ICM L19-1: Amplitude
Amplitude
Amplitude is the measurement of the degree of change in atmospheric
pressure caused by sound waves. It is directly related to the acoustic energy
or intensity of a sound. Both amplitude and intensity are related to sound’s
power.
Amplitude can be measured as deviation from the centre of the wave (peak
deviation); from the top of the wave to the bottom (peak-to-peak); or as
average amplitude over time (root-mean-squared).1
pressure
peak
deviation
peak-to-peak
amplitude
peak
deviation
seconds
Figure 1. A time domain plot showing two different measures of amplitude.
Amplitude is the amount of force applied over an area and can be measured
in Newtons per square metre (N/m2).2
Power & Intensity
The power of a sound is the rate at which an instrument radiates acoustic
energy. As such, power is a measurement of the amount of acoustic energy a
sound wave carries each second and is measured in watts.
The intensity of a sound is a measurement of the amount of acoustic energy a
sound wave carries every second through an area of 1 m2 (orientated
perpendicular to the propagation direction of a wave). Intensity is measured in
watts per square metre (watt/m2).
Since the energy of a sound wave spreads out (in all directions) as it moves
along, a sound wave can be viewed as an expanding sphere. The amount of
energy in any given square metre of the expanding sphere’s surface
decreases as the distance from the sound source increases (or as the sphere
enlarges). As such, intensity gets less as the distance from the sound source
is increased.3
1
If the values of a sine wave were just averaged the result would be zero. For this reason root-meansquared (rms) amplitude is calculated by taking several readings of amplitude, squaring each reading
(removes negative values), taking the average, then taking the square root of the lot (reverses squaring).
2
One Newton is the amount of force required to accelerate an object weighing one kilogram by one
metre per second.
3
The surface area of a sphere = 4r2. Intensity at the surface of a sphere = source intensity / 4r2. So:
doubling the distance from the source decreases the intensity by a factor of 4; whilst tripling it
increases intensity by a factor of 9.
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ICM L19-1: Amplitude
The range of sound intensities that human beings can perceive is huge. For a
tone of 1000Hz, it has been found that the lowest sound intensity the average
person can perceive (the average threshold intensity) is 0.000000000001
watt/m2 (10-12 watt/m2); whilst the highest we can distinguish (the limit of
feeling) is 1 watt/m2.
These numbers are very awkward to carry out calculations with and so
‘intensity’ in watts/m2 is converted into ‘sound intensity level’ in decibels (dB).
Decibels
The decibel is the unit used to express relative differences in signal strength.
When a signal is said to be, say, 10dB it is usually being measured with
reference to the threshold of hearing perception at 0dB (i.e. a sound that can
be just about perceived by an average human4).
Sound Intensity Level (SIL)
Measuring differences in intensity (watts/m2) means using the following
equation:
R =10 log I/Iref 5
A logarithm of a number is the exponent to which 10 must be raised to yield
that number. So log 1000 = 3 (because 103 = 1000).
Using a logarithmic scale allows the whole of the audibility range to be
‘compressed’ into a much smaller range of values. Using a relative value also
allows the intensity of sounds to be talked about with reference to human
hearing.
Sound Pressure Level (SPL) (Amplitude)
Differences in amplitude (amplitude being a measure of pressure) are
calculated using a similar formula:
R = 20 log A/Aref 6
This formula yields the pressure difference in dBSPL (usually abbreviated to
dB).
4
Actual average threshold of hearing at 1000Hz is more like 2.5 * 10 -12 watts/m2 or around 4dB. The
threshold of hearing varies with frequency (see ICM 15).
5
Since intensity at a fixed distance is directly proportional to power the same formula (R = 10 log
P/Pref) can be used for comparing the power of two signals (in watts).
6
The intensity of a sound causes differing pressure waves according to temperature and atmospheric
pressure. Under normal conditions it has been calculated that: I = 0.00234 * pressure variation squared.
This means that intensity is proportional to the square of pressure variation. As such, the formula for
calculating differences in amplitude is: 10 log A2/Aref2 = 20 log A/Aref
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ICM L19-1: Amplitude
How Decibels Are Used in Music Software
With music software, and gain/amplitude (or ‘volume’) measures on music
equipment in general, 0dB is not set at the threshold of human perception,
rather it means 0dBFS (dB Full Scale). 0dBFS is the absolute maximum that
an audio interface can reproduce before clipping (i.e. before the sound is
distorted). This means that other levels are measured in negative decibels in
comparison to it.
Figure 2. Increasing the amplitude of an audio file in Cubase SX. Notice that 199.5% increase in
amplitude is equal to 6dB.
That amplitude is measured in decibels is significant for users of music
software: it means that users do not have to spend time working out ratios of
amplitudes to adjust all amplitude levels in the same proportion. Indeed, if
they did they would be making a mistake!
For example, imagine that several instruments have been created and are
being used to play MIDI information in tracks on a music sequencer, and one
of the instruments is too quiet. This instrument cannot simply be turned up
since it is already at its maximum level. The only way to deal with this is to
turn down the amplitude of all the other sounds, reducing their amplitudes, for
example, by one half. Instead of looking at each amplitude level and dividing it
by two, the decibel scale means that the user should instead simply turn all
the amplitudes down by the same number of decibels (in this case 6dB).
If the user decided to turn down the amplitude by 40%, calculating how many
decibels to reduce the sound by would be more difficult…
Amplitude (SPL) Calculation 1 (percentage to dB)
To carry out a conversion of amplitude percentage to change in dB requires
the use of the formula for SPL.
For example, to work out what happens when amplitude doubles A ref is set to
the original intensity, and A the new intensity. Since it is known that the new
intensity is double that of the old intensity (the ratio between the new and old
intensities is 2:1) A/Aref becomes 2.
SPL = 20 log A/Aref = 20 log (2/1) = 20 log 2  20 (0.3) = 6dB
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ICM L19-1: Amplitude
Any percentage change in amplitude can be calculated similarly by dividing
percentage by 100 and substituting the result in as A/Aref. For example, to
reduce an amplitude to 40% of its original size: 20 log 0.4  -8dB.
Such calculations might be used to determine the increase in decibels (i.e.
sound intensity, NOT loudness) when two waves are added one on top of
another. Say three identical waves are added on top of one another: A/A ref =
3, so 20 log 3  9.5dB.
Amplitude (SPL) Calculation 2
To carry out a conversion of dB to amplitude percentage (to do the conversion
Cubase is doing in the above example) is to carry out the calculation the other
way round. This just requires the formula to be rewritten:
log A/Aref = SPL/20 therefore: 10SPL/20 = A/Aref
And then substituting in the value for SPL:
106/20 = A/Aref = 1.9952…  199.5 %
This can be done for any dB value. For 4 dB: 104/20 = 1.5848…  158.5 %
Miking Distances
Amplitude calculations can also be used to calculate miking distances. So, for
example, halving the distance between the microphone and the source will
increase the amplitude by 6dB.
SIL Calculations
The formula for SIL has been already been defined as:
SL= 10 log I/Iref
To calculate SIL with reference to human hearing Iref is set to the threshold of
hearing: 10-12 watts/m2.
Sound intensity levels with reference to human hearing can now be defined:
SL of an intensity = 1 watt/m2 (the limit of feeling)
SL = 10 log I/Iref = 10 (log 1/10-12) = 10 (12) = 120 dB
SL of an intensity = 10-12 watt/m2 (the threshold of hearing)
SL = 10 log I/Iref = 10 log 10-12/10-12 = 10 (0) = 0 dB
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ICM L19-1: Amplitude
Other intensities can also be converted
SL of an intensity = 10-5 watt/m2
SL = 10 log I/Iref = 10 log 10-5/10-12 = 10 (107) = 70 dB
SL of an intensity = 10-4 watt/m2
SL = 10 log I/Iref = 10 log 10-4/10-12 = 10 (108) = 80 dB
These calculations are used to create a scale for the intensity of sound, as
humans perceive it from 0dB to 120dB (the limit of feeling is variously given
as 120-140dB depending upon age of person).
Threshold of hearing
Breathing
Rustling of leaves
Quiet Whisper
Quiet Home
Quiet Street
Normal Conversation
10-12 watt/m2
10-11 watt/m2
10-10 watt/m2
10-9 watt/m2
10-8 watt/m2
10-7 watt/m2
10-6 watt/m2
0dB
10dB
20dB
30dB
40dB
50dB
60dB
Vacuum Cleaner
Automobile (25 feet)
Food Blender (3 feet)
Diesel Truck (3 feet)
Amplified Music (6 feet)
Jet Plane (500 feet)
10-5 watt/m2
10-4 watt/m2
10-3 watt/m2
10-2 watt/m2
1 watt/m2
1 watt/m2
70dB
80dB
90dB
100dB
120dB
120dB
Table 1: A table showing sound intensities for well known sounds.
Whenever intensity is increased by a factor of 10 (i.e. by 10 times), SIL
increases by 10dB. So, for example: a sound of intensity 10dB is 10 times as
intense as a sound of 0dB7; a sound of intensity 20dB is 10 times as intense
as a sound of 10dB; and so on.
Ear Damage
It is very easy for someone to damage their ears when they listen to music for
long periods of time. People working with sound regularly need to ensure that
they do not damage their ears. Such damage can manifest itself as a loud
ringing in the ears. This loud ringing can be temporary or permanent.
US health and safety at work legislation limits exposure to the following:




8 hrs at 90dB
4 hrs at 100dB
2 hrs at 100dB
15 min at 115dB
115dB is the highest permissible sound intensity for the unprotected ear.
Noise above 140dB is not permitted at all.
People working with sound for long periods at a time should ensure that the
sound intensity level is below 80dB.
7
Remember: 0dB in this context is the lowest sound intensity the average human can perceive; it is not
no sound at all.
5