According to the Doppler effect, whenever there is a relative motion between a source of sound and the listener, the apparent frequency of the sound heard by the listener is different from the actual frequency of sound emitted by the source.

Let the source be S and the listener be L initially. Both are at rest initially. Let f be the frequency of sound emitted by the source and λ be the wavelength of the sound emitted. If v is the velocity of sound in still air, then λ = v/f . In one second, v waves from source reach the listener which is also the frequency.

Now let us consider a situation where the medium M, source S and the listener L move with velocities v_m , v_s and v_l respectively. As a result of motion of the medium, the sound waves in one second travel a distance v + v_m along SL.

Relative velocity of sound with respect to source = (v + v_m) – v_s . This is the distance covered by the sound wave in one second, relative to the source. The apparent wavelength λ₁ is [(v + v_m) – v_s ] / f  --------------Eqn (1)

The above waves with wavelength λ₁ , travel towards the listerner who has moved a distance LL₁ = v_l in one second. Hence, relative velocity of sound waves with respect to the listener is  [(v + v_m) – v_l ]

Apparent frequency of sound waves heard by the listener is f₁ = [(v + v_m) – v_l ] / λ₁   -----Eqn (2)

Substituting, Eqn (1) in Eqn (2),  f₁ =  [(v + v_m) – v_l ] /  [(v + v_m) – v_s ]  * f

This is the general Doppler effect equation. We can deduce formula for specific cases based on the above equation.

Consider a case of a stationary source. An observer moves towards the source. Hence, the distance between the source and the listener is decreasing. Hence, there will be an increase in the apparent frequency.

Here source is at rest and hence v_s = 0 and medium is stationary v_m = 0 and v_l is negative. ) Hence,  f₁ =  [v  – (-v_l )] /  v    * f = (v +  v_l ) /  v    * f