# Doppler Shift Formula

Doppler Shift Formula

The Doppler Shift, when associated with sound, is the change in frequency of a source as it moves: the frequency will appear to increase as the source comes towards a listener and will appear to decrease as the source moves away from a listener. (This formula is also used to calculate the motion of stars.)

f = f_{s} (v + v_{L})/(v – v_{s})for sound

f = frequency heard by listener

f_{s} = frequency of the source

v = velocity of sound

v_{s} = velocity of the source

(positive if moving towards listener, negative if moving away from listener)

v_{L} = velocity of listener

(positive if moving toward the source, negative if moving away from the source)

Doppler Shift Formula Questions:

1) An ambulance has a velocity of 50 m/s and its siren produces a steady frequency of 250 Hz. What is the frequency of sound heard by an observer who is in front of the ambulance, assuming the velocity of sound equals 343 m/s ?

Answer: The velocity of sound, v = 343 m/s, and the velocity of the ambulance, the source, v_{s} = 50 m/s. The frequency of the source, f_{s} = 250 Hz. The velocity of the listener is v_{L} = 0. Use the formula.

f = f_{s} (v + v_{L})/(v – v_{s})for sound

f = (250 Hz)(343 m/s + 0 m/s) / (343 m/s – 50 m/s)

f = (250 Hz)(343 m/s)/293 m/s

f = (85750 Hz m/s)/ 293 m/s

f = 292.66 Hz

Note, the frequency heard by the listener is higher than that actually being emitted by the ambulance.

2) An ambulance comes to a stop, but its siren is still on with a steady frequency of 250 Hz. What is the frequency of sound heard by an observer who driving away from the ambulance at 50 m/s, assuming the velocity of sound equals 343 m/s ?

Answer: Because the listener is moving away from the ambulance, his velocity, v_{L} = – 25 m/s. The velocity of the ambulance, v_{s} = 0.

f = f_{s} (v + v_{L})/(v – v_{s})for sound

f = (250 Hz) (343 m/s – 50 m/s)/(343 m/s – 0)

f = (250 Hz)(293 m/s) / (343 m/s)

f = 73250 Hz m/s / 343 m/s

f = 213.56 Hz

Note: the sound heard by the driver is at a lower frequency than that actually being emitted by the ambulance.

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