__What is called “Doppler effect”?__

If there is a *relative motion* between the source of sound and the observer, the *frequency* of sound felt by the observer is different from the *frequency* of sound emitted by the source. This phenomenon is popularly known as Doppler effect.

*The apparent change in frequency of sound, when the sound source and the observer are in relative motion is called Doppler effect.*

Let, us consider five different situations where Doppler effect is noticed –

- Sound source is moving towards an stationary observer.
- Source is moving away from an stationary observer.
- Observer is moving towards an stationary sound source.
- The observer is moving away from an stationary sound source.
- Observer and sound source, both are moving towards each other.

__Source is moving towards observer__

Consider about the figure shown below.

A sound source S is moving with a speed ( v_s ) towards an stationary observer at O . Let –

- ( \nu ) is the
*frequency*of vibration of the source. - ( v ) is the
*velocity*of sound in the medium. - ( v_s ) is the velocity of sound source.
- ( T ) is the
*time period*of the vibrations of sound source.

Assume that, at time ( t = 0 ) *, *the sound source is at a distance ( L ) from the observer.

Then, first sound pulse will reach the observer at time –

t_1 = \left ( \frac { L }{ v } \right )

The sound source will emit second pulse after time ( T ) of the first pulse*. *In the mean time, the source will move a distance ( s = v_s T ) towards the observer.

The distance between source and observer is now –

L - s = L - v_s T .

So, the second pulse will reach the observer at time –

t_2 = T + \left ( \frac {L - v_s T}{v} \right )

Therefore, time interval between two successive pulses felt by the observer will be –

T' = ( t_2 - t_1 )

= \left [ T + \left ( \frac {L - v_s T}{v} \right ) - \left ( \frac {L}{v} \right ) \right ]

= \left [ T - \left ( \frac {v_s T}{v} \right ) \right ]

= T \left ( 1 - \frac {v_s}{v} \right ) = T \left ( \frac {v - v_s}{v} \right )

Therefore, the apparent frequency of the sound felt by the observer will be –

\nu' = \left ( \frac {1}{T'} \right ) = \left ( \frac {1}{T} \right ) \left ( \frac {v}{v - v_s} \right )

= \nu \left ( \frac {v}{v - v_s} \right ) ……… (1)

So, \quad \nu' > \nu \quad \text {because} \quad ( v - v_s ) < v

*Hence, the frequency **of sound appears to increases when the source moves towards the stationary observer.*

__Source is moving away from observer__

If the sound source* *is moving with speed ( v_s ) away from the stationary observer then the apparent frequency of sound can be obtained by replacing ( v_s ) by ( - v_s ) in equation (1).

Therefore, \quad \nu' = \nu \left ( \frac {v}{v + v_s} \right ) ……… (2)

So, \quad \nu' < \nu \quad \text {because} \quad ( v + v_s ) > v

*Hence, the frequency of sound appears to decreases when the source moves away from the stationary observer.*

__Observer is moving towards source__

Consider about the figure shown below.

A sound source S is stationary and a observer O is moving towards the sound source. Let, ( v_o ) is the velocity of moving observer.

At any instant the distance between two successive pulses will be ( \lambda = v T ) . When the observer receives one pulse, the next pulse is at distance ( v T ) away from him. The second pulse is moving towards observer with sound velocity ( v ) . Also the observer is moving towards the pulse with a speed of ( v_o ) .

*Relative velocity* of pulse with respect to observer will be ( v + v_o ) . Therefore, time between two successive pulses received by the observer will be –

T' = \left [ \frac {v T}{(v + v_o)} \right ]

= T \left [ \frac {v}{(v + v_o)} \right ]

This is the *time period* of pulses felt by the observer. Hence, the apparent frequency of the sound heard by the observer will be –

\nu' = \left ( \frac {1}{T'} \right ) = \left ( \frac {1}{T} \right ) \left ( \frac {v + v_o}{v} \right )

= \nu \left ( \frac {v + v_o}{v} \right ) ………. (3)

Clearly \quad \nu' > \nu \quad \text {because} \quad ( v + v_0 ) > v

*Hence, the frequency of sound appears to increases when the observer moves towards the stationary source.*

__Observer is moving away from source__

If the observer* *is moving with speed ( v_o ) away from the stationary source, then the apparent frequency of sound can be obtained by replacing ( v_o ) by ( - v_o ) in equation (3).

Therefore, \quad \nu' = \nu \left ( \frac {v - v_o}{v} \right ) ……… (4)

So, \quad \nu' < \nu \quad \text {because} \quad ( v - v_o ) < v .

*Hence, the frequency of sound appears to decreases when the observer moves away from the stationary source.*

__Observer and source both are moving to each other__

Consider about the figure shown below. Observer and sound source both are moving towards each other. Let –

- ( v_o ) is the velocity of moving observer towards source.
- ( v_s ) is the velocity of moving sound source towards observer.

At time ( t = 0 ) , the observer and the source are at positions ( O_1 ) and ( S_1 ) respectively. Assume that, the distance between them is ( L ) . Since, the observer is moving towards the source, so the relative speed of the first pulse with respect to the observer will be ( v + v_o ) .

Therefore, the observer will receive the first pulse at time –

t_1 = \left ( \frac {L}{v + v_o} \right )

Source will emit the second pulse after time ( T ) . In time ( T ) , the source has moved a distance of ( v_s T ) and the observer has moved a distance of ( v_o T ) . Hence, the new distance between them is now –

L - T ( v_s + v_o )

So, the second pulse will reach the observer after time –

t_2 = T + \left [ \frac { L - T ( v_s + v_o ) }{ ( v + v_o ) } \right ]

Thus, the time interval between two successive pulse received by the observer is –

T' = ( t_2 - t_1 )

= T + \left [ \frac { L - T \left ( v_s + v_o \right ) }{ ( v + v_o ) } \right ] - \left ( \frac {L}{v + v_o} \right )

= T \left [ 1 - \left ( \frac { v_s + v_o }{ v + v_o } \right ) \right ]

= T \left ( \frac { v - v_s }{ v + v_o } \right )

Hence, the apparent frequency of the sound felt by the observer is –

\nu' = \left ( \frac { 1 }{ T' } \right )

= \left ( \frac { 1 }{ T } \right ) \left ( \frac { v + v_o }{ v - v_s } \right )

= \nu \left ( \frac { v + v_o }{ v - v_s } \right ) ……… (5)

So, \quad \nu' > \nu \quad \text {because} \quad ( v + v_o ) > ( v - v_s ) .

*Hence, the frequency of sound appears to increases when the observer and source both moves towards each other.*

__Asymmetrical Doppler effect__

Doppler effect in case of sound wave is asymmetric in nature. From equations (1) and (2), we concluded that –

- If the observer is stationary and sound source is moving with a speed of ( v' ) towards the observer, then the apparent frequency is \left [ \nu' = \nu \left ( \frac {v}{v - v'} \right ) \right ]
- In case the sound source is stationary and observer moves with the same speed of ( v' ) towards the source, then the apparent frequency is \left [ \nu " = \nu \left ( \frac {v + v'}{v} \right ) \right ]
- So we notice that [ \nu' \ne \nu" ]

This is called asymmetric Doppler effect.

*For this reason the Doppler effect in sound is said to be asymmetric while Doppler effect in light is symmetric.*

__No Doppler effect__

It is noticed that there is no change in observed frequency by an observer if –

- Both the source and the observer move in the same direction with the same speed.
- Either the source or the observer is at centre of a circle and the other is moving along the circumference with
*uniform speed*. - Both source and observer are at rest and the medium moves ( i.e. when wind is blowing ).

*This is called “No Doppler Effect.”*

See numerical problems based on this article.