Acoustics and Psychoacoustics: Introduction to sound - Part 5

1.5 Sound interactions
So far we have only considered sound in isolation and we have seen that sound has velocity, frequency, wavelength and reduces in intensity in proportion to the square of the distance from the source. However, sound also interacts with physical objects and other sound waves, and is affected by changes in the propagating medium. The purpose of this section is to examine some of these interactions as an understanding of them is necessary in order to understand both how musical instruments work and how sound propagates in buildings.

1.5.1 Superposition
When sounds destructively interfere with each other they do not disappear. Instead they travel through each other. Similarly, when they constructively interfere they do not grow but simply pass through each other. This is because, although the total pressure, or velocity component, may lie anywhere between zero and the sum of the individual pressures or velocities, the energy flow of the sound waves is still preserved and so the waves continue to propagate.

Thus the pressure or velocity at a given point in space is simply the sum, or superposition, of the individual waves that are propagating through that point, as shown in Figure 1.14. This characteristic of sound waves is called linear superposition and is very useful as it allows us to describe, and therefore analyse, the sound wave at a given point in space as the linear sum of individual components.

1.5.2 Sound refraction
This is analogous to the refraction of light at the boundary of different materials. In the optical case refraction arises because the speed of light is different in different materials, for example it is slower in water than it is in air. In the acoustic case refraction arises for the same reasons, because the velocity of sound in air is dependent on the temperature, as shown in Equation 1.5.

Figure 1.14 Superposition of a sound wave in the golf ball and spring model.

Consider the situation shown in Figure 1.15 where there is a boundary between air at two different temperatures. When a sound wave approaches this boundary at an angle, then the direction of propagation will alter according to Snell's law, that is, using Equation 1.5:

sin θ₁/sin θ₂ = v_T1/v_T2 =

(20.1√T1)/(20.1√T2) = √(T1/T2)                                    (1.19)

where θ₁, θ₂ = the propagation angles in the two media
v_T1, v_T2 = the velocities of the sound wave in the two media
and T1, T2 = the absolute temperatures of the two media

Thus the change in direction is a function of the square root of the ratio of the absolute temperatures of the air on either side of the boundary. As the speed of sound increases with temperature one would expect to observe that when sound moves from colder to hotter air that it would be refracted away from the normal direction and that it would refract towards the normal when moving from hotter to colder air.

This effect has some interesting consequences for outdoor sound propagation. Normally the temperature of air reduces as a function of height and this results in the sound wave being bent upwards as it moves away from

Figure 1.15 Refraction of a sound wave (absolute temperature in medium₁ is T1 and in medium₂ is T2; velocity in medium₁ is v_T1 and in medium₂ is v_T2).

Figure 1.16 Refraction of a sound wave due to a normal temperature gradient.

a sound source, as shown in Figure 1.16. This means that listeners on the ground will experience a reduction in sound level, as they move away from the sound, which reduces more quickly than the inverse square law would predict. This is a helpful effect for reducing the effect of environmental noise nuisance.