Interference of waves

Consider two or more waves propagating in the same medium. If the waves are travelling in the same direction, they tend to add up forming a new wave of large amplitude. If waves of the same frequency and amplitude travelling with the same speed but in opposite directions meet, they tend to cancel each other. That is, no new wave is formed. The addition of waves is called wave superposition and is governed by the principle of superposition of waves.

The principle of superposition of waves states that, the resultant disturbances at any point is equal to the algebraic sum of the disturbances of individual waves at that point.

When two or more waves combine to form a resultant wave, the waves are said to have interfered. Therefore, wave interference is a combination of two or more waves to form a resultant wave in which the particle displacement is either reinforced or cancelled. When two waves interfere and if a crest of one wave meets the crest of another wave at the same point, then the resultant wave has a larger amplitude, as shown in Figure 1.20. This is referred to as constructive interference. Constructive interference occurs at any location along the medium where two or more interfering waves have displacements in the same direction (the waves are in phase). On the other hand, if a crest of a wave meets a trough of another wave, then the resultant wave has a smaller amplitude, as shown in Figure 1.21. This is referred to as dectructive interference. Destructive interference occurs at any point along the medium where the interfering waves have displacements in opposite directions (the waves are out of phase).

Figure 1.20: Constructice interference

Figure 1.21: Destructive interference

Examples of the resultant displacement due to the interference of two waves of different amplitudes are given in Table 1.1.

Table 1.1: Resultant displacements from interference

In constructive interference, we get lines of increased disturbance. These lines are called antinodal lines. In destructive interference, we get lines of zero disturbance. These lines are called nodal lines. Figure 1.22 shows circular waves from two dippers that are close together in a ripple tank. The waves cross through one another and result in both constructive and destructive interference.

Figure 1.22: Interference of circular waves

The red dots indicate where crests or troughs meet giving rise to maximum amplitude which results to maximum intensity. The blue dots indicate where crest meets a trough giving rise to minimum or zero amplitude resulting to minimum intensity. The lines of maximum constructive interference are labelled A₀, A₁ and A₂. Points on these lines move up and down with higher amplitudes much than they would if the waves came from one source alone. The lines labelled N₁ and N₂ represent bands along which there is maximum destructive interference. Points on these lines move up and down with much less amplitude than they would if the waves came from either source alone.

Note that, if the two waves from S₁ and S₂ have the same amplitude, then the points along the lines A₀, A₁ and A₂ will have the amplitute twice the amplitude of the wave. On the other hand, the points along the lines N₁ and N₂ will have zero amplitude.

Water waves from two identical point sources add up at certain points (where a crest meets a crest) and cancel out at certain points (where a crest meets a trough). Where the water waves add up, constructive interference occurs. The water waves have increased amplitudes along lines of constructive interference. On the other hand, destructive interference occurs where the water waves cancel out. Along the lines of destructive interference, the water is observed to be still.

Applications of interference of waves

1. Wave interference is applied when creating holograms. A hologram is a photograph of an interference pattern which is able to produce a three-dimensional image.
2. Destructive interference is used in noise-reduction systems such as earphones and car mufflers. The systems capture sound from the environment and produces a second wave, which interferes with the first wave destructively leading to the reduction in the loudness of the noise.
3. Another application of interference of waves is in Active Noise Control. This is based on the fact that the wave generated by a primary source (such as an engine) can be cancelled by the wave emitted by a secondary source (loudspeakers) driven at the same frequency as the primary source, so that the two waves cancel out each other. This technique is applied to reduce the annoying propeller noise inside the cabin of an aircraft.