Acoustics and Psychoacoustics: Introduction to sound - Part 2
[Part 1 discusses pressure waves and sound transmission.]
1.2 Sound intensity, power and pressure level
The energy of a sound wave is a measure of the amount of sound present. However, in general we are more interested in the rate of energy transfer, instead of the total energy transferred. Therefore we are interested in the amount of energy transferred per unit of time, that is the number of joules per second (watts) that propagate.
Sound is also a three-dimensional quantity and so a sound wave will occupy space. Because of this it is helpful to characterise the rate of energy transfer with respect to area, that is, in terms watts per unit area. This gives a quantity known as the sound intensity which is a measure of the power density of a sound wave propagating in a particular direction, as shown in Figure 1.7.
1.2.1 Sound intensity level
The sound intensity represents the flow of energy through a unit area. In other words it represents the watts per unit area from a sound source and this means that it can be related to the sound power level by dividing it by the radiating area of the sound source. As discussed earlier, sound intensity has a direction which is perpendicular to the area that the energy is flowing through, see Figure 1.7.
The sound intensity of real sound sources can vary over a range which is greater than one million-million (10^{12}) to one. Because of this, and because of the way we perceive the loudness of a sound, the sound intensity level is usually expressed on a logarithmic scale. This scale is based on the ratio of the actual power density to a reference intensity of 1 picowatt per square metre (10-12 Wm^{-2}).^{1} Thus the sound intensity level (SIL) is defined as:
SIL = 10 log_{10}(I_{actual}/I_{ref}). (1.10)
where I_{actual} = the actual sound power density level (in W m^{-2})
and I_{ref} = the reference sound power density level (10^{-12} Wm^{-2})
The factor of 10 arises because this makes the result a number in which an integer change is approximately equal to the smallest change that can be perceived by the human ear. A factor of 10 change in the power density ratio is called the bel; in Equation 1.10 this would result in a change of 10 in the outcome. The integer unit that results from Equation 1.10 is therefore called the decibel (dB). It represents a ^{10}v10 change in the power density ratio, that is a ratio of about 1.26.
Example 1.6 A loudspeaker with an effective diameter of 25 cm radiates 20 mW. What is the sound intensity level at the loudspeaker?
Sound intensity is the power per unit area. Firstly, we must work out the radiating area of the loudspeaker which is:
A_{speaker} = pr^{2} = p(0.25 m/2) = 0.049 m^{2}
Then we can work out the sound intensity as:
I = (W/A_{speaker}) = (20 x 10^{-3} W/0.049 m^{2}) = 0.41 W m^{-2}
This result can be substituted into Equation 1.12 to give the sound intensity level, which is:
SIL = 10 log_{10}(I_{actual}/I_{ref}) = 10 log_{10}(0.41 W m^{-2}/10^{-12} W m^{-2}) = 116 dB |