Tuesday, February 26, 2019
Doppler Effect in sound
1. If an observer is situated at a fixed distance from a sound-source, the frequency of sound heard by him is the same as produced by the source. But if the sound-source, or observer, or both, are in a state of motion; then the frequency of the sound appears to be changed to the observer. This phenomenon of the apparent change in the frequency of the source due to a relative motion between the source and the observer is called as Doppler’s effect.
2. It is observed that frequency of the sound appears to be increased, when the source of the sound and the observer are approaching each other but appears to be decreased, when the two move away from each other. For example, to an observer standing on a railway platform, the sound of the whistle of an engine appears shrill or of high frequency, when the engine is approaching him. But as soon as the engine crosses him and starts receding, the sound of the whistle appears suddenly grave or of low frequency.
3. It should be noted here that frequency of the sound always appear to increase, when source of sound moves towards a stationary observer or an observer moves towards a stationary source of sound are both move towards each other. On the other hand, the frequency of sound is found to decrease, when source of sound or both move away from each other.
4. We shall study Doppler’s effect in three parts:
A) Effect of motion of sound-source alone
B) Effect of motion of the observer alone
C) Effect of motion of of both together
A) When sound–source is in motion and observer is at rest:
Suppose S and O be the positions of the sound-source and the observer respectively. Let n be the actual frequency of the source and v the velocity of sound. Then n waves will be emitted from the source in one second which will travel with velocity v. If the source is stationary then these n waves will spread in the distance SO, where SO= v [fig (1a)]. To the observer O the wavelength of the waves will be λ= v/n.
a) When the source moves towards stationary observer: If the source S is in motion with velocity vs towards the stationary observer, then after 1 sec, the source will reach the point S’, such that SS’ = vs.Then the n waves emitted by the source in 1 sec will now spread in a distance (v-vs) only, because in 1 sec the other source itself moves a distance vs towards the observer [fig(1b)]. Therefore, to the observer, the apparent wave length of the sound will be
λ’ =(v-vs)/n
if n’ is the apparent frequency, then
n’=v/ λ’=v/[(v-vs)/n]=[(v/(v-vs)] × n
Thus we find that n’ is greater than n i.e., Pitch (or frequency) of the sound appears to increase, when the source of sound cannot be greater than the velocity of sound otherwise n’ will become negative.
b) When source moves from the stationary observer:
If the source moves away from the stationary observer with velocity v, then apparent frequency of the sound can be obtained by simply replacing vs by –vs. Then apparent frequency will be given by
n’=[v/{v-(-vs)}]n=[v/(v+vs)]n
Thus now n’ is less than n or pitch of the sound appears to decrease, when the source of sound moves away from the stationary observer.
B) When sound source is at rest and the observer is in motion:
In the adjoining figure 1, S and O again represent the positions of source and observer respectively. The source S is emitting n number of waves per second, each heaving wavelength λ = v/n. Of the n waves crossing the ear of the observer in 1 sec, the wave emitted right in the beginning will reach some point A; while that emitted right in the last, will just reach at the point O where observer is situated. Thus distance OA contains n waves which cross the ear of the observer in 1 .
a) When observer moves towards the source: Suppose the observer is moving towards the stationary source with velocity v0. After 1 second it will reach the point O’ such that OO’ =v0.Because of this motion, the number of waves crossing the ear of observer will be waves contained in distance OA plus the number of waves contained in distance OO’ i.e., v0/λ. Therefore, apparent frequency of sound will be
n’ = n+(v0/λ)=n+[v0/(v/n)]=n+(nv0/v)
n’ = [(v+v0)/v)n]....(3)
Thus, n’ is greater than n i.e., pitch of sound appears to increase, when the listener moves towards the stationary source.
b) When observer moves away from the source: In this case the number of waves crossing the ear of the observer in one second will be less than n. However, the apparent frequency can be easily determined by simply replacing v0 by –v0 in equation (3). Therefore, apparent frequency of the sound, when observer is moving away from the source is given by
n’ = [{v+(-v)}/v]n=[(v-v0)/v]n.........(4)
Now, n’ is less than n i.e., frequency of sound appears to decrease, when the observer moves away from the source. It may be noted here again that velocity of the observer cannot be greater than the velocity of sound otherwise n’ will become negative.
C) When source and observer both are in motion:
From the above discussion, it follows that
i) When source is in motion and observer is stationary, Doppler effect is due to change in the size of the wavelengths of waves and
ii) When source is at rest and observer is in motion, Doppler Effect is due to change in the number of waves crossing the ear of the observer.
We shall now use above conclusions to derive the expressions for apparent frequency of sound in different cases of both the source and observer in motion.
a) When both the source and observer move towards each other:
Suppose n’ is the apparent frequency, when source alone is in motion with velocity vs towards the observer (which is stationary). Then, from equation (1), we get:
n’ =[v/(v-v0)]n......(5)
where n is the actual frequency and v, the velocity of sound.
Now, suppose observer is also moving with velocity v0 towards the source, which appears to be emitting waves of frequency n’. Due to motion of the observer, the frequency will change from n’ to n’’, which according to equation (3) is given by
n’’=[(v+v0)/v]n’........(6)
From equations (5) and (6), we get
n’’ = [(v+v0)/v]× [v/(v-vs)]n=[(v+v0)/(v-vs)]n ....(7)
Above equation shows that
i) The apparent frequency will be greater than the actual frequency.
ii) The observer may move with velocity greater than that the velocity of sound but the velocity of source cannot be greater than velocity of sound otherwise n’ will become negative which is not possible.
b) When the source moves towards the observer and observer moves away from the source : In this case apparent frequency can be obtained by
replacing v0 by in equation (7). Thus, apparent frequency is given by
n’’ = [{v+(-v0)}/(v-vs)]n=[(v-v0)/(v-vs)]n...........(8)
In such a case, n’’ may be greater or less than n depending upon v0 is less than or greater vs. Further, in such a case Doppler Effect will be observable only if either both v0 and vs are less than v or both greater than v.
Similarly, expression for apparent frequency of the sound, when both the source and the observer move away from each other or when the source moves away from the observer but the observer moves towards the source can be obtained by changing signs of v0 or vs or both in equation (7) and (8).