The Doppler effect

When a source of waves moves towards you, the waves get squashed together so you receive a higher frequency; when it moves away they are stretched and the frequency drops. You hear it as a siren’s pitch falls as it passes. The same effect in light – red shift and blue shift – tells astronomers how fast stars and galaxies are moving.

Key facts

  • Source approaching: higher observed frequency (shorter wavelength). Receding: lower frequency.
  • Source moving, listener still: f_L = v f_s ÷ (v ∓ v_s) (− when approaching, + when receding).
  • Listener moving, source still: f_L = (v ± v_L) f_s ÷ v (+ when approaching).
  • The speed of the sound itself does not change; only the received frequency and wavelength do.
  • Light: red shift z = Δλ ÷ λ ≈ v ÷ c for speeds much less than c.
  • Uses: speed cameras (radar), measuring blood flow with ultrasound, finding exoplanets and galaxy speeds.

Worked examples

Example 1

An ambulance siren of 700 Hz approaches at 25 m/s. Speed of sound 340 m/s. What frequency does a pedestrian hear? And after it passes?

  1. Approaching f_L = 340 × 700 ÷ (340 − 25) = 755.6 Hz
  2. Receding f_L = 340 × 700 ÷ (340 + 25) = 652.1 Hz

Answer: 755.6 Hz approaching; 652.1 Hz receding

Example 2

A car horn is 500 Hz. A stationary listener hears 530 Hz as it approaches. How fast is the car? (v = 340 m/s)

  1. 530 = 340 × 500 ÷ (340 − v_s) 340 − v_s = 170 000 ÷ 530 = 320.8
  2. So v_s = 340 − 320.8 = 19.2 m/s

Answer: about 19.2 m/s

Example 3

A hydrogen line emitted at 656.3 nm is observed at 662.9 nm from a galaxy. How fast is it moving, and which way?

  1. Δλ 662.9 − 656.3 = 6.6 nm
  2. z = Δλ ÷ λ z = 6.6 ÷ 656.3 = 0.0101
  3. v = zc v = 0.0101 × 3.0 × 10⁸ ≈ 3.0 × 10⁶ m/s
  4. Longer wavelength: red shift, so it is receding.

Answer: about 3.0 × 10⁶ m/s, away from us