What is Doppler shift ? Obtain an expression for the apparent frequency of sound heard when the observer is in motion with respect a source at rest ?

Doppler Shift: The difference between apparent frequency heard by the observer and actual frequency produced by the sources is called Doppler shift.
If source is at rest observe is moving :
Consider source observer are separated by distance ‘L’
Let source emits sound of frequency ‘n’, ‘V’ be the velocity of sound.
If observer is moving towards source

Let source releases 1^st crest at ‘S’. This 1^st crest was received by observer after a time
${\mathrm{t}}_1=\frac{\mathrm{L}}{\mathrm{V}+{\mathrm{V}}_0} ......\left(1\right)$
Where ‘V+V₀’ is relative velocity of sound w.r.t observer.
2^nd crest is released by source after a time interval T (Time period). During this time observer covers a distance (OO^|) which is V_ot
This 2^nd crest was received by observer at a time
${\mathrm{t}}_2=\mathrm{T}+\frac{\mathrm{L}-{\mathrm{V}}_0\mathrm{T}}{\mathrm{V}+{\mathrm{V}}_0}$
Time difference between ‘t₂’ and ‘t₁’ is apparent time period w.r.t observer
$\begin{array}{l}{\mathrm{T}}^1={\mathrm{t}}_2-{\mathrm{t}}_1=\mathrm{T}+\frac{\mathrm{L}-{\mathrm{V}}_0\mathrm{T}}{\mathrm{V}+{\mathrm{V}}_0}-\frac{\mathrm{L}}{\mathrm{V}+{\mathrm{V}}_0}\\ {\mathrm{T}}^1=\mathrm{T}+\frac{\mathrm{L}}{\mathrm{V}+{\mathrm{V}}_0}-\frac{{\mathrm{V}}_0\mathrm{T}}{\mathrm{V}+{\mathrm{V}}_0}-\frac{\mathrm{L}}{\mathrm{V}+{\mathrm{V}}_0}\\ {\mathrm{T}}^1=\mathrm{T}-\frac{{\mathrm{V}}_0\mathrm{T}}{\mathrm{V}+{\mathrm{V}}_0}\\ {\mathrm{T}}^1=\mathrm{T}\left[1-\frac{{\mathrm{V}}_0}{\mathrm{V}+{\mathrm{V}}_0}\right]\\ {\mathrm{T}}^1=\mathrm{T}\left[\frac{\mathrm{V}+{\mathrm{V}}_0-{\mathrm{V}}_0}{\mathrm{V}+{\mathrm{V}}_0}\right]\\ {\mathrm{T}}^1=\mathrm{T}\left[\frac{\mathrm{V}}{\mathrm{V}+{\mathrm{V}}_0}\right]\end{array}$
Reciprocal of time period is frequency
${\mathrm{n}}^1=\mathrm{n}\left(\frac{\mathrm{V}+{\mathrm{V}}_0}{\mathrm{V}}\right)$
Similarly moving away ${\mathrm{n}}^1=\mathrm{n}\left(\frac{\mathrm{V}-{\mathrm{V}}_0}{\mathrm{V}}\right)$