Saved Bookmarks
| 1. |
Obtain the equatin of frequency observed by observer for moving source and moving observer at different velocities. |
|
Answer» Solution :Let us take the observer to the source as the positive direction. Let the source and the observer be moving with velocities `v _(s) and v _(0)` respectively as shown in figure.  Suppose at time `t =0,` the observer is at `O_(1)` and the source is at `S_(1), O _(1)` being to the left of `S_(1).` The source EMITS a wave of velocity v of frequency v and period `T_(0)` all measureed by an observer at rest with respect to the medium. Let L be the distance between `O_(1) and S_(1)` at `t=0,` when the source emits the first crest. Now, since the observer is moving, the velocity of the wave relative to the observer is `v+v_(0).` Therefore, the first crest reaches the observer at time `t _(1) =(L)/(v + v _(0)).` At time `t = T_(0),` both the ovserver and the source have MOVED to their new positions `O_(2) and S_(2)` respecitvely. THe new DISTNACE between the observer and the source `O _(2) S _(2)` would be `L + (v _(s) -v _(0)) T _(0).` At `S_(2)` the source emits a second crest. THis reaches the observer at time . `t _(2) =T_(0) +([ L + (v _(s) - v _(0)) T _(0)])/((v + v _(0)))` At time `n T _(0)` the source emits its `(n +1) ^(th)` crest and this reaches the observer at time, `t _(n +1) = nT _(0) + (L + n (v _(s) -v _(0)) T _(0))/(v + v _(0))` Hence, in a time intervel, `t _(n +1) -t _(1) = n T _(0) + (L + n (v _(s - v _(0)) T _(0)))/(v +v_(0)) - (L)/(v + v _(0)) ,` the observer counts n crests. The observer records the period of the wave as equal to T given by, ` T = (t _(n)+1_ -t _(1))/(n)` `therefore = n T _(0) + (n (v _(s -v _(0)) T _(0))/(v + v _(0)))/(n )` ` therefore T = T _(0) + ((v _(s) -v _(0)) T _(0))/(v + v _(0))` `therefore T = T _(0) [1 + (v _(s) -v _(0))/(v + v _(0))]` `thereore T= T_(0) [1 + (v _(s) -v _(0))/(v + v _(0)) ]= T _(0) ((v + v _(s))/( v+ v _(0)))` The frequency v observed by the observer is given by, `V = 1/T` `therefore v =v _(0) ((v + v _(s))/( v + v _(0))) ^(-1)` ` therefore v =v _(0) ((v + v _(0))/(v +v _(s)))` This is general equation of DOPPLER effect. |
|