Doppler Effect Formula
Reference for the Doppler effect f' = f(v+vo)/(v-vs).
Calculate frequency shift for moving sources and observers with examples for sirens and ultrasound.
Need to calculate, not just reference? Use the interactive version.
Open Doppler Effect Frequency Calculator →
The Formula
f' = f × (v + vₒ) / (v + vₛ)
The Doppler effect causes the perceived frequency to change when a source or observer is moving. Approaching objects sound higher in pitch. Receding objects sound lower.
Variables
| Symbol | Meaning |
|---|---|
| f' | Observed (perceived) frequency (Hz) |
| f | Emitted (source) frequency (Hz) |
| v | Speed of sound in the medium (m/s) |
| vₒ | Observer velocity (positive = toward source, negative = away) |
| vₛ | Source velocity (positive = away from observer, negative = toward) |
Example 1
An ambulance siren emits 700 Hz while approaching you at 30 m/s. Speed of sound = 343 m/s.
Observer is stationary: vₒ = 0
Source approaching: vₛ = -30 m/s
f' = 700 × (343 + 0) / (343 + (-30)) = 700 × 343/313
f' ≈ 767 Hz (higher pitch)
Example 2
Same ambulance moving away from you at 30 m/s
Source receding: vₛ = +30 m/s
f' = 700 × 343 / (343 + 30) = 700 × 343/373
f' ≈ 644 Hz (lower pitch)
When to Use It
Use the Doppler effect formula when:
- Predicting how frequency changes with motion
- Understanding radar and sonar speed detection
- Analyzing astronomical redshift and blueshift
- Designing ultrasound and medical imaging equipment
Key Notes
- Full formula: f' = f × (v + v_o) / (v − v_s), where v is wave speed, v_o is observer speed, and v_s is source speed. Use + for approaching and − for receding in the appropriate numerator/denominator position.
- Only affects frequency, not speed: The speed of sound in air is constant (at a given temperature). The Doppler effect changes the perceived frequency (pitch) — the actual wave speed stays the same.
- Applies to all wave types: The Doppler effect works for sound, light, and water waves. For light (electromagnetic waves), the relativistic Doppler formula must be used at high speeds.
- Red shift and blue shift: In astronomy, galaxies moving away show red-shifted light (lower frequency). Approaching galaxies show blue-shifted light. This is the key evidence for the expanding universe.
- Practical uses: Radar speed guns, medical Doppler ultrasound, weather radar (detecting precipitation movement), and air traffic control all rely on the Doppler effect.