Put the Observer in the Numerator and the Source in the Denominator
The Doppler effect changes the frequency you hear when the source, the observer or both move. One formula covers all the sound cases, and a different, simpler rule covers light.
Motion and force questions become simpler when direction, starting condition, time and the acting forces are marked first.
The formula and sign rules
f' = f (v + vₒ)/(v - vₛ) with v the speed of sound, vₒ the observer's speed toward the source and vₛ the source's speed toward the observer; use a minus sign for the observer moving away and a plus sign in the denominator for a source moving away. For a source and observer each moving away at 15 m/s with v = 340 m/s, f' = f (340 - 15)/(340 + 15), so a heard frequency of 1950 Hz means f = 1950 x 355/325 = 2130 Hz. An observer moving toward a stationary source at v/5 hears 1.2 f while the wavelength in the medium is unchanged. If the source and the observer move with the same velocity in the same direction, there is no relative motion and f' = f.
Echoes and circular motion
For an echo, treat the reflector as a stationary observer and then as a stationary source. A siren of 900 Hz moving at 30 m/s toward a cliff, with the observer behind it, sends a frequency 900 x 330/(330 - 30) = 990 Hz to the cliff, which reflects exactly that frequency to the stationary observer.
the lowest frequency heard far away is when it moves directly away: 385 x 340/(340 + 10) = 374 Hz.
Light and red shift
For sound, it matters whether the source or the observer moves relative to the medium, but for light there is no medium, so only the relative motion matters; this follows from the theory of relativity. For speeds much smaller than c, Δλ/λ = v/c: a 0.4% longer wavelength from a galaxy means v = 0.004 x 3 x 10⁸ = 1.2 x 10⁶ m/s, and a 1 nm shift on 600 nm gives v = c/600 = 5 x 10⁵ m/s. A longer wavelength (red shift) means the source is receding.
OlympIQ exam lens
- ✓f' = f (v ± vₒ)/(v ∓ vₛ): observer toward the source takes plus on top; source toward the observer takes minus below
- ✓Motion perpendicular to the line of sight does not change the frequency; use the instant of nearest approach or recession
- ✓Echo: reflector acts first as an observer, then as a stationary source
- ✓Same velocity of source and observer in the same direction means no Doppler shift
- ✓Light: only relative motion matters, Δλ/λ = v/c for small v; red shift means receding
Do not apply the sound formula to light. For sound the shift depends on which of source or observer is moving through the medium; for light only the relative speed counts, and the formula is relativistic.
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