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Acoustic book

Frequency and intensity


Frequency and amplitude are two very important terms for their psycho-physiological meaning. Indeed, the selection of colours depends on the frequency of light waves while the height of a sound depends on the frequency of sound waves if provided at sufficient amplitude.
Our ear is a very selective and sensitive organ: it can perceive amplitudes in the order of 0.0000001 cm and periods less than 0.00007 seconds.
The table below shows some wavelengths in centimetres of some characteristic sounds in a standard air ambient. The table consists of three columns and 4 lines, where in the first column there is a sound mark, in the second the frequency and in the third the wavelength.

Sound Frequency Wavelength
Low-frequency noises: 100Hz 340cm
Note the third octave: 440Hz 77cm
Medium to high frequency: 1000Hz 34cm
High frequencies: 4000Hz 3.5cm

The ear of a young person can hear sounds with frequencies ranging from a minimum of 16 Hz to a maximum of 20,000 Hz that is about 10 music octaves. We are going to talk about the concept of an octave later.
Very interesting is the comparison to the range of visible waves that do not exceed an octave. The sounds below 16 Hz are not heard as true sound, but they can be perceived, if of high intensity, such as vibrations. There are animals able to hear very high-pitched sounds like dogs, that can hear up to 30,000 Hz and bats, able to hear up to 90,000 Hz. Below 16 Hz, sound waves are called infrasound, while above 20,000 Hz they are called ultrasound.
In practice, the minimum reproducible frequency is given by pipe organ and it is of 16.35 Hz up to a maximum of about 4,000 Hz.
Here is a table showing temperate frequencies throughout the audible range.

C 16.35 32.70 65.40 130.8 261.6 523.2 1046 2093 4186 8372
C diesis 17.32 34.64 69.29 138.6 277.2 554.4 1108 2217 4434 8869
D 18.35 36.70 73.41 146.8 293.6 587.3 1175 2350 4699 9398
D diesis 19.44 38.89 77.78 155.5 311.1 622.2 1244 2489 4978 9956
E 20.60 41.20 82.80 164.5 329.6 659.2 1318 2637 5274 10548
F 21.82 43.65 87.30 174.7 349.2 698.5 1396 2793 5587 11175
F diesis 23.12 46.24 92.50 185.0 370.0 740.0 1480 2960 5920 11840
G 24.50 49.00 98.00 196.0 392.0 784.0 1568 3136 6272 12544
G diesis 25.96 51.91 103.8 207.6 415.3 830.6 1661 3322 6644 13289
A 27.50 55.00 110.0 220.0 440.0 880.0 1760 3520 7040 14080
A diesis 29.13 58.27 116.5 233.0 466.2 932.3 1864 3729 7458 14917
B 30.86 61.73 123.4 246.9 493.8 987.7 1975 3951 7902 15804


To illustrate the mechanism of sound waves propagation let’s simply consider an impulse moving along a cylindrical shape mean, as for example the air inside a cylinder.
Figure 14 shows a flywheel to which is connected a connecting rod and a piston sliding within a cylinder.

Figure 14 Figure 14
Figure 14 Figure 14 for embossed printing

The impulse generated by a movement and return of a piston consists of a localized region R in which the pressure p is greater than the undisturbed pressure p0 of the mean. As the impulse propagates along the cylinder, the air elements perform a simple oscillating around their equilibrium position, and they do not propagate through the impulse.
Therefore, the impulse propagates from a region R to a region R1 compressing the air in the region R0 while not moving air in the region R. If the piston performs continuous movements there is a succession of compressions that move along the duct in the form of longitudinal waves.

A sound wave is composed by a compression and a rarefaction in a very similar way to longitudinal waves with springs. The vibrations of a sound body spread in the spherical sense, that is, symmetrically in all directions. The speed transmission of sound into air at a temperature of 15 degrees is of 340 meters per second and depends neither on the wave frequency nor by the intensity, shape or density, but it slightly varies with changes in temperature and air humidity. Here is a table showing the speed of sound into various substances.

GAS Grade Centigrade Temperature Speed m/s
Carbon Dioxide 0 259
Oxigen 0 316
Air 0 331
Air 20 343
Azote 0 334
Helium 0 965

LIQUIDS Grade Centigrade Temperature Speed m/s
Mercury 25 1450
Water 25 1498
Sea water 25 1531

SOLIDS Speed m/s
Rubber 1800
Lead 2100
Plastic 2700
Gold 3000
Iron 5000-6000
Glass 5000-6000
Granito 6000


The sensation you have of a sound is related to the energy carried by the sound wave. It is a subjective impression that a hearer link to a particular sound, while the sound wave energy is an objective physical quantity. A psychology field called psychophysics studies the connections between these two quantities. The intensity I of a sound wave is the energy that passes through a unit area per unit time. It can be experimentally determined by measuring the energy E recording on a microphone or during a time interval T. The intensity is equal to:

I = E / A * T
Where I = intensity, E = Energy, A = Area, T = Time

In the mks system the intensity unit is given by J/m^2*S or W/m^2.

Even if the sensation you perceive of a sound increases with intensity, the connection between sensation and intensity is not linear. For example, in a reading room, the intensity of a speaker's voice can be of value 10 greater in the front rows in comparison with the back of the room, but a listener who moves from the front rows to the back of the room just feels a slight decrease in loudness. A young adult can detect sounds ranging from a minimum intensity of 10E-12W/m^2 to a maximum intensity of 1W/m^2.

By convention, we assume that the minimum intensity that we can reveal is the point 0 (I) of a scale of sound intensity level called scale of decibels (dB). On this scale, an increase of a factor intensity of 10 corresponds to an increase of the intensity level of 10 dB. Therefore, since I = 10E- 12 W/mE2 correspond to = 0 dB, 10E-11W/mE2 = correspond to = 10 dB, and so on.

The intensity level is mathematically defined by the following formula:
i = 10 log (I'/l)

For higher intensities to 1W/mE2 you go from sound to pain sensation. The intensity level of this threshold is:
i = 10 log ( 1W/mE2 ) / ( 10E-12W/mE2 ) = 120 dB

The range of human hearing is therefore between 0 and 120 dB.

Here is a table of the intensity of some of the most common sounds

Sound level in dB Intensity W / m ^ 2 Example
0 10E-12 Threshold of hearing
10 10E-11 Swish leaves
20 10E-10 Whisper at one metre distance
30 10E-9 Quite house sounds
40 10E-8 Middle house sounds
50 10E-7 Middle office sounds
60 10E-6 Normal conversation
70 10E-5 Loud office
80 10E-4 Rush hour traffic
90 10E-3 Underground noises
100 10E-2 Vending machine
120 10E 0 Jackhammer
140 10E 2 30 metres jet aircraft

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