# Acoustics and Psychoacoustics: Introduction to sound - Part 4

*[Part 1 discusses pressure waves and sound transmission. Part 2 covers sound intensity, power and pressure level. Part 3 looks at adding sounds together.]*

**1.4 The inverse square law**

So far we have only considered sound as a disturbance that propagates in one direction. However, in reality sound propagates in three dimensions. This means that the sound from a source does not travel on a constant beam, instead it spreads out as it travels away from the radiating source, as shown in Figure 1.7.

As the sound spreads out from a source it gets weaker. This is not due to it being absorbed but due to its energy being spread more thinly. Figure 1.11 gives a picture of what happens. Consider a half blown-up spherical balloon, which is coated with honey to a certain thickness. If the balloon is blown up to double its radius, the surface area of the balloon would have increased fourfold. As the amount of honey has not changed it must therefore have a quarter of the thickness that it had before.

Figure 1.11 The honey and balloon model of the inverse square law for sound.

The sound intensity from a source behaves in an analogous fashion in that every time the distance from a sound source is doubled the intensity reduces by a factor of four, that is there is an inverse square relationship between sound intensity and the distance from the sound source. The area of a sphere is given by the equation:

*A*_{sphere} = 4p*r*^{2}

The sound intensity is defined as the power per unit area. Therefore the sound intensity as a function of distance from a sound source is given by:

*I* = *W*_{source}/*A*_{sphere} = *W*_{source}/4p*r*^{2} (1.16)

where *I* = the sound intensity (in W m^{-2})

*W*_{source} = the power of the source (in W)

and *r* = the distance from the source (in m)

Equation 1.16 shows that the sound intensity for a sound wave that spreads out in all directions from a source reduces as the square of the distance. Furthermore this reduction in intensity is purely a function of geometry and is not due to any physical absorption process.

In practice, there are sources of absorption in air, for example impurities and water molecules, or smog and humidity. These sources of absorption have more effect at high frequencies and as a result, sound not only gets quieter but also gets duller as one moves away from a source. The amount of excess attenuation is dependent on the level of impurities and humidity and is therefore variable.