Asource emitting a sound of frequency f0 is moving to the right while an observer is moving to the left. x vobserver vsource observer source with respect to the ground, let the speed of the source be vsource, the speed of the sound be vsound, and the speed of the observer be vobserver. find the detected frequency f ′ .

When a wave and/or the observer are moving, the frequency changes according to the Doppler Effect.

When the source is at rest and the observer moves away from it, the wave looks slower and its velocity will be: 
v' = vsound - vobserver
In this case,
f₁ = v'/λ = (vsound - vobserver) / λ = f₀ · (vsound - vobserver) / vsound
   = f₀ · (1 - vobserver/vsound)

On the contrary, when the source is moving away from an observer at rest, the wave appears with a different wavelengh: 
λ' = (vsound + vsource) / f₀
In this case,
f₂ = v/λ' = vsound / ((vsound + vsource) / f₀) 
   = f₀ · 

Now, let's put together these two cases:
f' = v' / λ' =
   =  
   = f₀ ·
   = f₀ ·