WAVES

Ans. Doppler shift : The difference between actual frequency and apparent frequency is known as Doppler shift .

Case(1):Consider Source of sound at rest and Observer $O$ approaching source of sound $S$ with avelocity' $V_{o}$ .Let the speed of sound in air. $v$ frequency $\nu_{o}$ and time period $T_{o}$ .As shown in the figure at the time $t=0$ source is at distance $L $ from the observer .

Therefore. At $t=0$ first crest reaches the observer at time $$t_{1}=\frac{L}{V+V_{o}}$$  $t=T_{o}$ second crest reaches the observer at time $$t_{2}=T_{o}+\biggl(\frac{L-V_{o}T_{o}}{V+V_{o}}\biggr )$$ $t=nT_{o}, \text{then}\left(\mathbf{n}+1\right)$ crest reachs the observer at tiime $$t_{n+ 1}=nT_{o}+\biggl(\frac {L- nV_{o}T_{o}}{V+ V_{o}}\biggr )$$ Time interval 

$$\begin{gathered}
 \:t_{_{n+1}}-t_{_1}=\:nT_{_o}+\biggl(\frac{L-\:nV_{_o}T_{_o}}{V+V_{_o}}\biggr )-\biggl(\frac{L}{V+V_{_o}}\biggr ) \\
nT\:=\:nT_{_o}\left(1\:-\:\frac{V_{_o}}{V+V_{_o}}\right) \\
T=T_{o}\left({\frac{V}{V+V_{_o}}}\right) \\
\mathrm{apparent\:\: frequency}\: \mathbf{\nu}={\frac{1}{T}}={\frac{1}{T_{o}\left({\frac{V}{V+V_{o}}}\right)}} \\
\nu\:=\:\nu_{o}\biggl(1\:+\:\frac{V_{o}}{V}\biggr) \\
\end{gathered}$$
The above equation represents the apparent frequency while an observer approaching the source of sound.

Case (2): when Observer moving away from the source of sound we replaced $V_{\circ}$ by $-\:V_{\circ}$in the above equation$$\nu^{\prime}\:=\:\nu_{o}\biggl(1\:-\:\frac{V_{o}}{V}\biggr)$$