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

Doppler Effect : The apparent change in frequency of sound of source due to relative motion between source and observes is known as Doppler effect.
Expression for apparent frequency when source is in motion and observer at rest.
Consider source ‘s’ and observer ‘o’ are separated by a distance ‘L’ let source releases sound of frequency ‘n’ and ‘V’ be the velocity of the sound.
Source moving towards observes with a velocity ‘V_s’ source releases 1^st crest at ‘s’. This crest was received by observer.

After a time ${\mathrm{t}}_1=\frac{\mathrm{L}}{\mathrm{V}}\dots \text{ (1) }\left(\text{ time }=\frac{\text{ dist }}{\mathrm{vel}.}\right)$
Source releases 2^nd crest after a time interval T (time period), during this time source covers a distance SS¹ (V_sT)
This 2^nd crest received by observer after a time
${\mathrm{t}}_2=\mathrm{T}+\frac{\mathrm{L}-{\mathrm{V}}_{\mathrm{S}}\mathrm{T}}{\mathrm{V}} ....\left(2\right)$
The time difference between ‘t₂’ and ‘t₁’ is apparent time period (w.r. to person)
$\mathrm{i}.\mathrm{e}., {\mathrm{T}}^1={\mathrm{t}}_2-{\mathrm{t}}_1$
${\mathrm{T}}^1=\mathrm{T}+\frac{\mathrm{L}-{\mathrm{V}}_{\mathrm{S}}\mathrm{T}}{\mathrm{V}}-\frac{\mathrm{L}}{\mathrm{V}}$
$\begin{array}{l}{\mathrm{T}}^1=\mathrm{T}+\frac{\mathrm{L}}{\mathrm{V}}-\frac{{\mathrm{V}}_{\mathrm{S}}\mathrm{T}}{\mathrm{V}}-\frac{\mathrm{L}}{\mathrm{V}}\\ {\mathrm{T}}^1=\mathrm{T}-\frac{{\mathrm{V}}_{\mathrm{S}}\mathrm{T}}{\mathrm{V}}\\ {\mathrm{T}}^1=\mathrm{T}\left(1-\frac{{\mathrm{V}}_{\mathrm{S}}}{\mathrm{V}}\right)\\ {\mathrm{T}}^1=\mathrm{T}\left(\frac{\mathrm{V}-{\mathrm{V}}_{\mathrm{S}}}{\mathrm{V}}\right)\end{array}$
$\therefore \text{ reciprocal of time period is frequency }$
${\mathrm{n}}^1=\left(\frac{\mathrm{V}}{\mathrm{V}-{\mathrm{V}}_{\mathrm{S}}}\right)\mathrm{n}$
Similarly if source moving away from observer. The apparent frequency is
${\mathrm{n}}^1=\left(\frac{\mathrm{V}}{\mathrm{V}+{\mathrm{V}}_{\mathrm{s}}}\right)\mathrm{n}$