 

dotted lines are in the 

same place) we get full 

reinforcement. 

At 180° we get complete 

phase cancellation. 

In between we have 

different degrees of 

reinforcement or 

cancellation. 


• If you examine the topic web page at:http://www.ling.mq.edu.au/speech/acoustics/waveforms/adding_waveforms.html 

you can see some examples of how 

adding different waves with different phase 

angles (or phase relationships) can result 

in some very different wave shapes. 


• On the following pages, all of the diagrams 

represent the waveforms of the speech of a 

single male speaker of Australian English. 

• Time Scales: For figures 5 to 13 the horizontal 

time scale is indicated by the vertical dashed 

lines which are 100 milliseconds (ms) or 1/10 

second apart and the waveforms on these 

diagrams represent 800 ms (0.8 secs) of 

speech. For figures 1 to 4 the waveforms are 40 

ms (0.04 secs) long and there are no vertical 

time scale markers. 

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• A much more detailed discussion of the 

following speech waveforms can be found 

on the topic web site at:- 

http://www.ling.mq.edu.au/speech/acoustics/waveforms/speech_waveforms.html 


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2. Here we compare 

the weak murmur in 

a voiced stop /d/ 

occlusion with the 

waveforms of /n/ 

and /l/ and /j/, two 

approximates. Note 

the fairly simple 

waveforms for /d/ 

and /n/ and the more 

complex waveforms 

for /l/ and /j/. 


40 ms 


1. One fricative /h/ 

and three vowels 

with the start of 

each pitch cycle 

indicated by a 

vertical bar. 

The different 

vowels have 

different 

waveform 

patterns 

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40 ms 

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4. Here we can 

see different parts 

of the same /z/. 

Initially the 

amount of noise 

(randomness) is 

slight but it 

gradually 

strengthens to 

compete in 

strength with the 

regular (voiced) 

component. 


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3. /f/ greatly amplified 

(is actually about the 

same amplitude as 

the low intensity part 

of /s/ marked with α). 

Note the random 

pattern for /f/ and /s/ 

and the slight 

amounts of random 

pattern superimposed 

on a regular pattern 

for /v/ and /z/. 

5. Here we can 

see three CVC 

syllables each 

starting with /h/ 

and ending with 

/d/. Its difficult to 

differentiate the 

three vowels 

from their pattern 

(which looks a 

little like a fish 

skeleton). 

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6. In this figure we 

can compare the 

three voiceless 

stops of English 

(followed by the 

same vowel). 

Notice that the 

burst is strongest 

for /t/ and that they 

all have a 

significant random 

aspiration phase 

following the burst. 


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8. Here we 

compare the 

affricates of 

English. The burst 

and aspiration of 

/tS/ is similar to 

the voiceless 

stops. The /dZ/ 

aspiration is much 

longer than for the 

voiced stops and 

the VOT is 

negative. 


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10. For these two 

voiced fricatives 

we can see that 

the regular voiced 

component is 

strong and the 

irregular fricative 

component is 

relatively weak 

only becoming 

fairly strong near 

the end of the /z/ 

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7. Here we 

compare the 

three English 

voiced stops. 

The are all prevoiced 

(negative 

VOT) and the 

bursts can be 

seen just before 

the vowel. 


9. /f/ is a weak 

fricative and /s/ 

is a strong 

fricative. Both 

are random. We 

can clearly 

differentiate the 

strengths of 

these two types 

of stops here. 


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11. The /m/ and /n/ 

waveforms are 

very similar (ignore 

the creak). Their 

waveforms can’t 

be discriminated 

but they can be 

discriminated from 

the more complex 

and intense pattern 

of the following 

vowel. 

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12. /l/ and /r/ 

waveforms are 

also very similar 

to each other 

and to /m/ and 

/n/ but, again, 

are simpler in 

pattern than the 

following vowel. 


Waveform Visual Classes 

• For the purposes of this course we will define a number 

of "waveform visual classes". These classes are used 

purely for the practical purpose of distinguishing between 

pairs of waveforms. 

• These waveform visual classes are not true phonetic 

classes, but they are sets consisting of one or more 

complete phonetic class. 

• For the present, we will only consider consonant 

phonemes in simple CV syllables. In VC and VCV 

contexts the visual characteristics of some of the 

consonants can be quite different from their visual 

characteristics in CV context (this is particularly true for 

the oral stops). 


b) Ambiguous, difficult to distinguish consonant visual classes: 

• WVC#5: Affricates: Appearance intermediate between the 

stops and fricatives /tS dZ/. /tS/ often looks like an intense 

stop, but sometimes its burst is not clear and so it will look 

more like a short voiceless fricative. /dZ/ often looks quite 

similar to a voiced strong fricative /z Z/ but usually the 

aspiration phase is much stronger. 

• WVC#6: Voiced fricatives (strong and weak) /v D z Z/: Can 

look a lot like WVC#4, but the mixture of periodicity and 

aperiodicity is sometimes clearly seen just before the vowel, 

especially for the strong fricatives /z Z/. 

• WVC#7: Voiced stops /b d g/: When the burst is clear 

(especially for /d/) the identification of this class is possible 

but often the burst is unclear and these sounds look very 

much like voiced strong fricatives. 

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13. The 

semivowels /w/ 

and /j/ (“y”) are 

also similar to 

each other and 

the preceding 4 

nasals and 

approximants but, 

again, are simpler 

in pattern than the 

following vowel. 


Clearly separable consonant waveform visual classes:- 

• WVC#1: Strong Voiceless fricatives /s S/: Gradually 

increasing aperiodic or random pattern, no bursts, 

relatively intense frication. 

• WVC#2: Weak Voiceless fricatives /f T h/: Gradually 

increasing aperiodic or random pattern, no bursts, 

relatively weak frication. 

• WVC#3: Voiceless oral stops /p t k/: Aperiodic sound 

commencing abruptly with a burst. 

• WVC#4: Approximants and nasal consonants /l r w j m n 

N/: Relatively simple periodic (voiced) pattern and weaker 

in intensity than the vowel but rising smoothly to the vowel 

intensity over several glottal cycles during the consonantvowel 

transition. 


• You are expected to be able to 

discriminate vowel waveforms from 

consonant waveforms (at least in simple 

CV contexts) and to reasonably reliably 

discriminate pairs of tokens belonging to 

the clearly separable CV consonant 

waveform visual classes (WVC#1 to 4). 

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Readings 

• The main readings for this topic are the 

topic web pages reached via:- 

http://www.ling.mq.edu.au/speech/acoustics/waveforms/index.html 

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Acoustic Waveforms Acoustic Waveforms Simple Harmonic Motion ...

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