The inverse square law (cont.)
|Example 1.13 A loudspeaker radiates one hundred milliwatts (100 mW). What is the sound intensity level (SIL) at a distance of 1 m, 2 m and 4 m from the loudspeaker? How does this compare with the sound power level (SWL) at the loudspeaker?
The sound power level can be calculated from Equation 1.11 and is given by:
SWL = 10 log10 (wactual/wref) = 10 log10 (100 mW/1 x 10-12 W)
= 10 log10 (1 x 1011) = 110 dB
The sound intensity at a given distance can be calculated using Equations 1.10 and 1.16 as:
SIL = 10 log10 (Iactual/Iref) = 10 log10 ((Wsource/4πr2)/Iref)
This can be simplified to give:
SIL = 10 log10 (wsource/wref) - 10 log10(4π) - 10 log10(r2)
which can be simplified further to:
SIL = 10 log10 (wsource/wref) - 20 log10(r) - 11 dB (1.17)
This equation can then be used to calculate the intensity level at the three distances as:
SIL1 m = 10 log10 (100 mW/10-12 W) - 20 log10(1) - 11 dB
= 110 dB - 0 dB - 11 dB = 99 dB
SIL2 m = 10 log10 (100 mW/10-12 W) - 20 log10(2) - 11 dB
= 110 dB - 6 dB - 11 dB = 93 dB
SIL4 m = 10 log10 (100 mW/10-12 W) - 20 log10(4) - 11 dB
= 110 dB - 12 dB - 11 dB = 87 dB
From these results we can see that the sound at 1 m from a source is 11 dB less than the sound power level at the source. Note that the sound intensity level at the source is, in theory, infinite because the area for a point source is zero. In practice, all real sources have a finite area so the intensity at the source is always finite.
We can also see that the sound intensity level reduces by 6 dB every time we double the distance; this is a direct consequence of the inverse square law and is a convenient rule of thumb. The reduction in intensity of a source with respect to the logarithm of distance is plotted in Figure 1.12 and shows the 6 dB per doubling of distance relationship as a straight line except when one is very close to the source. In this situation the fact that the source is finite in extent renders Equation 1.16 invalid. As an approximate rule the nearfield region occurs within the radius described by the physical size of the source. In this region the sound field can vary wildly depending on the local variation of the vibration amplitudes of the source.
Equation 1.16 describes the reduction in sound intensity for a source which radiates in all directions. However, this is only possible when the sound source is well away from any surfaces that might reflect the propagating wave. Sound radiation in this type of propagating environment is often called the free field radiation, because there are no boundaries to restrict wave propagation.
Figure 1.12 Sound intensity as a function of distance from the source.