1. Doppler Effect

2. Abstract
In 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.

3. One might wonder why the siren on a moving ambulance seems to produce sound with a higher pitch when it passes an observer and decreases when it recede the observer.
Is this simply because of the relative distance between the observer and the ambulance (sound)?
Or is it because of the loudness of the sound produced by the siren?

4. 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.

5. 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.

6. SITUATION 1 Stationary Source and Observers (NO DOPPLER EFFECT)
A stationary sound source S emits a spherical wavefronts of one λ apart spread out at speed v relative to the medium air.
In time t, the wavefronts move a distance vt toward the observers, O1 & O2.
The number of wavelengths detected by the observer infront and behind the source are the same and equal to vt/λ.

7. Thus, the frequency f heard by both stationary observers is given by,
f - frequency of sound source
v - speed of sound waves
t - time
λ - wavelength

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

9. If observer 1 moves towards the sound source, the distance traveled by the wavefronts with respect to O1 in time t, is vt + vot.
Consequently, there would be a increase in the frequency heard by O1 as given by,

10. From equation (1), we have λ = v/ f, f’ becomes
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)
From equation (1), we have λ = v/ f, f’ becomes
(3)

11. 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)

12. However the wavelength of sound remains constant.
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.

13. Sound source moving toward observer
Observer hears
increased pitch
(shorter wave length)
Frequency fo
Frequency fs
source
observer
at rest

14. Sound source moving away from observer
Observer hears
decreased pitch
(longer wave length)
Frequency fo
Frequency fs
observer
at rest
source

15. SITUATION 3 Moving Source; Stationary Observers
As the source moves a distance vST (T=1/f period of wave) toward O there is a decrease in the wavelength of sound by a quantity of vsT.
The shortened wavelength λ’ becomes
λ’ = λ – vsT

16. The frequency f’ of sound wave heard by O1 increases as given by,
(6)

17. The frequency f’ of sound wave heard by O2 decreases as given by,
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)

18. (MOVING SOURCE; STATIONARY OBSERVER)
Combining Equations (6) and (7), we have
(8)
(MOVING SOURCE; STATIONARY OBSERVER)

19. 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,
(9)
(MOVING SOURCE AND OBSERVER)

20. (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.

21. If vo> vs , increase in observed frequency
(APPROACHING OBSERVER; RECEEDING SOURCE )
If vo> vs , increase in observed frequency
If vo< vs , decrease in observed frequency
(RECEEDING OBSERVER; RECEEDING SOURCE )
Decrease in observed frequency

22. Increase in observed frequency
(APPROACHING OBSERVER; APPROACHING SOURCE)
Increase in observed frequency
(RECEEDING OBSERVER; APPROACHING SOURCE)
If vo> vs , decrease in observed frequency
If vo< vs , increase in observed frequency