FIGURE 17.1a shows two baseball players, Alan and Bill, at batting practice. Unfortunately, someone has turned the pitching machines so that pitching machine A throws baseballs toward Bill while machine B throws toward Alan. If two baseballs are launched at the same time, and with the same speed, they collide at the crossing point. Two particles cannot occupy the same point of space at the same time.

FIGURE 17.1 Unlike particles, two waves can pass directly through each other.

But waves, unlike particles, can pass directly through each other. In FIGURE 17.1b Alan and Bill are listening to the stereo system in the locker room after practice. Because both hear the music quite well, the sound wave that travels from loudspeaker A toward Bill must pass through the wave traveling from loudspeaker B toward Alan.

What happens to the medium at a point where two waves are present simultaneously? If wave 1 displaces a particle in the medium by $D_1$ and wave 2 simultaneously displaces it by $D_2$, the net displacement of the particle is simply $D_1+D_2$. This is a very important idea because it tells us how to combine waves. It is known as the principle of superposition.

Mathematically, the net displacement of a particle in the medium is

where $D_i$ is the displacement that would be caused by wave i alone. We will make the simplifying assumption that the displacements of the individual waves are along the same line so that we can add displacements as scalars rather than vectors.

To use the principle of superposition you must know the displacement caused by each wave if traveling alone. Then you go through the medium point by point and add the displacements due to each wave at that point to find the net displacement at that point.

To illustrate, FIGURE 17.2 shows snapshot graphs taken 1 s apart of two waves traveling at the same speed $(1\text{m}\text{/}\text{s})$ in opposite directions. The principle of superposition comes into play wherever the waves overlap. The solid line is the sum at each point of the two displacements at that point. This is the displacement that you would actually observe as the two waves pass through each other.

FIGURE 17.2 The superposition of two waves as they pass through each other.

Notice how two overlapping positive displacements at $t=2\text{s}$ add to give a displacement twice that of the individual waves. This is called constructive interference. Similarly, destructive interference is occurring at $t=3\text{s}$ when positive and negative displacements add to give a superposition with zero displacement. We will defer the main discussion until later in this chapter, but you can already see that interference is a consequence of superposition.