27.1

The Principle of Linear Superposition

CONCEPTS AT A GLANCE Chapter 17 examines what happens when several sound waves are present at the same place at the same time. The pressure disturbance that results is governed by the principle of linear superposition, which states that the resultant disturbance is the sum of the disturbances from the individual waves. Light is also a wave, an electromagnetic wave, and it too obeys the superposition principle. The Concepts-at-a-Glance chart in Figure 27-1 illustrates that this principle is used to explain interference phenomena associated with light, including Young’s double-slit interference, thin-film interference, and the interference effects that occur in the Michelson interferometer. This chart is analogous to the one in Figure 17-17 for the interference of sound waves. When two or more light waves pass through a given point, their electric fields combine according to the principle of linear superposition and produce a resultant electric field. According to Equation 24.5b, the square of the electric field strength is proportional to the intensity of the light, which, in turn, is related to its brightness. Thus, interference can and does alter the brightness of light, just as it affects the loudness of sound.

Figure 27-1 CONCEPTS AT A GLANCE The principle of linear superposition explains various interference phenomena associated with electromagnetic waves. The beautiful iridescent blue color of this butterfly is due to thin-film interference of light. A transparent cuticle-like material on the upper wing surface acts like a thin film. (Douglas E. Walker/Masterfile).

Figure 27-2 illustrates what happens when two identical waves (same wavelength and same amplitude) arrive at the point P in phase—that is, crest-to-crest and trough-to-trough. According to the principle of linear superposition, the waves reinforce each other and constructive interference occurs. The resulting total wave at P has an amplitude that is twice the amplitude of either individual wave, and in the case of light waves, the brightness at P is greater than that due to either wave alone. The waves start out in phase and are in phase at P because the distances and between this spot and the sources of the waves differ by one wavelength . In Figure 27-2, these distances are wavelengths and wavelengths. In general, when the waves start out in phase, constructive interference will result at P whenever the distances are the same or differ by any integer number of wavelengths—in other words, assuming is the larger distance, whenever , where .

Figure 27-2 The waves emitted by source 1 and source 2 start out in phase and arrive at point P in phase, leading to constructive interference at that point.

Figure 27-3 shows what occurs when two identical waves arrive at the point P out of phase with one another, or crest-to-trough. Now the waves mutually cancel, according to the principle of linear superposition, and destructive interference results. With light waves this would mean that there is no brightness. The waves begin with the same phase but are out of phase at P because the distances through which they travel in reaching this spot differ by one-half of a wavelength ( and in the drawing). In general, for waves that start out in phase, destructive interference will take place at P whenever the distances differ by any odd integer number of half-wavelengths—that is, whenever , where is the larger distance. This is equivalent to .

Figure 27-3 The waves emitted by the two sources have the same phase to begin with, but they arrive at point P out of phase. As a result, destructive interference occurs at P.

Examples illustrating the application of the principle of linear superposition to explain the interference of light waves can be found throughout this chapter. For relatively straightforward examples that deal with two sources of sound waves and the resulting constructive or destructive interference see Examples 1 and 2 in Chapter 17 and Concepts & Calculations Example 11 in this chapter.
If constructive or destructive interference is to continue occurring at a point, the sources of the waves must be coherent sources. Two sources are coherent if the waves they emit maintain a constant phase relation. Effectively, this means that the waves do not shift relative to one another as time passes. For instance, suppose that the wave pattern of source 1 in Figure 27-3 shifted forward or backward by random amounts at random moments. Then, on average, neither constructive nor destructive interference would be observed at point P because there would be no stable relation between the two wave patterns. Lasers are coherent sources of light, whereas incandescent light bulbs and fluorescent lamps are incoherent sources.

Check Your Understanding 1

Two separate coherent sources produce waves whose wavelengths are 0.10 m. The two waves spread out and intersect at a certain point. Does constructive or destructive interference occur at this point when (a) one wave travels 3.20 m and the other travels 3.00 m, (b) one wave travels 3.20 m and the other travels 3.05 m, and (c) one wave travels 3.20 m and the other travels 2.95 m? (The answers are given at the end of the book.)
Background: Whether constructive or destructive interference occurs at a given point depends on how the difference in path lengths is related to the wavelength of the waves.
For similar questions (including calculational counterparts), consult Self-Assessment Test 27.1, which is described at the end of Section 27.4.

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