The speed of the vehicle which acts as the source of the reflected wave determines the change in frequency, which can be detected with the box.
Imagine the sound a race car makes as it rushes by, whining high pitched and then suddenly lower. The spectrum of light can tell us what elements are in the stars in the galaxies because each element has unique energy levels and therefore emits or absorbs light at particular wavelengths.
As the sound travels towards you, the ambulance is also moving towards you, albeit at a much slower pace. Similar applications can be used to measure wind velocities in the atmosphere, which is the " Doppler radar " of which meteorologists are so fond. Not understanding this point has led to confusion and misunderstanding.
The circles represent the sound waves fronts emitted from the siren. Both these formulas are non-relativistic approximations that are true as long as the velocity of the moving object is much less than the speed of light. It is much easier, almost trivial, to establish the lack of Doppler shift between emitter and absorber in the laboratory frame.