# The Doppler Effect

Perhaps you have noticed how the sound of a vehicle’s horn changes as the vehicle moves past you. The frequency of the sound you hear as the vehicle approaches you is higher than the frequency you hear as it moves away from you. This is one example of the Doppler effect.

To see what causes this apparent frequency change, imagine you are in a boat that is lying at anchor on a gentle sea where the waves have a period of T=3.0 s. This means that every 3.0 s a crest hits your boat. Figure shows this situation, with the water waves moving toward the left. If you set your watch to just as one crest hits, the watch reads 3.0 s when the next crest hits, 6.0 s when the third crest hits, and so on. From these observations you conclude that the wave frequency is f=(1/T) = (1/3.0) Hz. Now suppose you start your motor and head directly into the oncoming waves, as shown in Figure. Again you set your watch to as a crest hits the front of your boat. Now, however, because you are moving toward the next wave crest as it moves toward you, it hits you less than 3.0 s after the first hit. In other words, the period you observe is shorter than the 3.0-s period you observed when you were stationary. Because f = 1/T, you observe a higher wave frequency than when you were at rest.

If you turn around and move in the same direction as the waves (see Fig.), you observe the opposite effect. You set your watch to as a crest hits the back of the boat. Because you are now moving away from the next crest, more than 3.0 s has elapsed on your watch by the time that crest catches you. Thus, you observe a lower frequency than when you were at rest.

These effects occur because the relative speed between your boat and the waves depends on the direction of travel and on the speed of your boat. When you are moving toward the right in Figure, this relative speed is higher than that of the wave speed, which leads to the observation of an increased frequency. When you turn around and move to the left, the relative speed is lower, as is the observed frequency of the water waves.

Let us now examine an analogous situation with sound waves, in which the water waves become sound waves, the water becomes the air, and the person on the boat becomes an observer listening to the sound. In this case, an observer O is moving and a sound source S is stationary. For simplicity, we assume that the air is also stationary and that the observer moves directly toward the source. The observer moves with a speed vO toward a stationary point source (vS = 0)  In general, at rest means at rest with respect to the medium, air.

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