Doppler Effect

IntroductionIn our everyday life, we are used to perceive

sound by our sense of hearing. Sounds are the vibrations that travel through the air. It is characterized by the wave quantities which include frequency, wavelength, period and speed.

The paper aims to present basic concepts on Doppler Effect in sound. It would help us understand the changes in the relative motion of the source and the observer.

Questions?

• Why does the siren on a moving ambulance seem to produce sound with a higher pitch when it approaches an observer then decreases when it recede the observer.

• Is this simply because of the relative distance between the observer and the ambulance (sound)?

• Is it because of the loudness of the sound produced by the siren?

Definition

• Doppler Effect is the change in the frequency (or wavelength) of any emitted waves, such as a wave of light or sound as the source of the wave approaches or moves away from an observer.

• This effect was named from the Austrian physicist, Christian Johann Doppler, who first stated the physical principle in 1842.

Consider the Following

• Doppler’s principle explains why, if the source of waves and the observer are approaching each other, the sound heard by the observer becomes higher in pitch, whereas if the source and observer are moving apart the pitch becomes lower.

• For the sound waves to propagate it requires a medium such as air, where it serves as a frame of reference with respect to which motion of source and observer are measured.

SITUATION 1 Stationary Source and Observers (NO DOPPLER EFFECT)

• A stationary sound source S A stationary sound source S emits a spherical emits a spherical wavefronts of one λ apart wavefronts of one λ apart spread out at speed spread out at speed vv relative to the medium air.relative to the medium air.

• In time In time tt, the wavefronts , the wavefronts move a distance move a distance vtvt toward toward the observers, Othe observers, O11 & O & O2.2.

• The number of wavelengths The number of wavelengths detected by the observer detected by the observer infront and behind the infront and behind the source are the same and source are the same and equal to equal to vtvt/λ. /λ.

So, the frequency So, the frequency f f heard by both stationary heard by both stationary observers is given by equation (1),observers is given by equation (1),

f - frequency of sound sourcev - speed of sound waves

t - timeλ - wavelength

Consider the Following

• What if both of the observers in figure 1 are What if both of the observers in figure 1 are moving, is there any change in the frequency moving, is there any change in the frequency and wavelength of the source? and wavelength of the source?

SITUATION 2 Stationary Source; Moving Observers

• Observer 1 moves a Observer 1 moves a distance distance vvoott toward the toward the source at speed source at speed vvoo

• We know that wavefronts We know that wavefronts also move at speedalso move at speed v v towards Otowards O11 in time t at in time t at distance distance vtvt. The relative . The relative speed of the wavefronts speed of the wavefronts with respect to Owith respect to O11 becomes becomes ((vv + + vvOO)t.)t.

• The number of The number of wavelengths intercepted by wavelengths intercepted by OO11 at this distance is at this distance is

((vv + + vvOO)t)t/ λ. / λ.

From equation (1), we have λ = v/ f, f’ becomes

(3)

This shows that there is an increase in the frequency f’ heard by O1 as it goes nearer to the sound source as given by,

(2)

• If observer 2 moves away from the sound source, the distance traveled by the wavefronts with respect to O2 in time t, is vt – vot.

• Consequently, there would be a decrease in the frequency heard by O2 as given by,

(4)

Combining Equations (3) and (4), we have

(5)

(STATIONARY SOURCE; MOVING OBSERVER)

• In these situations only the frequency heard by the observers changes due to there motion relative to the source.• However the wavelength of sound remains constant.

SITUATION 3 Moving Source; Stationary Observers

• As the source moves a distance vST (T=1/f period of wave) toward O1 there is a decrease in the wavelength of sound by a quantity of vsT.• The shortened wavelength λ’ becomes

λ’ = λ – vsT

The frequency The frequency f’ f’ of sound wave heard by Oof sound wave heard by O1 1 increases increases

as given by,as given by,

(6)

• With respect to observer 2, the wavelength of sound increases, where λ’ becomes λ + vsT.

•The frequency f’ of sound wave heard by O2 decreases as given by,

(7)

Combining Equations (6) and (7), we have

(8)

(MOVING SOURCE; STATIONARY OBSERVER)

SITUATION 4 Moving Source and Observer

From the equations (5) and (8), we can now derive the equation of general Doppler Effect by replacing f in equation (5) with f’ of equation (8). This result to,

(MOVING SOURCE AND OBSERVER)

(9)

(9)

The ± signs correspond to the direction of the source or observer when they are moving relative to the other. These would determine whether there is an increase or decrease on the frequency heard by the observer during the motion.

(APPROACHING OBSERVER; RECEEDING SOURCE )(APPROACHING OBSERVER; RECEEDING SOURCE )

• If vIf voo> v> vs s , increase in observed frequency, increase in observed frequency

If vIf voo< v< vss , decrease in observed frequency , decrease in observed frequency

(RECEEDING OBSERVER; RECEEDING SOURCE )(RECEEDING OBSERVER; RECEEDING SOURCE )

• Decrease in observed frequencyDecrease in observed frequency

(APPROACHING OBSERVER; APPROACHING SOURCE)(APPROACHING OBSERVER; APPROACHING SOURCE)

• Increase in observed frequencyIncrease in observed frequency

(RECEEDING(RECEEDING OBSERVER; APPROACHING SOURCE)OBSERVER; APPROACHING SOURCE)

• If vIf voo> v> vs s , decrease in observed frequency, decrease in observed frequency

If vIf voo< v< vss , increase in observed frequency , increase in observed frequency