Lesson: Chapter - 17

The Doppler Effect

So far we have only discussed cases where the source of waves is at rest. Often, waves are emitted by a source that moves with respect to the medium that carries the waves, like when a speeding cop car blares its siren to alert onlookers to stand aside. The speed of the waves, v, depends only on the properties of the medium, like air temperature in the case of sound waves, and not on the motion of the source: the waves will travel at the speed of sound (343 m/s) no matter how fast the cop drives. However, the frequency and wavelength of the waves will depend on the motion of the wave’s source.

This change in frequency is called a Doppler shift.Think of the cop car’s siren, traveling at speed v_s, and emitting waves with frequency f and period T = 1/f. The wave crests travel outward from the car in perfect circles (spheres actually, but we’re only interested in the effects at ground level). At time T after the first wave crest is emitted, the next one leaves the siren. By this time, the first crest has advanced one wavelength, ?, but the car has also traveled a distance of v_sT. As a result, the two wave crests are closer together than if the cop car had been stationary.

The shorter wavelength is called the Doppler-shifted wavelength, given by the formula ?_D = ? - v_sT = ?(v - v_s) / v. The Doppler-shifted frequency is given by the formula:

ƒ_D = ƒ  v/v - v_s

Similarly, someone standing behind the speeding siren will hear a sound with a longer wavelength, ?_D = ? + v_sT = ?(v + v_s)v , and a lower frequency, ƒ_D = ƒ  v/  (v + v_s).

You’ve probably noticed the Doppler effect with passing sirens. It’s even noticeable with normal cars: the swish of a passing car goes from a higher hissing sound to a lower hissing sound as it speeds by. The Doppler effect has also been put to valuable use in astronomy, measuring the speed with which different celestial objects are moving away from the Earth.

Example

A cop car drives at 30 m/s toward the scene of a crime, with its siren blaring at a frequency of 2000 Hz. At what frequency do people hear the siren as it approaches? At what frequency do they hear it as it passes? The speed of sound in the air is 343 m/s.

As the car approaches, the sound waves will have shorter wavelengths and higher frequencies, and as it goes by, the sound waves will have longer wavelengths and lower frequencies. More precisely, the frequency as the cop car approaches is:

The frequency as the cop car drives by is:

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