Question

please provide me the derivation of doppler 's effect when source is stationary and observer is moving.

Answer 1

Let S and O denote the initial positions of a source of sound and an observer. For the sake of simplicity, we shall assume that the source, the observer and the medium are all moving along the positive direction. Let the velocity of sound in still air be = v The velocity of the source = a The velocity of the observer = b and the velocity of the medium (wind blowing) = w Let Sl and Ol represent the positions of the source and the observer after 1 second. Distance travelled in 1 second is nothing but the velocity. The waves produced by the source travel a distance SA in 1 second, but as the wind is blowing with a velocity w, it carries the wavefront from A to Al where A Al= w. The distance travelled by the waves relative to the source in 1 second. If f is the frequency of the waves produced by the source, then f waves are accommodated in a distance SlAl. Since the observer recedes by a distance b in 1 second, the relative velocity of the waves with respect to the observer is (v + w - b). Therefore, the apparent frequency is given by the number of waves of wavelength ll contained within the above distance. Substituting for ll from equation (i), we get This is the general expression for the apparent frequency of the sound when the source of sound, observer and the medium are in motion, in the same direction. Discussion of equation (1-38) for particular casesBack to Top Case (i)Back to Top Source moving towards a stationary observerBack to Top Let us assume that the wind velocity is zero. Then w = 0 and b = 0. Equation (1-38) becomes The apparent frequency will be greater than the actual frequency. Case (ii)Back to Top Source moving away from a stationary observerBack to Top Assuming the wind velocity to be zero, putting -a in the place of a and b = 0 in equation (1-38), we get The apparent frequency will be lesser than the actual frequency. Case (iii)Back to Top Observer moving away from a stationary sourceBack to Top Putting w = 0 and a = 0 in equation (1-38), we get The apparent frequency will be lesser than the actual frequency. Case (iv)Back to Top Observer moving towards a stationary sourceBack to Top Putting w = 0, a = 0 and -b in the place of b in equation (1-38), we get The apparent frequency will be greater than the actual frequency. Case (v)Back to Top Observer and source moving in the same direction as sound in a stationary mediumBack to Top Putting w = 0, in equation (1-38), we get If b < a, then (v - b) > (v - a) and fl > f. If b > a, then (v - b) < (v - a) and fl < f. If b = a, then fl = f. Thus, for a passenger sitting in a train the frequency of the whistle of the train appears to be the same, when the train moves, as when it was at rest. Case (vi)Back to Top Observer and source moving towards each other in a stationary mediumBack to Top Now the velocity of the observer is opposite to the velocity of sound, while that of the source is the same as that of sound. 'b' is to be replaced by -b in equation (1-38). Case (vii)Back to Top Observer and source moving away from each other in a stationary mediumBack to Top Considering the direction of the velocity of sound reaching the observer as positive, a is negative and b is positive. Then The apparent frequency is less than the actual frequency. Case (viii)Back to Top Wind blowing opposite to the direction of the velocity of soundBack to Top In this case, w is to be replaced by -w in equation (1-38)