Doppler Effect (Approaching Source)

As a source rushes toward you, each successive wavefront is emitted a little closer than the last, squeezing the waves together and raising the pitch you hear. An ambulance siren emitting 700 Hz while driving at 30 m/s through 343 m/s air reaches you at 700 × 343/(343 − 30) ≈ 767 Hz — about a semitone and a half sharp. The instant it passes, the geometry flips, the waves stretch, and the pitch drops: the classic "nee-naw... nyooow" every child imitates.

The denominator tells a dramatic story of its own. As vₛ approaches the speed of sound, the wavefronts pile up on top of each other and f′ grows without bound — the physical wall of compressed air that early jet pilots called the sound barrier. Cross it and the piled-up fronts trail behind as a shock cone: the sonic boom. Police radar and medical Doppler ultrasound run the same relation in reverse, converting a measured frequency shift into the speed of a car or of blood cells in an artery.