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What is meant by Doppler effect ? Discuss the following case. (i) Source in motion and Observer at rest (a) Source moves towards observer (b) Source moves away from the observer (ii)Observer in motion and Source at rest. (a)bserver moves towards Source (b) Observer resides away from the Source (iii) Both are in motion (a) Source and Observer approach each other (b) Source and Observer resides from each other (c) Sourcechases Observer (d) Observer chases Source |
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Answer» Solution :When the source and the observer are in relative motion with respect to each other and to the medium in which sound in propagated, the frequency of the sound waves observed is different from the frequency of the source. This phenomeon is called Doppler Effect. Source in motion and the observer at rest. Source MOVES TOWARDS the observer.  Suppose a source S moves to the right (as shown in figure) with a velocity `v_(s)` and let the frequency of the sound waves produced by the source be `f_(s)`. it is assumed that the velocity of sound in a medium is v. The compression (sound wave front) produced by the source S at three successive instants of time are shown in the figure. When S is at position `x_(1)` the compression is at `C_(1)`. When s is at position `x_(2)` the compression is at `C_(2)` and similary for `C_(3) and C_(2)` . It is assumed that if `C_(1)` reaches the observer's position A then at the instant `C_(2)` reaches the point B and `C_(3)` reaches the point C as shown in the figure. Obvisouly it is seen that the distance between compresions, `C_(2) and C_(3)` is shorter than distance between `C_(1) and C_(2)` . It is meant by the wavelength decreases when the source S moves towards related to wavelength and therefore, frequency increases. Calculation : Let `LAMBDA` be the wavelength of the souce s as measured by the observer when S is at position `x_(1) and lambda` be wavelength of the source boserved by the observer when S moves to position `x_(2)`. Then the change in wavelength is `Delta lambda= lambda- lambda =v_(s)` t, where t is the time taken by the source to travel between `x_(1) and x_(2)`. Therefore, `lambda= lambda-vt ""....(1)` But `t=(lambda)/(v) ""...(2)` On substituting equations (2) in equation (1), we get. `lambda'= lambda (1-(v_(s))/(v))` Since frequency is inversely propprtional to wavelength, we have `f'=(v_(s))/(lambda') and f(v_(s))/(lambda)` Hence `f'=(f)/((1-(v_(s))/(v)))""...(3)` Since `(v_(s))/(v) lt lt 1`, by using the binomial expansion and retaining only first order in `(v_(s))/(v)` we get `f'=f(1+(v_(s))/(v)) ""....(4)` Source in motion and the observer at rest. Source moves away from the observer. Since the velocity of the source is opposite in DIRECTION when compared to case(a), hence by changing the sign of the velocity of the source in the above case i.,e by substituting `(v_(s) to -v_(s))` in equation (1), we get `f'=(f)/((1+(v_(s))/(v))) ""...(5)` Using binomial expansion again, we get `f'=f(1-(v_(s))/(v)) ""...(6)` Observer in motion and source at rest : Observer moves towards Source :  We can assume that the obsever O moves towards the source S with velocity, `v_(0)`. The source S is at rest and the velocity of sound wave (with respect to the medium) produced by the source is v. From the Figure, it is observed that both `v_(0) and v` arein opposite direction. Then, their relative velocity is `v_(r)=v+v_(0)`. The wavelenght of the sound is `v_(r)=v+v_(0)`. The wavelength of the sound wve is `lambda=(v)/(f)` which means the frequency observed by the observerO is `f'=(v_(r))/(lambda)`. Then `f'=(v_(r))/(lambda)=((v+v_(0))/(v))f` `=f(1+(v_(0))/(v))""...(7)` Observer in motion and source at rest : Observer receds away from the Source : If the observer O is moving away (receding away) from he source s, then velocity `v_(0)` andv moves in the same direction. Hence. their relative velocity is `v_(r)=v-v_(0)` Hence the frequency observerd by the observer O is `f'=(v_(r))/(lambda)=((v-v_(0))/(v))f` `=f(1-(v_(0))/(v)) ""...(8)` Both are in motion. : Both are in motion. : Source and observer approach each other  Let `v_(s) and v_(0)` be the respective velocities of source and observer approaching each other as shown in Figure. In order to calculate the apparent frequency observed by the obsever, let us have a dummy (behaving as observer or source) in between the source and observer. Since the dummy is at rest, dummy (observer) observes the apparent frequency due to approaching soure as given in equation `f'(f)/((1-(v_(s))/(v))` as. `f_(d)=(f)/((1-(v_(s))/(v))) ""....(1)` The true observer approaches the dummy from the other side at that instant of time. Since the source (true source) comes in a direction opposite to true observer, the dummy (source) is treated as stationary source for the true observer at that instant. Hence, apparent frequency when the true observer approaches the stationary source (dummy source), `f=f(1+(v_(0))/(v))` `f'=f_(d)(1+(v_(0))/(v))` `rArr f_(d)=(f)/((1+(v_(0))/(v))) ""....(2)` Since this is true any arbiitrary time, therefore, comparing equation (1) and equation (2), we get `(f)/((1-(v_(s))/(v)))=(f')/((1+(v_(0))/(v)))` `rArr(vf')/((v+v_(0)))=(vf)/((v-v_(s)))` Hence the apparent frequency as seen by the observer is `f'=((v+v_(0))/(v-v_(0)))f""...(3)` Both are in motion. : Source and observr recede from each other :  It is NOTICED that the velocity of the source and the observer each point in opposite directions with respect to the case in (a) and hence, we substitute `(v_(s) to -v_(s)) and (v_(0) to -v_(0))` in equation, (3), and therefore, the apparent frequency observed by the observer when the source and observer reced from each other is `f'=((v-v_(0))/(v+v_(s)))f` Both are in motion. : Source chases the obserer :  Only the observer's velocity is oppositely directed when compared to case (a) . Therefore, substituting `(v_(0) to -v_(0))` in equation (3), we get `f'=((v-v_(0))/(v-v_(s)))f` Observer chases the source :  Only the source velocity is oppositely directedwhen compared to case (a). therefore substituting `(v_(s) to -v_(s))` in equation (3), we get `f'=((v+v_(0))/(v+v_(s)))f` |
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