Sound Waves: Doppler Effect

R. Field 04/19/2012

Sound Waves:

Doppler Effect

s sound D sound s sound D sound s D sound s D sound pulse obs f v v v f v v v v v t t v t v t N f = = + = + = + = + / / ) ( λ λ s s sound s sound pulse obs f v t t v t N f = = = = λ λ / s sound D sound obs

f

v

v

v

f

=

±

• Doppler Shift:

If either the detector or the source of sound is moving, or if both are moving, then the emitted frequency, f_s, of the source and the detected frequency, f_obs, are different. If both the source and the detector are at rest with respect to the air and v_sound = λλλλ_sf_s is the

speed of sound in air then

(Source and detector at rest)

(source stationary, detector stationary)

• Detector Moving, Source Stationary:

If source is at rest and the detector is moving toward the source with speed v_D with respect to the air and v = λλλλ_sf_s is the speed of sound in air then

(Source at rest, Detector moving toward the source)

(source stationary, detector moving toward the source)

(source stationary, detector moving toward (+) or away (-) from the source)

Sound Waves:

Doppler Effect

s s sound D sound obs f v v v v f − + = s s sound sound obs f v v v f m = s s sound sound s s s sound sound s s s sound sound obs sound obs f v v v f v f v v T v T v v v f − = − = − = = / /

λ

• Detector Stationary, Source Moving:

If detector is at rest and the source is moving toward the

detector with speed vs with respect to the air and vsound = λλλλsfs is the speed of sound in air and Ts = 1/fs is the time between emitted pulses then

(detector stationary, source moving toward the detector)

(detector stationary, source moving toward the detector)

• Detector and Source Moving:

If detector is moving with speed v_D and the source is moving with speed v_s with respect to the air then there are the

following four possibilities:

s s sound D sound obs f v v v v f + − = Detector Detector D sound f v v f = − Source Source D sound f v v f = +

(detector stationary, source moving toward (-) or away (+) from the source)

vsound

vsound

vsound

R. Field 04/19/2012

Summary:

Doppler Effect

s s sound D sound obs

f

v

v

v

v

f

−

−

=

• Detector and Source Moving:

If detector is moving with speed v_D and the source is moving with speed v_s, and v_sound = λλλλ_sf_s is the speed of sound in air then the observed frequency at the detector is

Take vD positive if the detector is moving in the direction of the propagation of the sound wave.

Take v_D negative if the detector is moving opposite the direction of propagation of the sound wave.

Take v_s positive if the source is moving in the direction of propagation of the sound wave.

Take v_S negative if the source is moving opposite the direction of propagation of the sound wave.

Doppler Effect:

Examples

Hz

Hz

f

v

v

f

v

v

v

f

_s sound s s s sound sound obs

959

)

343

/

200

(

1

400

/

1

1

≈

−

=

−

=

−

=

Hz

Hz

f

v

v

f

v

v

v

f

_{s D sound s} sound D sound obs

633

)

400

)(

343

/

200

1

(

)

/

1

(

≈

+

=

+

=

+

=

• A low flying airplane skims the ground at a speed of 200 m/s as it approaches a stationary observer. A loud horn whose fundamental frequency is 400 Hz is carried on the plane. What frequency does the ground observer hear? (Assume that the speed of sound in the air is 343 m/s.)

• If instead the horn were on the ground, what frequency would the airplane pilot hear as she approached?

Answer: 959 Hz

R. Field 04/19/2012

Doppler Effect:

Example

• The pitch of the sound from a race car engine drops the musical interval of a fourth when it passes the spectators. This means the frequency of the

sound after passing is 0.75 times what it was before. How fast is the race can moving? (Assume that the speed of sound in the air is 343 m/s.)

s

m

s

m

v

v

_{s sound}

(

343

/

)

49

.

0

/

75

.

0

1

75

.

0

1

1

1

≈













+

−

=













+

−

=

ξ

ξ

s s sound sound toward f v v v f − = _s s sound sound away f v v v f + = s sound s sound toward away v v v v f f + − = =

ξ

Answer: 49.0 m/s s sound s sound v v v v + ) = − (

ξ

v_s(1+

ξ

) = v_sound(1−

ξ

)

Doppler Effect:

Example

• A stationary motion detector sends sound waves of frequency of 600 Hz toward a truck that is speeding away. The waves sent out by the detector are reflected off the truck and then are received back at the detector. If the frequency of the waves received back at the detector is 400 Hz, what is the speed of the receding truck (in m/s)? (Take the speed of sound to be 343 m/s.) 0 f v v v f sound truck sound truck − = 0 f v v v v f v v v f truck sound truck sound truck truck sound sound obs + − = + =

Part 1: The truck is the detector and ftruck is the frequency observed by the truck and f0 is the original frequency emitted by the motion detector which is the source.

Part 2: The truck is now the source emitting frequency f_truck and the motion detector is the detector which observes frequency f_obs.

s m s m Hz Hz v f f

v_truck 0 obs _sound 600 400 (343 / ) = 68.6 /

+ − =     + − = truck sound truck sound obs v v v v f f + − = 0 Answer: 68.6 m/s

R. Field 04/19/2012

Doppler Effect:

Example

• In the figure, a French submarine and a U.S. submarine move toward each other during maneuvers in motionless water in the North Atlantic. The French sub

moves at speed v_F = 100 km/h, and the U.S. sub at v_US = 200 km/h. The French

sub sends out a sonar signal (sound wave in water) at 1,000 Hz. Sonar waves travel at 5000 km/h. What frequency is detected by the French sub in the signal reflected back to it by the U.S. sub?

0 f v v v v f F sound US sound US − + = Hz Hz h km h km h km h km h km h km h km h km f v v v v v v v v f v v v v f F sound US sound US sound F sound US US sound F sound obs 6 . 1127 ) 1000 ( ) / 100 / 5000 )( / 200 / 5000 ( ) / 200 / 5000 )( / 100 / 5000 ( ) )( ( ) )( ( 0 = − − + + = − − + + = − + =

Part 1: The US sub is the detector and f_US is the frequency observed by the US sub and f₀ is the original frequency emitted by the French sub which is the source.

Part 2: The US sub is now the source emitting frequency fUS and the French sub is the detector which observes frequency fobs.