VIBRATIONS MAKING WAVES

High-level of noise would not be suitable in some jobs, such as working in a ship’s engine room or looking after airplanes landing and lifting off at an airport. The simple solution would be to wear headphones. These will significantly reduce the intensity of the noise reaching the ears. Wearing noisecancelling headphones will do a better job at protecting the ears. Electronics within such headphones create their own sound that is an exact copy of the incident noise, except it is always in antiphase (phase difference of ${180^ \circ }$) with the noise. The addition of these two waves has the effect of reducing the intensity of the sound reaching the ears to almost zero.
Noise-cancelling headphones are useful in some situations, but they are not ideal if you are at a concert!
Can you think of other jobs where such headphones would be useful?
In this chapter, we will study how waves add-up and cancel-out. The principle of superposition of waves is an excellent starting point.

In Chapter 12, we studied the production of waves and the difference between longitudinal and transverse waves. In this chapter, we are going to consider what happens when two or more waves meet at a point in space and combine together (Figure 13.2).
So what happens when two waves arrive together at the same place? We can answer this from our everyday experience. What happens when the beams of light waves from two torches cross over? They pass straight through one another. Similarly, sound waves pass through one another, apparently without affecting each other. This is very different from the behaviour of particles. Two marbles meeting in midair would ricochet off one another in a very un-wave-like way. If we look carefully at how two sets of waves interact when they meet, we find some surprising results.

When two waves meet, they combine, with the displacements of the two waves adding together. Figure 13.3 shows the displacement–distance graphs for two sinusoidal waves (blue and green) of different wavelengths. It also shows the resultant wave (red), which comes from combining these two. How do we find this resultant displacement shown in red?
Consider position A. Here, the displacement of both waves is zero, and so the resultant displacement must also be zero. At position B, both waves have positive displacement. The resultant displacement is found by adding these together. At position C, the displacement of one wave is positive while the other is negative. The resultant displacement lies between the two displacements. In fact, the resultant displacement is the algebraic sum of the displacements of waves A and B; that is, their sum, taking account of their signs (positive or negative).

We can work our way along the distance axis in this way, calculating the resultant of the two waves by algebraically adding them up at intervals. Notice that, for these two waves, the resultant wave is a rather complex wave with dips and bumps along its length.
The idea that we can find the resultant of two waves that meet at a point simply by adding up the displacements at each point is called the principle of superposition of waves. This principle can be applied to more than two waves and also to all types of waves. A statement of the principle of superposition is:
When two or more waves meet at a point, the resultant displacement is the algebraic sum of the displacements of the individual waves.