Obtain the equatin of frequency observed by observer for moving source and moving observer at different velocities.

Explanation

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.