Wiggling both ends of a very long string is not a practical way to generate standing waves. Instead, as in the photograph in Figure 17.3, standing waves are usually seen on a string that is fixed at both ends. To understand why this condition causes standing waves, we need to examine what happens when a traveling wave encounters a discontinuity.

FIGURE 17.7a shows a discontinuity between a string with a larger linear density and one with a smaller linear density. The tension is the same in both strings, so the wave speed is slower on the left, faster on the right. Whenever a wave encounters a discontinuity, some of the wave’s energy is transmitted forward and some is reflected.

FIGURE 17.7 A wave reflects when it encounters a discontinuity or a boundary.

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Light waves exhibit an analogous behavior when they encounter a piece of glass. Most of the light wave’s energy is transmitted through the glass, which is why glass is transparent, but a small amount of energy is reflected. That is how you see your reflection dimly in a storefront window.

In FIGURE 17.7b, an incident wave encounters a discontinuity at which the wave speed decreases. In this case, the reflected pulse is inverted. A positive displacement of the incident wave becomes a negative displacement of the reflected wave. Because $\text{sin}(\varphi +\pi )=-\text{sin}\varphi$, we say that the reflected wave has a phase change of $\pi$ rad upon reflection. This aspect of reflection will be important later in the chapter when we look at the interference of light waves.

The wave in FIGURE 17.7c reflects from a boundary. This is like Figure 17.7b in the limit that the string on the right becomes infinitely massive. Thus the reflection in Figure 17.7c looks like that of Figure 17.7b with one exception: Because there is no transmitted wave, all the wave’s energy is reflected. Hence the amplitude of a wave reflected from a boundary is unchanged.