# Relativistic Doppler Effect Formula

## Doppler Effect

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.

For a moving source and a moving observer, the formula of frequency according to Doppler’s effect is given by:

$f =(\frac{c+v_{r}}{c+v_{s}})f_{0}$

where:
is the velocity of waves in the medium

is the velocity of the receiver relative to the medium

is the velocity of the source relative to the medium

### Solved Example

Problem 1:

A source is moving towards observer with a speed of 40 m/s and having frequency 240 Hz and observer is moving towards the source with a velocity of 40 m/s. What is the apparent frequency heard by the observer, if the velocity of sound is 340 m/s?

given:

Vs=40 m/s

Vr=40 m/s

c=340 m/s

f0= 240 Hz

The source and observer are approaching each other, therefore the velocity of the source V is negative and the velocity of the listener Vr is positive.

$f =(\frac{c+v_{r}}{c+v_{s}})f_{0}$

$f =(\frac{340+40}{340-40})240$
f = 304Hz
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