Solved Examples on Doppler Effect and Doppler Shift

Example 1: Two vehicles A and B are moving toward each other at a speed of 80 m/s. If one of the vehicles honks with a frequency of 250 Hz, then what will be the frequency of the honk observed by the passenger sitting inside vehicle A? (The velocity of sound in air is 343 m/s).

Solution: 

Given data:

The velocity of vehicles A and B are 80m/s, i.e.,

v_r = v_s = 80 m/s

Emitted frequency (f₀) = 250 Hz

As the observer and the source are moving toward each other, the apparent frequency (f) is given as follows:

f = [(c + v_r)/ (c – vs)] f₀

f = [(343 + 80)/ (343 – 80)] × 250

= (423/263) × 250

f = 402.0912 Hz ≈ 402 Hz

Hence, the apparent frequency (f) is approximately equal to 402 Hz.

Example 2: A man is standing at the railway crossing, and the frequency of the whistle emitted by the train is 1.05 kHz. What will be the apparent frequency heard by the man as the train approaches him at a speed of 360 km/h? (The velocity of sound in air is 343 m/s).

Solution: 

Given data:

Speed of the train (v_s) = 360 km/h = 100 m/s

v_r = 0

The velocity of sound in air (c) = 343 m/s

Emitted frequency (f₀) = 1.05 kHz

Here, the source is moving toward the observer at rest. So, the apparent frequency (f) is given as follows:

f = f₀

= [343/ (343 − 100)] × 1.05

= (343/243) × 1.05

= 1.48 kHz

Hence, the apparent frequency (f) is 1.48 kHz.

Example 3: A traffic policeman who was standing on a road whistled at a frequency of 480 Hz. What will be the apparent frequency heard by the car driver who is approaching the policeman at a speed of 90 km/h? (The velocity of sound in air is 343 m/s).

Solution: 

Given data:

The velocity of sound in air (c) = 343 m/s

Speed of the car (v_r) = 90 km/h = 25 m/s

v_s = 0

Emitted frequency (f₀) = 480 Hz

Here, the observer is moving toward the source at rest. So, the apparent frequency (f) is given as follows:

f = [(c + v_r)/c] f₀

= [(343 + 25)/ 343] × 480

= (368/343) × 480

f = 514.985 Hz ≈ 515 Hz

Hence, the apparent frequency (f) is approximately equal to 515 Hz.

Example 4: Two trains A and B are moving away from other at a speed of 540 km/h. If the frequency of the whistle emitted by train B is 1.65 kHz, then what will be the apparent frequency of the whistle heard by the passenger sitting inside train A? (The velocity of sound in air is 343 m/s).

Solution:

Given data:

The velocity of sound in air (c) = 343 m/s

The velocity of trains A and B is 540 km/h, i.e., 

v_s = v_r = 540 km/h = 150 m/s

Emitted frequency (f₀) = 1.65 kHz = 1650 Hz

As the observer and the source are moving away from each other, the apparent frequency (f) is given as follows:

f = [(c – v_r)/ (c + vs)] f₀

= [(343 – 150)/ (343 + 150)] ×1650

= (193/493) ×1650

= 645.943 ≈ 646 Hz

Hence, the apparent frequency (f) is approximately equal to 646 Hz.

Example 5: A traffic policeman who was standing on a road whistled at a frequency of 550 Hz. What will be the apparent frequency heard by the person driving a motorbike who is moving away from the policeman at a speed of 72 km/h? (The velocity of sound in air is 343 m/s).

Solution:

Given data:

The velocity of sound in air (c) = 343 m/s

Speed of the motorbike (v_r) =72 km/h = 20 m/s

v_s = 0

Emitted frequency (f₀) = 550 Hz

Here, the observer is moving away from the source at rest. So, the apparent frequency (f) is given as follows:

f = [(c − v_r)/c] f₀

= [(343 − 20)/ 343] × 550

= (323/343) × 550

f = 517.930 Hz ≈ 518 Hz

Hence, the apparent frequency (f) is approximately equal to 518 Hz.

Doppler effect or Doppler shift phenomenon was described in 1842 by an Austrian physicist, Christian Doppler, and it is named after him. The Doppler effect or Doppler shift is a change (increase or decrease) in the frequency of a wave as the source and the observer move (towards or away from) each other relative to the medium. Based on the direction of the source and the observer and the magnitudes of their velocities, the observed frequency can be less or more than the source frequency. For example, the pitch of the sound of an ambulance siren changes as it passes us. It happens because of the relative velocity between the source and the observer. When the ambulance is approaching us, the relative velocity is negative, and the relative velocity is positive when it is moving away. So, that is the reason behind the difference in the pitch of the sound of the ambulance siren while it is approaching and while it is moving away. The Doppler effect applies to all types of waves, including sound and light. The Doppler effect is responsible for the Blue Shift or Red Shift phenomenon observed in light waves. The Doppler effect is used in various fields such as radar, astronomy, satellite communication and navigation, medical imaging, etc.