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What is meant by Doppler effect ? Discuss the following cases (1) Source in motion and Observer at rest (b) Source moves away from the observer (2) Observer in motion and Source at rest . (a) Observer moves towards Source (b) Observer resides away from the Source (3) Both are in motion (a) Source and Observer approach each other (b) Sources and Observer resides from each other (c)Source chases Observer (d) Observer chases Source |
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Answer» Solution :Doppler Effect: When the source and the observer are in relative motion with respect to each other and to the medium in which sound propagates, the frequency of the sound wave observed is different from the frequency of the source. This phenomenon is called Doppler Effect. 1. Source in motion and the observer at rest  (a) 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)` . We assume 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 similarly for `x_(3)` and `C_3`. Assume that if `C_1` reaches the observer.s position A then at that instant `C_2` reaches the point B and `C_3` reaches the point C as shown in the figure. It is obvious to see that the DISTANCE between compressions `C_2` and `C_3` is shorter than distance between `C_1` and `C_2` This means the wavelength decreases when the source S moves towards the observer O (since sound travels longitudinally and wavelength is the distance between two consecutive compressions). But frequency is inversely related to wavelength and therefore, frequency increases. Let `lambda` be the wavelength of the source S as measured by the observer when S is at position `x_1` and `lambda` be wavelength of the source observed 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 - v_(s) t "" .... (1) ` But `t = (lambda)/(v)"" ...... (2)` On substituting equation (2) in equation (1) , we get`lambda. = lambda (1 - (v_(s))/(v))` Since frequency is inversely proportional 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` , we use the binomial expansion and retaining only first order in `(v_(s))/(v)` , we get `f. = f (1 + (v_(s))/(v)) v "" .... (4)` (b) Source moves away from the observer: Since the velocity here of the source is opposite in direction when compared to case (a), therefore, 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)` 2. Observer in motion and source at rest: a) Observer moves towards Source:Let us assume that the observer O V moves towards the source S with velocity `v_(o)` .The source S is at rest and the velocity of sound waves (with respect to the medium) produced by the source is v. From the figure, we observe that both `v_(o)`and v are in opposite direction. Then, their relative velocity is `v_(r) = v + v_(o)` .The wavelength of the sound wave is `lambda = (v)/(f)` , which means the frequency observed by the observer O is `f. = (v_(r))/(lambda)` . Then `f. = (v_(r))/(lambda) = ((v + v_(o))/(v)) f= f ( 1+(v_(o))/(v)) "" .... (7)` (b)Observer recedes away from the Source: If the observer O is moving away (receding away) from the source S, then velocity `v_(o)`and v moves in the same direction. Therefore, their relative velocity is `v_(r) = v - v_(0)` Hence, the frequency observed by the observer O is `f. = (v_(r))/(lambda) = ((v-v_(0))/(v)) f = f (1 - (v_(0))/(v))` (3) Both are in motion : (a) 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 observer, as a simple calculation, let us have a dummy (behaving as observer or source) in between the source and observer. Since the dummy is at rest, the dummy (observer) observes the apparent frequency due to approaching source as given in equation (3) as `f_(d) = (f)/(1 - (v_(s))/(v)) "" .... (9)` At that instant of time, the true observer approaches the dummy from the other side. 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), from equation (7) is `f. = f_(d) (1 + (v_(0))/(v)) implies f_(d) = (f.)/((1 +(v_(0))/(v))) ""..... (10)` Since this is true for any arbitrary time , therefore , comparing equation (9) and equation (10) , we get `(f)/(( 1 - (v_(s))/(v))) = (f.)/((1 + (v_(0))/(v))) implies (vf.)/((v + v_(0))) = (vf)/((v- v_(s)))` Hence , the apparent frequency as seen by the observer is `f. = ((vv )/(vv)) f` (b) Source and observer recede from each other :  Here , we can derive the result as in the PREVIOUS case . Instead of a detailed calculation , by inspection from figure , we notice 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 (11) , and therefore , the apparent frequency observed by the observer when the source and observer recede from each other is `f. = ((v - v_(0))/(v + v_(s))) f "" .... (12)` (c) Source chases the observer  Only the observer.s velocity is oppositely directed when compared to case (a) . Therefore , substituting (`v_(0) to - v_(0))` in equation (11) , we get `f. = ((v- v_(0))/(v + v_(s)))f "" .... (13)` (d) Observer chases the source Only the source velocity is oppositely directed when compared to case (a) . Therefore , substituting `v_(s) to - v_(s)` in equation (12) , we get `f. = ((v+ v_(0))/(v + v_(s))) f "" .... (14)` |
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