# Relativistic Doppler Effect Formula

The Doppler Effect is the effect perceived when energy waves like light waves or sound waves travel with regards to an observer. There will be an alteration in frequency. An upward shift in frequency for an approaching source and a downward shift in frequency in the case of a retreating source. The relativistic Doppler Effect is some alteration in frequency caused when there is relativistic motion between observer and source.

The relativistic doppler effect formula is articulated as,

$v’=v\sqrt{\frac{1+\frac{v}{c}}{1-\frac{v}{c}}}$

if β = v/c then relativistic doppler effect formula is articulated as,

$v’=v\sqrt{\frac{1+\beta }{1-\beta }}$

Where,
The Apparent frequency (the frequency of observer) is ν’,

The Real frequency (frequency of the source) is ν.

The observer’s velocity v is,

The speed of light is c.

The Doppler effect of a moving source can also be articulated as,

$v’=\frac{v}{1\pm \frac{v_{s}}{v}}$

Where,
The source velocity is vs,

The observer velocity v is,

The Apparent frequency (the frequency of observer) is ν’,

The Real frequency (frequency of the source) is ν.

Solved Problems Relativistic Doppler Effect

Grounded on Relativistic Doppler Effect, some of the solved samples are provided underneath:

Problem 1: A bus is moving towards john at 1.5 ×× 1088 m/s. The bus flashes a headlight of frequency 4200 Hz on john. At what frequency will John receive the light?

Source velocity vs =  1.5 × 1088 m/s = 0.5 c, Observer velocity v = c, frequency νν = 4200 Hz

The frequency of light john receives is,

ν’ = ν/(1 ± vs/v)

ν’ = 4200/(1 – 0.5 c/ c)

ν’ = 2100 Hz.

Thus, the frequency of light john receives is 2100 Hz.

Problem 2: A train leaves the station with a velocity of 40 m/s. What will be its frequency if the passenger standing still perceives the frequency at 440 Hz?

Observed frequency ν’ = 40 m/s,

Real frequency ν = ?

The speed of sound is v = 343 m/s.

The frequency of train is,

ν’ = ν/(1 ± vs/v)

440 = ν/(1 ±± 40/343)

∴ frequency ν = 498.08 Hz

Thus, the frequency of train is 498.08 Hz.

#### Practise This Question

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