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

(N/A) When the observer moves with velocity $v_{0}$ towards a stationary source,we analyze the situation in the reference frame of the moving observer. In this frame,the source and the medium approach the observer at speed $v_{0}$,and the speed at which the wave crests approach the observer is $v + v_{0}$.
The time interval between the arrival of the first and the $(n+1)^{\text{th}}$ crests is given by:
$t_{n+1} - t_{n} = n T_{0} - \frac{n v_{0} T_{0}}{v + v_{0}}$
The observer measures the period of the wave as:
$T = \frac{t_{n+1} - t_{1}}{n} = T_{0} \left( 1 - \frac{v_{0}}{v + v_{0}} \right) = T_{0} \left( \frac{v}{v + v_{0}} \right)$
Since frequency $\nu = \frac{1}{T}$ and original frequency $\nu_{0} = \frac{1}{T_{0}}$,the observed frequency is:
$\nu = \nu_{0} \left( \frac{v + v_{0}}{v} \right) = \nu_{0} \left( 1 + \frac{v_{0}}{v} \right)$
When the observer moves away from the stationary source,we replace $v_{0}$ with $-v_{0}$ in the equation:
$\nu = \nu_{0} \left( 1 - \frac{v_{0}}{v} \right)$