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Doppler Shift Formula

Doppler Shift was given by Christian Johann Doppler in 1842. This apparent change in the frequency of sound as a result of relative motion between the source and the observer is the Doppler effect.
There are eight Doppler Effect Formulas for frequency depending on cases:

(i) When the source is moving towards a observer at rest

(ii) When the source is moving away from the observer at rest

(iii) When observer is moving towards the stationary source

(iv) When observer moving away from a stationary source

(v) When both Source and observer moves towards each other

(vi) When both Source and observer move away from each other

(Vii) When the Source is approaching the Stationary observer and observer moving away from it

(Viii) When the Observer is approaching the Stationary source and source moving away from it
Where, vs = Velocity of the Source,
vo = Velocity of the Observer,
v = Velocity of sound or light in medium,
f = Real frequency,
f' = Apparent frequency.

Doppler effect formula is used to find the apparent frequency and wavelength for the source moving towards the observer and away from the observer or observer moving towards the source or away from the source.

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Doppler Shift Problems

Solved problems based on Doppler effect are given below:

Solved Examples

Question 1: A source and listener are moving towards each other with the speed of 54 km/hr. If the true frequency of sound emitted by the source is 500 Hz, calculate the observed frequency when both source and listener are moving towards each other.
Velocity of sound in air = 330 ms-1.
Solution:

Given: True frequency, f = 500 Hz,
Velocity of sound, vs = 54 km/hr = $\frac{54 \times 1000}{3600}$ = 15 ms-1,
Source and observer moving towards each other
$\therefore$ Apparent frequency is given by f' = $\frac{v + v_o}{v - v_{s}}$ f
= $\frac{330 + 15}{330 - 15}$ $\times$ 500
= 547.62 Hz.

Question 2: A fixed source emits sound of frequency 1000 Hz. What is the frequency as heard by a observer
(i) at rest
(ii) Moving towards the source at a constant speed of 20 ms-1 and
(iii) Moving away from the source at the same rate.
Solution:

Velocity of sound in air, v = 340 ms-1,
True frequency, f = 1000 Hz,
Velocity of observer, vo = 20 ms-1
(i) When both the source and observer are at rest, apparent frequency is same as true frequency
$\therefore$ Frequency as heard by listener at rest = f = 1000 Hz
(ii) Observer is moving towards the source
$\therefore$ Apparent frequency f1 = $\frac{v + v_{o}}{v}$ f

= $\frac{340 + 20}{340}$ $\times$ 1000

= 1059 Hz
(iii) Observer is moving away from the source
$\therefore$ Apparent frequency f2 = $\frac{v - v_{o}}{v}$ f
= $\frac{340 - 20}{340}$ $\times$ 1000
= 941 Hz.

 More topics in Doppler Shift Formula Doppler Effect Formula
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