24.5.  The Doppler Effect and Electromagnetic Waves

Section 16.9 presents a discussion of the Doppler effect that sound waves exhibit when either the source of a sound wave, the observer of the wave, or both are moving with respect to the medium of propagation (e.g., air). This effect is one in which the observed sound frequency is greater or smaller than the frequency emitted by the source. A different Doppler effect arises when the source moves than when the observer moves.

Electromagnetic waves also can exhibit a Doppler effect, but it differs from that for sound waves for two reasons. First, sound waves require a medium such as air in which to propagate. In the Doppler effect for sound, it is the motion (of the source, the observer, and the waves themselves) relative to this medium that is important. In the Doppler effect for electromagnetic waves, motion relative to a medium plays no role, because the waves do not require a medium in which to propagate. They can travel in a vacuum. Second, in the equations for the Doppler effect in Section 16.9, the speed of sound plays an important role, and it depends on the reference frame relative to which it is measured. For example, the speed of sound with respect to moving air is different than it is with respect to stationary air. As we will see in Section 28.2, electromagnetic waves behave in a different way. The speed at which they travel has the same value, whether it is measured relative to a stationary observer or relative to one moving at a constant velocity. For these two reasons, the same Doppler effect arises for electromagnetic waves when either the source or the observer of the waves moves; only the relative motion of the source and the observer with respect to one another is important.

When electromagnetic waves and the source and the observer of the waves all travel along the same line in a vacuum (or in air, to a good degree of approximation), the single equation that specifies the Doppler effect is

(24.6)

In this expression, f_o is the observed frequency, and f_s is the frequency emitted by the source. The symbol v_rel stands for the speed of the source and the observer relative to one another, and c is the speed of light in a vacuum. Equation 24.6 applies only if v_rel is very small compared to c—that is, if v_relc. The plus sign in Equation 24.6 applies when the source and observer are moving toward one another, and the minus sign applies when they are moving apart.

It is essential to realize that v_rel is the relative speed of the source and the observer. Thus, if the source is moving due east at a speed of 28 m/s with respect to the earth, while the observer is moving due east at a speed of 22 m/s, the value for v_rel is 28 m/s–22 m/s=6 m/s. Because v_rel is the relative speed, it has no algebraic sign. The direction of the relative motion is taken into account by choosing the plus or minus sign in Equation 24.6. The plus sign is used when the source and the observer come together, and the minus sign is used when they move apart. Example 6 illustrates one familiar use of the Doppler effect for electromagnetic waves.

Police use radar guns and the Doppler effect to catch speeders. One such gun emits an electromagnetic wave with a frequency of fs=8.0×109 Hz, as Figure 24.17 illustrates. In this picture, a car is approaching a police car parked on the side of the road. The direction of approach is essentially head-on. The wave from the radar gun reflects from the speeding car and returns to the police car, where on-board equipment measures its frequency to be greater than that of the emitted wave by 2100 Hz. Find the speed of the car with respect to the highway. Figure 24.17  The radar gun used by police emits an electromagnetic wave in the radio frequency region of the spectrum. The Doppler effect exhibited by the wave after it reflects from a moving vehicle is used to determine the vehicle’s speed. Reasoning  The Doppler effect depends on the relative speed vrel between the speeding car and the police car. We will find this speed and then relate it to the speed of the moving car with respect to the highway by using the fact that the police car is at rest. There are two Doppler frequency changes in this situation. First, the speeder’s car “observes” the wave frequency coming from the radar gun to have a frequency fo that is different from the emitted frequency fs. According to Equation 24.6 (with the plus sign, since the two cars are coming together), fo–fs=fs(vrel/c). Then, the wave reflects and returns to the police car, where it is observed to have a frequency fo' that is different than its frequency fo at the instant of reflection. Again using Equation 24.6, we find that fo'–fo=fo(vrel/c). Adding the two previous equations gives the following result for the total Doppler change in frequency: where we have assumed that fo and fs differ by only a negligibly small amount, since vrel is small compared to the speed of light c. Solving for the relative speed vrel gives Solution The speed of the moving car relative to the police car is Since the police car is at rest, its speed is vP=0 m/s, and the speeder has a speed of .

The Doppler effect of electromagnetic waves provides a powerful tool for astronomers. For instance, Example 5.10 discusses how astronomers have identified a supermassive black hole at the center of galaxy M87 by using the Hubble space telescope. They focused the telescope on regions to either side of the center of the galaxy (see Figure 5.15). From the light emitted by these two regions, they were able to use the Doppler effect to determine that one side is moving away from the earth, while the other side is moving toward the earth. In other words, the galaxy is rotating. The speeds of recession and approach enabled astronomers to determine the rotational speed of the galaxy, and Example 5.10 shows how the value for this speed leads to the identification of the black hole. Astronomers routinely study the Doppler effect of the light that reaches the earth from distant parts of the universe. From such studies, they have determined the speeds at which distant light-emitting objects are receding from the earth.

The drawing shows three situations A, B, and C in which an observer and a source of electromagnetic waves are moving along the same line. In each case the source emits a wave of the same frequency. The arrows in each situation denote velocity vectors relative to the ground and have magnitudes of either v or 2v. Rank the frequencies of the observed waves in descending order (largest first) according to magnitude. Background: The electromagnetic wave frequency that is observed is not necessarily the same as the frequency emitted by the source because of the Doppler effect. The observed frequency depends on the speed of the source and the observer relative to one another. For similar questions (including calculational counterparts), consult Self-Assessment Test 24.1, which is described at the end of Section 24.6.

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