Hong Life CalculatorsUnderstanding the Doppler effect calculator — concept and principle
The Doppler effect is the change in frequency you hear when a sound source and a listener move relative to each other: a siren sounds higher as it approaches and drops in pitch as it passes. Enter the source frequency, the speeds of source and observer toward each other, and the speed of sound, and the calculator returns the observed frequency while approaching and after passing.
How it is calculated
- Approaching: f′ = f · (c + v_o) ÷ (c − v_s)
- After passing (both receding): f″ = f · (c − v_o) ÷ (c + v_s)
- f is the emitted frequency (Hz), c the speed of sound in the air (m/s), v_s the source speed and v_o the observer speed, both entered as positive when moving toward the other. Speeds are relative to still air.
The principle
A moving source chases its own waves. During one period T it moves v_s·T forward, so the crests ahead are packed closer together: the wavelength becomes (c − v_s)/f, and since the waves still travel at c, the frequency rises. A moving observer does not change the wavelength but meets crests more often, at a rate (c + v_o)/λ. Combining both effects gives the formula, first proposed by Christian Doppler in 1842 and tested with sound by Buys Ballot in 1845 using musicians on a moving train. Because the air is the reference, source and observer motion at equal speed give slightly different shifts.
Worked example
- Inputs: f = 500 Hz, v_s = 30 m/s toward the observer, v_o = 0, c = 343 m/s.
- Approaching: 500 × (343 + 0) ÷ (343 − 30) = 500 × 343 ÷ 313 = 547.92 Hz (shown as 547.9233227).
- After passing: 500 × (343 − 0) ÷ (343 + 30) = 500 × 343 ÷ 373 = 459.79 Hz (shown as 459.7855228).
- The drop heard as the source passes is about 88 Hz.
| Source frequency (Hz) | Observed frequency (Hz) |
|---|---|
| 125 | 136.9808307 |
| 250 | 273.9616613 |
| 375 | 410.942492 |
| 500 | 547.9233227 |
| 625 | 684.9041534 |
| 750 | 821.884984 |
| 1,000 | 1,095.846645 |
How to read the result
The ratio 547.92 ÷ 459.79 ≈ 1.19 is about three semitones, a clearly audible fall in pitch for a vehicle at 30 m/s (108 km/h). If instead the observer moved toward a still source at 30 m/s, the result would be 500 × 373 ÷ 343 ≈ 543.7 Hz. As v_s approaches c the denominator goes to zero; at and above the speed of sound the formula no longer applies and a shock wave (sonic boom) forms.
Common mistakes
- Getting the signs wrong: here speeds toward the other party are positive.
- Using the sum of speeds as a single relative speed; for sound, the medium matters.
- Applying the formula to light, which needs the relativistic Doppler formula.
Frequently asked questions
How does radar measure speed with the Doppler effect?
The wave reflects off the moving target, so the shift happens twice; for speeds much smaller than the wave speed it is about 2v/c times the frequency.
Can I find the source speed from the two pitches?
Yes. With a stationary observer, v_s = c · (f′ − f″) ÷ (f′ + f″); here 343 × 88.14 ÷ 1007.71 ≈ 30 m/s.
Why does the pitch change gradually as a car passes?
Only the speed component along your line of sight counts; it falls to zero alongside you, then reverses.
Sources: C. Doppler, Über das farbige Licht der Doppelsterne (1842) · D. Halliday, R. Resnick and J. Walker, Fundamentals of Physics, Wiley