SOUND Laura Hyland cs.aue.auc.dk/~laura. What sound is?.
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Sound is a form of energy. When we give energy to a body (by hitting or exciting it) we set that body in motion – it will vibrate. This body in turn will set the air around it in motion, causing it to vibrate also. The vibrations in the air will reflect the vibrations of the body. These vibrations travel through the air in waves until they reach our ears where we perceive them as sound. For example, if we excite a tuning fork by striking it the energy given causes the tines to vibrate very quickly. The movement of the tines is too small for the eye to percieve but not for the ear. These vibrations can also be felt when the tuning fork is brought in contact with the skin.
As the tines move back and forth they exert pressure on the air around them.
(a) The first displacement of the tine compresses the air molecules causing high
(b) Equal displacement of the tine in the opposite direction forces the molecules to
widely disperse themselves and so, causes low pressure.
(c) These rapid variations in pressure over time form a pattern which propogates
itself through the air as a wave. Points of high and low pressure are sometimes
reffered to as ’compression’ and ’rarefaction’ respectively.
(c) wave propegation of a tuning fork
as seen from above
If we look at the way one tine moves we see that it only moves in one plane
back and forth. In addition to this it moves the same distance back and forth at
a constant speed. We could say that it is trying to reach equlibrium – it is
trying to get back to its original stable position. This kind of response of an
object to excitation is characteristic to many objects. A pendulum is another example of this movement – it is also easier to visualize than the tines of a tuning fork! Think of the movement of a pendulum when put into motion; it swings back and forth but will always come to rest at its original position. This is called Simple Harmonic Motion.
Note with the pendulum that even when it approaches equlibrium it doesnt
slow down – it simply travels a smaller distance from the point of rest. This
is also the case for the tine of the tuning fork. Thus, we can say that any
body undergoing simple harmonic motion moves periodicallywith uniform
speed. We can also say that if the tine is moving periodically then the
pressure variations it creates will also be periodic.
The time taken to get from position a to b in all three cases is the same
after say, 6 seconds
at 0 seconds
after say, 3 seconds
These pressure patterns can be represented using as a circle.
Imagine the journey of the pendulum or the tine in four stages:
1) from its point of rest to its first point of maximum displacement...
2) its first point of maximum displacement back through the point of rest...
3) ... to its second point of maximum displacement...
4) ... and back from there through its point of rest again
We can map that journey to a circle. This is called the Unit Circle. The sine wave
represents this journey around and around the unit circle over time.
The sine wave or sinusoid or sinusoidal signal is probably the most commonly used graphic representation of sound waves. The diagram below shows one cycle or ’period’ of a wave, i.e., the build-up from equilibrium to maximum high pressure, to maximum low pressure, to equilibrium again. A sine wave sounds like this...
Pressure or density
of air molecules;
Time in seconds
The specific properties of a sine wave are described as follows.
Amplitude = variations in air pressure (measured in decibels)
Phase = The starting point of a wave along the y-axis (measured in degrees)
Period = The time need to come to exact the same location
Frequency refers to the number of cycles of a wave per second. This is measured in Hertz.So if a sinusoid has a frequency of 100Hz then one period of that wave repeats itself every 1/100th of a second. The diagram below shows a 100Hz sine and an 800Hz sine. For every 8 periods of (a) there is one period of (b). Humans can hear frequencies between 20Hz and 20,000Hz (20Khz).
There are three important things to remember about frequency:
1) Frequency is closely related to, but not the same as!!!,pitch.
2) Frequency does not determine the speed a wave travels at. Sound waves travel at approximately 340metres/second regardless of frequency. f=c/l
3) Frequency is inherent to, and determined by the vibrating body – not the amount of energy used to set that body vibrating. For example, the tuning fork emits the same frequency regardless of how hard we strike it.
Wavelength describes the length of one period of a wave, or twice the distance between one zero crossing point and the next. (A zero crossing point refers to the point at which the wave crosses the x-axis. This represents the point at which there is no pressure variation, i.e., the point where air molecules return to their original position.)
It is important to have a sense of the actual physical size of a wave. The speed of sound in air is approximately 340 metres per second*. Consider a wave of frequency 20 Hz, i.e., a pressure pattern repeating itself 20 times a second. 20 periods back to back have a length of 340metres so 1 period = 340/20 = 17 metres. Similarly, a wave at a frequency of 20kHz will be 340/20,000 in length = 0.017 metres or 1.7mm.
This property is important for formants!
Wavelength = Speed of sound in air / Frequency
Zero crossing points
*This is dependant on air temperature – the higher the temp the more freely air molecules will move, therefore the faster the wave will travel.
Amplitude describes the size of the pressure variations. It is measured along the vertical y-axis. Think of the pendulum or the tuning fork; the wider the displacement of the pendulum or tine, the larger the amplitude is. Amplitude is closely related to but not the same as!!!,loudness. Hence the reason the tuning fork sounds louder when we strike it hard. We will examine the relationship between amplitude and loudness later...
(a) Two signals of equal frequency and
(b) Two signals of varying frequency and
The amplitude of a wave changes or ’decays’ over time as it loses energy.
These changes are normally broken down into four stages;
Attack , Decay, Sustain and Release. Each stages is measured in milliseconds. For example, the signal below has an attack of 100ms. That means its amplitude goes from 0dB to 0.8dB in 100ms. Similiarly, it has a decay of 90ms so its amplitude goes from 0.8dB to ~0.38dB in 90ms, etc.
Collectively, the four stages are described as the amplitude envelope.
Consider the Unit Circle again. So far we have mapped the journey around the
circle starting from the point corrosponding to an amplitude of 0 on the y-axis
but we can offset the starting point so that we begin mapping from another point
on the circle. The offset from the point 0 on the circle will determine the initial
phase of the sine wave, i.e., the starting point of the wave along the y-axis.
This offset is the phase of the sine wave and it is measured in degrees. Figure (b)
below has a phase shift of 170 degrees
Why offset the start time? As we will see later, it’s often neccessary to look at several
signals together, each one having a different start time. The easiest way to represent
this time difference is using phase. For example, take two 100Hz signals. Say (a)
starts at 0 seconds and (b) starts at 1.375secs. (a) will be ¾ ways through its 131st
cycle when (b) begins. We can represent this time difference by giving (a) a phase
shift of 260 degrees or ¾ of a cycle. That is, when (b) is starting out at amplitude 0,
(a) is at amplitude 1. So, phase can be defined as the representation of the time delay
between two signals.
If we add these two 100hz signals together we see that points of high pressure in
(a) correspond with points of low pressure in (b). Thus, they cancel each other out
and the result we get is no pressure variations at all! The picture of this?... (c)!
When two waves intereact with each other like this it is called interference.
In the previous example the two waves cancelled each other out, resulting in a
decrease of pressure variations. This is Destructive Interference. The following
example shows a case where two waves interacting result in an increase in pressure
variations. This is called Constructive Interference. We have the same two signals
but this time they start ’in phase’, i.e., at the same time.
Constructive and destructive interference due to phase cancellation of two waves of
equal amp, freq and direction.
Standing wave due to two waves of equal amplitude, frequency and phase travelling
in opposite directions.
If we take two sine waves which are very close in frequency we experience a
phenomena called beating. Beating can be described as a periodic variation in
amplitude. These variations occur at at rate of f1 - f2(where f1 the higher
frequency and f2 the lower). The frequency of the new signal will be the average of
f1 and f2. Forexample, when a 440 Hz. and 442 Hz. sinusoids are combined we
hear 2 beats per second and tone whose frequency is 441 Hz.
Beating sounds like this... (170hz + 174hz)
Fundamentals, Harmonics & Partials
So far we have investigated interference between two sinusoids of equal frequency and also that between very close frequencies. Now we need to consider some other relationships. What happens when one sine wave is exactly half the frequency of the other? In the diagram below we see two sinusoids with frequencies, 220hz and 440hzand both have a 0 degree phase so for every one period of (a) we get two periods of (b). We can hear this relationship as well as see it. Listen to both frequencies – we hear the same note but one is much ’higher’ than the other. If both are played together we hear one one tone, not two! We will investigate the reason for this later when we look at pitch. In the following diagram this pattern is extended to five sinusoids. In this case the 2nd sinusoid is twice the 1st, the 3rd is three times the 1st, the 4th is four times the 1st and so on....
Fundamentals, Harmonics & Partials
5th Harmonic or 6th partial
3rd Harmonic or 4th partial
2nd Harmonic or 3rd partial
1st Harmonic or 2nd partial
or ’1st partial’
Fundamentals, Harmonics & Partials
Visually, it is clear that there is a relationship between all these sine waves.
Aurally, it is also clear; when all five sines are played together we perceive it
as one tone. Numerically there is also a relationship – the frequencies are all
multiples of the first frequency. There is an integer relationship between
the frequencies of all these sinusoids. (integers are whole numbers like 3, 5, -8,
120, etc). In a set of sinusoids like this the first frequency is reffered to as the
fundamental frequency. Subsequent sinusoids are called harmonics. The
whole set together is called the Harmonic Series.
Another term often used in this context is a ’partial’. NOTE: a partial is a
generic term to describe any component of a sound, for example, a sinusoid of
318hz in this set is a partial but NOT a harmonic because 318 is not an even
multiple of 100. The fundamental frequency is also a partial.Thus, all harmonics
are partials but not all partials are harmonics!
The Harmonic Series
Notice that there is also a relationship between the amplitude of partials
comprising the harmonic series.
Amp f5 = amp f1/5
Amp f4 = amp f1/4
Amp f3 = amp f1/3
Amp f2 = amp f1/2
Amp f0 = 1
Wave Superposition of Harmonically Related Signals
Notice that the signals being added are the 1st, 3rd and 5th harmonics of a series where f0 = 100, i.e., the odd harmonics. Now look at the resultant diagram to the right. Notice that the emergent ’shape’ approaches a square. The combination of odd harmonics will always give a square wave.
Time Domain vs Frequency Domain
Time Domain vs Frequency Domain
Even though we cannot see sound we have to remember that it has physical
dimensions and exists in space. Thus, like any other physical body it is
3-dimensional. The previous figure shows a sine wave in the three dimensional
plane. Along the horizontal x - axis we have time; on the vertical y-axis we have
amplitude; on the diagonal z-axis we have frequency.
Up until now we have viewed sine waves in the TIME DOMAIN only.
So why use both views? What can we see in one view that we can’t see in the other?
First let’s look at the Time Domain again; Here we can see time, of course, and
phase and amplitude. We can also see low frequencies if its a single sine wave
and the diagram is big enough, however, only so far as the eye can count the resolutions of the pressure pattern. Also, if there is more than one sinusoid identifying frequency is practically impossible. Obviously this is not very a sophisticated or accurate way of identifying frequency! Hence the reason we need to be able to switch views between one axis and the other. (If you think of architectural drawings of a building in plan and elevation In might make it easier to conceptualize this switching between views.)
Time Domain vs Frequency Domain
Now, if we look at the sinusoid in the FREQUENCY DOMAIN what can we see? Obvoiusly we can see Frequency. We can also see amplitude. However, we cannot see time or phase at all. Look at the following diagrams to see the same signal represented in the time domain and frequency domain.
440hz tone at an amplitude of 0.87 and phase shift of 260 degrees
220hz tone at an amplitude of 0.4 and a phase shift of 90 degrees
where is the sampling interval, and assume N is even.
Sound Intesity = rate of energy radiation in watts per metre squared per second
Imagine a spherical sound source radiating the same amount of energy at the same rate in all directions at the same time.That energy is recieved or ’picked up’ by some surface – say our eardrum or a microphone.
Intensity falls off
Watts per second
When a body vibrates with more energy its displacement from equilibrum is greater. So too then is the displacement of air molecules. The result is cycles of densely and sparsely packed molecules. The figure below shows a representation of two sines at the same frequency with different amplitudes. We can see that when the amplitude is higher the molecules are more densely packed together and thus, create higher pressure. This is what ’sound pressure level’ refers to.
So, higher pressure means more densely packed molecules which means more ’impact’ on the ear. (think of the different between being hit by a sponge and being hit by a stone; stone = denser material = more pressure = more impact.)
Loudness - Not a Linear Scale!
Sound pressure Level
Sound Intensity Level
Threshold of pain
Threshold of hearing
The problem with trying to measure loudness it that it is not linear. That means a doubling in intensity or in pressure does not necessarily corrospond to a perceived doubling in loudness! Thus, loudness increases logarithmically
SPL or SIL
Relationship between SPL and SIL?
An increase in sound intensity is proportional to the square of the pressure amplitude. In other words, as the amplitude doubles the sound intensity is quadrupled. This relationship requires a bit of a mathematical detour but here it is sufficient to simply remember that when measuring Sound pressure level we use the equation:
20 Log P1/P0 = SPL (in decibels)
...and when measuring Sound Intensity Level we use the equation:
10 Log I1/I0 = SPL (in decibels)
The Decibel Scale
So, by converting the linear SPL and SIL scales to logarithmic scales we now have
an accurate loudness scale. We will see shortly that the relationship between pitch
and frequency is also non linear – logarithmic scales will reappear there too!
SPL or SIL
Log (SPL or SIL)
Other ways of measuring Loudness?
So far we’ve seen that there are two ways of measuring loudness – SIL and SPL. However, SPL is the most commonly used scale for loudness measurment. For example, SPL is used to measure the level of noise on a building site for example, or in a concert hall in the centre of a busy city.
Musicians use the following dynamic markings to indicate loudness;
ppp - quiet as possible
pp - very quiet
p - quiet
f - loud
ff - very loud
fff - loud as possible
Frequency: Fletcher Munson Curves
The most comprehensive study of loudness perception at different frequencies is shown in the Fletcher Munson Curves. These curves demonstrate the relationship between sound pressure and frequency and the resulting loudness we perceive.
Fletcher Munson Curves
Signal representation on time-frequency plane
Sonogram of speech signal –evolution of spectum
Formowanie sygnału mowy