Grade 11 Physics · Moving sources and beats
Pitch bends when something moves
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Pitch bends when something moves

The siren that sails past you drops its pitch the instant it passes — that is the Doppler effect, and the sanity rail you must never let go of: approach raises pitch, retreat lowers it. Any answer that disagrees is wrong before the arithmetic starts.

Two machines, depending on who moves. A moving SOURCE squeezes its own wavefronts: f=fvvvsf' = \dfrac{f v}{v - v_s} — read aloud: f-prime equals f v over v minus v-s (ff' is said f-prime, the pitch actually heard). A moving LISTENER meets wavefronts more often: f=f(v+vo)vf' = \dfrac{f (v + v_o)}{v}. Approach makes both fractions top-heavy — the rail, built into the algebra.

And when two nearly equal tones sound together, they drift in and out of step and the loudness pulses: beats, at fbeat=f1f2f_{\text{beat}} = f_1 - f_2 with f1f_1 the higher. A violinist tuning to a fork listens for the beats to slow to nothing — silence between two pitches is what "in tune" means.