Absolutely Small - Michael D. Fayer [29]
Figure 6.2 shows a plot of five waves with different wavelengths. The units of length do not matter. What is important is that the five waves have wavelengths, λ = 1.2, 1.1, 1.0, 0.9, and 0.8. The phase of the waves are adjusted so that they all match at the point x = 0, where x is the horizontal axis. The waves match at x = 0 in the sense that each wave has a positive going peak at x = 0. However, because the waves have different wavelengths, the peaks do not necessarily match at other points along the x axis. For example, at about x = 10 or - 10, the dark gray wave has a maximum but the dashed light gray wave has a minimum. In addition, at approximately 10, one wave has a negative value and another has a positive value. At approximately x = 16 or - 16, two waves have maxima, but another wave is at a minimum. The important point is that for waves with different wavelengths, at one point (x = 0 in the example), all of the waves can be matched, but in general, at other points, some of the waves will be positive and some of the waves will be negative.
FIGURE 6.2. Five waves are shown that have different wavelengths. The wavelengths are λ = 1.2, 1.1, 1.0, 0.9, and 0.8. The phases are adjusted so all of the peaks of the waves match at 0 on the horizontal axis. However, because the waves have different wavelengths, they do not match up at other positions, in contrast to Figure 3.2. Note that at a position of approximately 10 or - 10, the dark gray wave has a positive peak, but the dashed light gray wave has a negative peak.
Figure 6.3 shows the result of superimposing (adding up) the five waves in Figure 6.2. At x = 0 (horizontal axis) in Figure 6.2, all of the waves are exactly in phase. The superposition (adding the waves together) shown in Figure 6.3 yields a maximum. In Figure 6.2, the waves are all exactly in phase only at x = 0. Near x = 0, the difference in the wavelengths has not produced a large shift in the peaks of one wave relative to another, so the waves are still pretty much in phase. There is another set of maxima at about x = 6 and -6. However, these maxima are not as large as the one at x = 0, because the peaks of the waves are not all right on top of each other, as can be seen in Figure 6.2. Beyond x = ± 10, the amplitude of the superposition is getting small. At any point, some waves are positive and some waves are negative, and they tend to destructively interfere. Because there are only five waves, the destructive interference is only partial.
Figure 6.4 shows the superposition of 250 waves with different wavelengths. The waves have equal-sized steps in wavelength in the range of wavelengths from 0 to 4. As for the five waves and their superposition shown in Figures 6.2 and 6.3, each wave has the same amplitude. The phases of the 250 waves are adjusted to match at x = 0 (x is the horizontal axis). Because there are many more waves over a wider range of wavelengths than in the superposition shown in Figure 6.3, the peak around x = 0 is much narrower, and the rest of the superposition dies out much more rapidly. The little oscillations come from the fact that all of the waves in the superposition have the same amplitude. If the amplitude of the wave at the middle of the spread of wavelengths has the biggest amplitude and the amplitudes of the other waves get smaller and smaller for wavelengths further and further from the center wavelength, it is possible to create a superposition that decays smoothly to zero without the set of deceasing amplitude oscillations. This type of superposition will be discussed below.
FIGURE 6.3. The superposition of the five waves shown in Figure 6.2. At x = 0 (horizontal axis), all of the waves in figure 6.2 are in phase, so they add constructively. Near x = 0, the waves are still pretty much in phase, but the next set of maxima at about x = 6 and -6 are not as large as the maximum at x = 0. In the regions between 10 and 20 and - 10 and -20, the difference in wavelengths makes some of the waves positive, where others