The Doppler effect
The passing siren
- An ambulance races past and its siren suddenly drops in pitch.
- The siren itself never changed — your ear heard a different frequency.
- This is the Doppler effect.
Why the pitch changes
- A source moving toward you bunches the wavefronts ahead → shorter $\lambda$ → higher pitch.
- Moving away, the wavefronts spread out → longer $\lambda$ → lower pitch.
As a sound source moves toward you, the pitch you hear is:
Moving toward you bunches the wavefronts, shortening the wavelength and raising the frequency you hear.
A source moving toward an observer bunches the wavefronts ahead of it.
Yes — each new wavefront is sent from a point a little closer, so they crowd together in front.
The formula
- For a moving source and a still observer: $f_{\text{o}} = \dfrac{v\,f_{\text{s}}}{v \pm v_{\text{s}}}$.
- Use minus when approaching (higher pitch), plus when receding (lower pitch).
In $f_{\text{o}} = \dfrac{v\,f_{\text{s}}}{v \pm v_{\text{s}}}$, for a source moving toward the observer you use:
A smaller denominator gives a larger $f_{\text{o}}$ — the higher pitch you expect when approaching.
Worked example
- A horn at $f_{\text{s}} = 800\ \text{Hz}$ moves at $30\ \dfrac{\text{m}}{\text{s}}$ toward you; $v = 340\ \dfrac{\text{m}}{\text{s}}$.
- Approaching → minus sign: $f_{\text{o}} = \dfrac{340 \times 800}{340 - 30}$.
- $f_{\text{o}} = \dfrac{272000}{310} \approx 877\ \text{Hz}$ — higher, as expected.
A horn ($f_{\text{s}} = 400\ \text{Hz}$) moves at $20\ \dfrac{\text{m}}{\text{s}}$ toward you; $v = 340\ \dfrac{\text{m}}{\text{s}}$. What frequency do you hear?
$f_{\text{o}} = \dfrac{340 \times 400}{340 - 20} = \dfrac{136000}{320} = 425\ \text{Hz}$.
A source moving away from you gives a ____ frequency than the source.
Moving away spreads the wavefronts out, lengthening the wavelength and lowering the frequency.
You've got it
- a moving source changes the frequency you hear — the Doppler effect
- toward → bunched wavefronts → higher; away → spread out → lower
- $f_{\text{o}} = \dfrac{v\,f_{\text{s}}}{v \pm v_{\text{s}}}$ (minus for approaching)