Obtain the equation of frequency observed by stationary observer and moving source.

Explanation

For this, we have to use below given sign convention.
Let us choose the convention to take the direction from the observer to the source as the positive direction of velocity.
Consider a source S moving with velocity v, and an observer who is stationary in a frame in which the medium is also at rest.
Let the speed of a wave of angular frequency `omega` and period `T_(0),` both measured by an observer at rest with respect to the medium be v.

As shown in figure, at time `t =0` the source is at point `S_(1),` located at a distance L from the observe and emits a crest. This reaches the observer at time `t _(1) = (L)/(v)" "...(1)`
At time `t = T_(0)1 `the source has moved a distance `v _(s) T_(0) and ` is at point `S _(2),` located at a distance `L+v _(2) T _(0)` from the observer.
At `S _(2),` the source emits a second crest. This reaches the observer at,
`t _(2) = T _(0) + (L+ v _(s) T _(0))/(v )`
At time `n T _(0)` the source emits its `(n+1) ^(th)` crest and this reaches the observer at time,
`t _(2) = T_(0) = n T_(0) + (L+ n v _(s) T _(0))/( v ) " "...(2)`
`t _(n+1) -t _(1) = ( n T _(0) + (L + n v _(s) T _(0))/( v )- (L )/(v))/( n )`
`[because` From equation (1) and (2)]
`= (n T _(0))/( n ) + (L + nv _(s) T _(0) - L)/( nv)`
`= (n T_(0))/( n ) + (nv _(s) T _(0))/( nv)`
`T =T_(0) [1 + (v _(s))/(v ) ] " "...(3)`
Equation (3) may be rewritten in terms of the frequency `v _(0)` thta would be measured if the source and observer ware stationary and the frequency v observed wien the source is moving as,
`v = v _(0) (1+(v _(s))/( v )) ^(-1) [ (1)/(T) = v ]`
If `v _(s)` is small complared with the wave speed v, taking b inomial expansion to terms in first order in `v _(s) //v` nd neglecting higher power may be approximated, giveng,
`v = v _(1) [1- (v _(s))/(v ) ] " "...(4)`
For a source approching the observer, we replace `v _(s) ` by `- v _(s)` to get,
`therefore v = v _(0) [1 + (v _(s))/( v ) ]" "...(5)`
The observer thus measures a lower frequency when the source recedes from him than he does when it is at rest,
He measures a higher frequency when the source approaches him.