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In the Doppler effect equation, v is the velocity of the sound wave, vd is the velocity of the detector, vs is the velocity of the sound s source, fs is the frequency of the wave emitted by the source, and fd is the frequency received by the detector This equation applies when the source is moving, when the observer is moving, and when both are moving As you solve problems using the above equation, be sure to define the coordinate system so that the positive direction is from the source to the detector The sound waves will be approaching the detector from the source, so the velocity of sound is always positive Try drawing diagrams to confirm that the term (v vd )/(v vs ) behaves as you would predict based on what you have learned about the Doppler effect Notice that for a source moving toward the detector (positive direction, which results in a smaller denominator compared to a stationary source) and for a detector moving toward the source (negative direction and increased numerator compared to a stationary detector), the detected frequency, fd, increases Similarily, if the source moves away from the detector or if the detector moves away from the source, then fd decreases Read the Connecting Math to Physics feature below to see how the Doppler effect equation reduces when the source or observer is stationary

zero, the equation might reduce to a form that is easier to use Stationary detector, source in motion: vd 0 fd fs Stationary source, detector in motion: vs 0 fd fs

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The Doppler Effect A trumpet player sounds C above middle C (524 Hz) while traveling in a convertible at 246 m/s If the car is coming toward you, what frequency would you hear Assume that the temperature is 20 C

Sketch the situation Establish a coordinate axis Make sure that the positive direction is from the source to the detector Show the velocities of the source and detector Known:

Unknown:

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Are the units correct Frequency is measured in hertz Is the magnitude realistic The source is moving toward you, so the frequency should be increased

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6 Repeat Example Problem 1, but with the car moving away from you What frequency would you hear 7 You are in an auto traveling at 250 m/s toward a pole-mounted warning siren If the siren s frequency is 365 Hz, what frequency do you hear Use 343 m/s as the speed of sound 8 You are in an auto traveling at 55 mph (246 m/s) A second auto is moving toward you at the same speed Its horn is sounding at 475 Hz What frequency do you hear Use 343 m/s as the speed of sound 9 A submarine is moving toward another submarine at 920 m/s It emits a 350-MHz ultrasound What frequency would the second sub, at rest, detect The speed of sound in water is 1482 m/s 10 A sound source plays middle C (262 Hz) How fast would the source have to go to raise the pitch to C sharp (271 Hz) Use 343 m/s as the speed of sound

The Doppler effect occurs in all wave motion, both mechanical and electromagnetic It has many applications Radar detectors use the Doppler effect to measure the speed of baseballs and automobiles Astronomers observe light from distant galaxies and use the Doppler effect to measure their speeds and infer their distances Physicians can detect the speed of the moving heart wall in a fetus by means of the Doppler effect in ultrasound Bats use the Doppler effect to detect and catch flying insects When an insect is flying faster than a bat, the reflected frequency is lower, but when the bat is catching up to the insect, as in Figure 15-7, the reflected frequency is higher Not only do bats use sound waves to navigate and locate their prey, but they often must do so in the presence of other bats This means they must discriminate their own calls and reflections against a background of many other sounds of many frequencies Scientists continue to study bats and their amazing abilities to use sound waves