The Doppler Shift Formula

Let us make the effect quantitative, for a sound wave of speed v (the speed of sound), a source of emitted frequency f, and a source moving at speed vₛ along the source-observer line.

When the source approaches, the fronts get squeezed: the perceived frequency f' increases.

   approaching source :   f' = f · v / (v − vₛ)      (f' > f, higher pitch)
   receding source    :   f' = f · v / (v + vₛ)      (f' < f, lower pitch)

The sign in the denominator is the only difficulty: minus when the source approaches (the denominator decreases, so f' increases), plus when it moves away. A good reflex: approach → higher pitch → larger f', consistent with a smaller denominator.

We often reason with the shift Δf = f' − f. For speeds small compared with v, it is almost proportional to the speed:

   Δf / f ≈ vₛ / v          (low-speed approximation)

In other words: measuring the frequency shift means measuring a speed. That is the whole power of the Doppler effect: it turns a frequency observation (easy) into a speed measurement (valuable), without contact.

   measured Δf  ->  vₛ = v · Δf / f      (we work back to the speed of the source)

A traffic speed radar applies exactly this: it sends out a wave, measures the shift of the wave reflected by the car, and deduces its speed.