Relativistic Doppler Effect Formula

The Doppler effect is defined as a rise or fall in the frequency of sound, light, or other waves when the source and observer move closer or farther apart. It is seen when energy waves such as light waves or sound waves travel in relation to an observer. It asserts that there will be a change in frequency when an approaching source causes an upward movement in frequency or a receding source causes a downward shift in frequency. This is called the relativistic Doppler Effect. 

What is the Relativistic Doppler Effect?

It is a change in frequency produced by relativistic velocity between the observer and the source. It is only relevant when the velocities of sound and objects are substantially slower than the speed of sound in that medium. Furthermore, the velocity of the object and the source must be parallel.

Formula for Relativistic Doppler Effect

The formula for Relativistic Doppler Effect is given by,

f’ = f (c + v_r) / (c + v_s)

where,

• f’ is the apparent frequency,
• f is the actual frequency,
• c is the speed of sound waves in the medium,
• v_r​ is the velocity of the receiver/observer with respect to the medium,
• v_s​ is the velocity of the source with respect to the medium.

Sample Problems

Problem 1: Calculate the apparent frequency if the source moves towards the observer at a speed of 50 m/s with the frequency of 200 Hz and the observer moves towards the source with a speed of 50 m/s. The velocity of sound waves in the medium is 340 m/s.

Solution:

We have,

f = 200

c = 340

v_r = 50

v_s = -50

Using the formula we have,

f’ = f (c + v_r) / (c + v_s)

= 200 (340 + 50)/(340 – 50)

= 200 (390/290)

= 268.96 Hz

Problem 2: Calculate the apparent frequency if the source moves towards the observer at a speed of 30 m/s with the frequency of 180 Hz and the observer moves towards the source with a speed of 20 m/s. The velocity of sound waves in the medium is 340 m/s.

Solution:

We have,

f = 180

c = 340

v_r = 20

v_s = -30

Using the formula we have,

f’ = f (c + v_r) / (c + v_s)

= 180 (340 + 20)/(340 – 30)

= 180 (360/290)

= 223.44 Hz

Problem 3: Calculate the apparent frequency if the source moves towards the observer at a speed of 70 m/s with the frequency of 100 Hz and the observer moves towards the source with a speed of 50 m/s. The velocity of sound waves in the medium is 340 m/s.

Solution:

We have,

f = 100

c = 340

v_r = 50

v_s = -70

Using the formula we have,

f’ = f (c + v_r) / (c + v_s)

= 100 (340 + 50)/(340 – 70)

= 100 (390/270)

= 144.45 Hz

Problem 4: Calculate the speed of the observer if the source moves towards the observer with an actual frequency of 150 Hz and apparent frequency of 170 Hz and the source moves towards the observer with a speed of 20 m/s. The velocity of sound waves in the medium is 340 m/s.

Solution:

We have,

f = 150

f’ = 170

c = 340

v_s = -20

Using the formula we have,

f’ = f (c + v_r) / (c + v_s)

c + v_r = f’ (c + v_s)/f

340 + v_r = 170 (340 – 20)/150

340 + v_r = 170 (2.13)

v_r = 22.1 m/s 

Problem 5: Calculate the speed of the observer if the source moves towards the observer with an actual frequency of 250 Hz and apparent frequency of 300 Hz and the source moves towards the observer with a speed of 40 m/s. The velocity of sound waves in the medium is 340 m/s.

Solution:

We have,

f = 250

f’ = 300

c = 340

v_s = -40

Using the formula we have,

f’ = f (c + v_r) / (c + v_s)

c + v_r = f’ (c + v_s)/f

340 + v_r = 300 (340 – 40)/250

340 + v_r = 360

v_r = 20 m/s

Problem 6: Calculate the speed of the source if the source moves towards the observer with an actual frequency of 120 Hz and apparent frequency of 170 Hz and the observer moves towards the source with a speed of 25 m/s. The velocity of sound waves in the medium is 340 m/s.

Solution:

We have,

f = 120

f’ = 170

c = 340

v_r = 25

Using the formula we have,

f’ = f (c + v_r) / (c + v_s)

c + v_s = f (c + v_r)/f’

340 + v_s = 120 (340 + 25)/170

340 + v_s = 257.64

v_s = 82.35 m/s