Exploring Doppler's Effect: How Sound Waves Change Frequency

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What is Doppler’s Effect?

When there is a relative motion between the source of a sound and observer, the frequency or pitch of the sound received by the observer is different than the actual frequency or pitch. This phenomenon is called Doppler’s effect.

The apparent change in pitch or the frequency of sound due to relative motion between source of sound and observer is called Doppler’s effect. To study the Doppler’s effect, we consider the following cases.

A. Source in motion and observer at rest:

a. Source moving towards the observer:

Consider a source of sound is emitting sound wave of frequency, $f$ . If $v$ be the velocity of sound and the source is at rest, the $f$ waves produced in 1s occupy a distance, $v$ . Then, the wavelength of sound wave is given by,

\lambda = \frac {\text {distance occupied by waves}}{\text {number of waves}}

\therefore \space \lambda = \frac vf

Suppose the source of sound moves towards the observer with velocity, $u\scriptsize s$ . Then, out of $f$ waves produced in 1s, the first wave is produced from position, S’. i.e., the $f$ waves produced in 1s now occupy a distance, $(v-u\scriptsize s)$ . This causes apparent decrease in wavelength. The apparent wavelength is given by,

\lambda' = \frac {\text {distance occupied by waves}}{\text {number of waves}}

\therefore\space\lambda = \frac {v - u\scriptsize s}f

The velocity of sound remains unaffected. The apparent change in frequency due to apparent change in wavelength is given by,

f' = \frac {v}{\lambda'}

\therefore\space f' = \frac v{v-u\scriptsize s}\times f

Thus, when the source of sound moves towards an observer, the apparent frequency or pitch of the sound increases. This is the reason why whistle of an approaching train is shriller.

b. Source moving away from the observer:

Consider a source of sound emitting sound wave of frequency, $f$ . If $v$ be the velocity of sound and the source is at rest, the $f$ waves produced in 1s occupy a distance, $v$ . Then, the wavelength of sound waves is given by,

\lambda = \frac {\text{Distance occupied by waves}}{\text{Number of waves}}

\therefore\space\lambda = \frac vf

Suppose the source of sound moves away from the observer with velocity, $u\scriptsize s$ ,then out of $f$ waves produced in 1s, the first wave is produced from position ‘s’ and last wave is produced from position S’. i.e., the $f$ waves produced in 1s occupy a distance, $(v + u\scriptsize s)$ . This causes apparent increase in wavelength, which is given by,

\lambda' = \frac {\text{Distance occupied by waves}}{\text{Number of waves}}

\therefore\space\lambda' = \frac {v + u\scriptsize s}f

The velocity of sound remains unaffected. The apparent frequency due to apparent change in wavelength is given by,

f' = \frac v{\lambda'}

\Rightarrow\space f' = \frac v{v+u\scriptsize s} \times f

\therefore\space f' \lt f

Thus, when the source of sound moves away from an observer, the apparent frequency or pitch of the sound decreases.

B. Observer in motion and source at rest:

a. Observer moving towards the source:

Consider a source of sound emitting sound waves of frequency, $f$ . Let $v$ be the velocity of sound and the source and observer both are at rest, then in 1s the observer can receive the waves lying within the distance, $v$ . i.e., the frequency of sound wave received by the observer is given by,

f = \frac {\text{distance occupied by waves}}{\text{length of one wave}}

\Rightarrow\space f = \frac f\lambda

\therefore\space\lambda = \frac vf

Suppose the observer moves towards the source with velocity, $u\scriptsize o$ . Then in 1s, the observer can receive the waves lying in the distance, $(v + u\scriptsize o)$ . i.e., the apparent frequency of the sound received by the observer increases but wavelength remains same. The apparent frequency of sound received by the observer is given by,

f' = \frac {\text{distance occupied by waves}}{\text{length of one wave}}

\therefore\space\lambda = \frac {v+u\scriptsize o}\lambda

Thus, when the observer moves towards the source; the relative velocity of sound to the observer increases. From above equations,

f' = \frac {v+u\scriptsize o}v \times f

\therefore\space f' \gt f

Thus, when the observer moves towards the stationary source of sound; the apparent frequency or pitch of the sound increases.

b. Observer moving away from the source:

Consider a source of sound emitting sound waves of frequency, $f$ . Let $v$ be the velocity of sound and both source and observer are at rest. In one second, the observer can receive the waves lying within the distance, $v$ . i.e., the frequency of sound wave received by the observer is given by,

f = \frac {\text{distance occupied by waves}}{\text{length of one wave}}

\Rightarrow\space f = \frac v\lambda

Suppose the observer moves away from the source with velocity, $u\scriptsize o$ . Then in 1s, the observer can receive only the waves lying within the distance, $(v-u\scriptsize o)$ . i.e., the apparent frequency of the sound received by the observer decreases but wavelength remains same. The apparent frequency of sound received by the observer is given by,

f' = \frac {\text{distance occupied by waves}}{\text{length of one wave}}

\therefore\space f' = \frac {v-u\scriptsize o}\lambda

Thus, when the observer moves away from the source; the relative velocity of sound to the observer decreases. Using above equations,

f' = \frac {v-u\scriptsize o}v \times f

\therefore\space f' \lt f

Thus, when the observer moves away from the source of sound, the apparent frequency or pitch of the sound decreases.

C. Source and observer both are in motion:

From above discussion, it is clear that the motion of source effects wavelength and motion of observer effects relative velocity of sound. So, when both source and observer are in motion, the apparent frequency of wave can be expressed as,

f' = \frac {\text{relative velocity of sound}}{\text{apparent wavelength}}

\therefore\space f' = \frac {v'}{\lambda'}

a. Source and observer approaching each other:

When, the source moves towards the observer the apparent wavelength of sound decreases.

i.e.\space \lambda' = \frac {v-u\scriptsize s}f

Also, when the observer moves towards the source, the relative velocity of sound to the observer increases, which is given by the relation, $v' = v + u\scriptsize o$ . Then, the apparent frequency received by the observer is given by,

f' = \frac {v'}{\lambda'}

\therefore\space f' = \frac {v + u\scriptsize o}{v-u\scriptsize s}\times f

b. Source and observer receding each other:

When the source of sound moves away from the observer, the apparent wavelength of sound increases, which is given by the relation,

\lambda' = \frac {v+u\scriptsize s}f

Also, when the observer moves away from the source; the relative velocity of sound to the observer decreases. This is given by the relation, $v' = u - u\scriptsize o$ . Then, the apparent frequency received by the observer is given by,

f' = \frac {v'}{\lambda'}

\therefore\space f' = \frac {v - u\scriptsize o}{v+u\scriptsize s}\times f

c. Source leading the observer:

When the source of sound moves away from the observer the apparent wavelength of sound increases, which is given by,

\lambda' = \frac {v+u\scriptsize s}f

When the observer moves towards source, the relative velocity of sound to the observer increases which is given by, $v' = v + u\scriptsize o$ . Then, the apparent frequency received by the observer is given by,

f' = \frac {v'}{\lambda'}

\therefore\space f' = \frac {v + u\scriptsize o}{v+u\scriptsize s}\times f

d. Observer leading the source:

When the source of sound moves towards the observer, the apparent wavelength of the sound decreases.

\lambda' = \frac {v-u\scriptsize s}f

When the observer moves away from the source, the relative velocity of sound to the observer decreases, which is given by the relation, $v' = v - u\scriptsize o$ . Then, the apparent frequency received by the observer is given by,

f' = \frac {v'}{\lambda'}

\therefore\space f' = \frac {v - u\scriptsize o}{v-u\scriptsize s}\times f

Doppler’s Effect summarization

\therefore\space f' = \frac {v \pm u\scriptsize o}{v\mp u \scriptsize s}\times f

Where, the first sign stands for the motion towards the other and the second sign stands for the motion away from the other.

Doppler’s red shift:

Experimentally, it has been found that the wavelength of light received on the earth from a distant star is greater than the actual wavelength. For the apparent wavelength to increases (to decrease apparent frequency) the source of light i.e., star must be moving away from us. This gives the idea that the universe is expanding. The apparent increase in wavelength i.e., shifting of wavelength towards red colour due to the motion of stars is called red shift.

Limitations of Doppler’s effect:

The Doppler’s effect is not observed if the relative velocity of source of sound or observer is equal or greater than the velocity of sound.

It is not observed when source and observer are moving with same velocity in same direction.

It is not observed when the source or observer is moving in the direction perpendicular to the direction of propagation of sound.

Points to remember:

When a tuning fork is loaded (filled) its frequency decreases.

When the pongs of a tuning fork are filed (cut) to smaller lengths, its frequency increases.

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