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Lesson 31 · 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′=fvv−vsf' = \dfrac{f v}{v - v_s} — read aloud: f-prime equals f v over v minus v-s. The cast, in words: ff is the pitch the source actually emits, f′f' (said f-prime) is the pitch the listener hears — the one you are solving for — and vv is the speed of sound in the air, all in the usual hertz and m/s. The subscripts name who is moving: vsv_s is the source's speed, vov_o the observer's — the listener's. 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=f1−f2f_{\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.

vsfvf′

f′=fvv−vsf' = \frac{f v}{v - v_s}

  • f′f'= Observed frequency (frequency)
  • ff= Source frequency (frequency)
  • vv= Speed of sound (speed)
  • vsv_s= Source speed (speed)
Doppler Effect (Approaching Source) solver →
fvvof′

f′=f(v+vo)vf' = \frac{f (v + v_o)}{v}

  • f′f'= Observed frequency (frequency)
  • ff= Source frequency (frequency)
  • vv= Speed of sound (speed)
  • vov_o= Observer speed (speed)
Doppler Effect (Approaching Observer) solver →
f1f2fbeat

fbeat=f1−f2f_{\text{beat}} = f_1 - f_2

  • fbeatf_{\text{beat}}= Beat frequency
  • f1f_1= Higher frequency
  • f2f_2= Lower frequency

each variable a frequency

Beat Frequency solver →