[Part 1 discusses pressure waves and sound transmission. Part 2 covers sound intensity, power and pressure level. Part 3 looks at adding sounds together. Part 4 discusses the "inverse square law" for sound. Part 5 begins a look at sound interactions, including refraction, absorption and reflection.]
1.5.6 Sound interference
We saw earlier that when sound waves come from correlated sources then their pressure and associated velocity components simply add. This meant that the pressure amplitude could vary between zero and the sum of the pressure amplitudes of the waves that were being added together, as shown in Example 1.9.
Whether the waves add together constructively or destructively depends on their relative phases and this will depend on the distance each one has had to travel. Because waves vary in space over their wavelength then the phase will also spatially vary. This means that the constructive or destructive addition will also vary in space.
Consider the situation shown in Figure 1.21, which shows two correlated sources feeding sound into a room. When the listening point is equidistant from the two sources (P1), the two sources add constructively because they are in phase. If one moves to another point (P2) which is not equidistant, the waves no longer necessarily add constructively. In fact, if the path difference is equal to half a wavelength then the two waves will add destructively and there will be no net pressure amplitude at that point. This effect is called interference, because correlated waves interfere with each other; note that this effect does not occur for uncorrelated sources.
Figure 1.21 Interference from correlated sources.
The relative phases of the waves depend on their path difference or relative delays. Because of this the pattern of constructive and destructive interferences depends strongly on position, as shown in Figure 1.22. Less obviously the interference is also strongly dependent on frequency. This is because the factor that determines whether or not the waves add constructively or destructively is the relative distance from the listening point to the sources measured in wavelengths (λ).
Figure 1.22 Effect of position on interference at a given frequency.
Because the shape of the amplitude response looks a bit like the teeth of a comb the frequency domain effect of interference is often referred to as 'comb filtering'. As the wavelength is inversely proportional to frequency one would expect to see the pattern of interference vary directly with frequency, and this is indeed the case. Figure 1.23 shows the amplitude that results when two sources of equal amplitude but different relative distances are combined. The amplitude is plotted as a function of the relative distance measured in wavelengths (λ).
Figure 1.23 shows that the waves constructively interfere when the relative delay is equal to a multiple of a wavelength, and that they interfere destructively at multiples of an odd number of half wavelengths. As the number of wavelengths for a fixed distance increases with frequency, this figure shows that the interference at a particular point varies with frequency. If the two waves are not of equal amplitude then the interference effect is reduced, also as shown in Figure 1.23. In fact once the interfering wave is less than one eighth of the other wave then the peak variation in sound pressure level is less than 1 dB.
Figure 1.23 Effect of frequency, or wavelength, on interference at a given position. The ratios refer to the relative amplitudes of the two waves.
There are several acoustical situations which can cause interference effects. The obvious ones are when two loudspeakers radiate the same sound into a room, or when the same sound is coupled into a room via two openings which are separated. A less obvious situation is when there is a single sound source spaced away from a reflecting boundary, either bounded or unbounded. In this situation an image source is formed by the reflection and thus there are effectively two sources available to cause interference, as shown in Figure 1.24. This latter situation can often cause problems for recording or sound reinforcement due to a microphone picking up a direct and reflected sound component and so suffering interference.
Figure 1.24 Interference arising from reflections from a boundary.
|Example 1.15 Two loudspeakers are one metre apart and radiate the same sound pressure level. A listener is two metres directly in front of one speaker on a line which is perpendicular to the line joining the two loudspeakers, see Figure 1.25.
Figure 1.25 Interference at a point due to two loudspeakers.
What are the first two frequencies at which destructive interference occurs? When does the listener first experience constructive interference, other than at very low frequencies?
First work out the path length difference using Pythagoras' theorem:
Δpath length = √(1 m2 + 2 m2) - 2 m = 0.24 m
The frequencies at which destructive interference will occur will be at λ/2 and 3λ/2. The frequencies at which this will happen will be when these wavelengths equal the path length difference. Thus the first frequency can be calculated using:
λ/2 = Δpath length i.e. λ = 2Δpath length
fλ/2 = v/λ = v/2Δpath length = 344 ms-1/(2 x 0.24 m) = 717 Hz
The second frequency will occur at 3 times the first and so can be given by:
f3λ/2 = 3 x fλ/2 = 3 x 717 Hz = 2150 Hz
The frequency at which the first constructive interference happens will occur at twice the frequency of the first destructive interference which will be:
fλ = 2 x fλ/2 = 2 x 717 Hz = 1434 Hz
If the listener were to move closer to the centre line of the speakers then the relative delays would reduce and the frequencies at which destructive interference occurs would get higher. In the limit when the listener was equidistant the interference frequencies would be infinite, that is, there would be no destructive interference.