Library / Foundations: Physics / FPH-44
Foundations: Physics
Doppler shift (moving source)
Frequency heard from a source approaching at speed vₛ.
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- FPH-44
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- Foundations: Physics
- Result
- f′ [Hz]
f is the frequency the source emits, c the speed of the wave in the medium and vₛ the speed of the source towards you. The observer is at rest. The pitch is raised because the source follows its own wavefronts and squeezes them together. For a source moving away, change the minus sign in the denominator to a plus.
Frequency heard from a source approaching at speed vₛ. The pitch rises as it closes and falls once it passes.
Calculator
- Shift f′ − f95.85 Hz
- Frequency once the source has passed (receding)919.6 Hz
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- A 1000 Hz horn at 30 m/s (108 km/h) is heard at 1096 Hz coming and 919 Hz going: a drop of 177 Hz, roughly three semitones, as it passes.
- As vₛ approaches c the heard frequency grows without limit, which is the pile-up of wavefronts behind a sonic boom.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (Hz) | Local step | Change in result |
|---|---|---|---|---|---|---|
| f Emitted frequency | 1 to 1,000,000 Hz | 1.096 | 1,096,000 | 1,096,000 | +10% | +10 % |
| vs Source speed toward you | 0 to 250 m/s | 1,000 | 3,688 | 2,688 | +10% | +0.968 % |
| c Wave speed | 300 to 1,600 m/s | 1,019 | 1,111 | 92 | +10% | −0.864 % |
Sweep of Source speed toward you
| Source speed toward you (m/s) | Observed frequency (Hz) |
|---|---|
| 0 | 1,000 |
| 31.25 | 1,100 |
| 62.5 | 1,223 |
| 93.75 | 1,376 |
| 125 | 1,573 |
| 156.3 | 1,837 |
| 187.5 | 2,206 |
| 218.8 | 2,761 |
| 250 | 3,688 |
Sweep one variable
Variables and how each one contributes
Observed frequency
What the equation gives you. Everything else feeds it.
Emitted frequency
The frequency the source gives out. Directly proportional: every heard frequency scales with the emitted one.
Wave speed
The speed of the wave in the medium (about 343 m/s for sound in air at 20 °C). Small effect at the default: a faster wave gives a smaller shift, because the source is a smaller fraction of the wave speed.
Source speed toward you
How fast the source is moving towards the observer. The shift grows more than in proportion as vₛ approaches c: a 10 % rise in vₛ here raises the heard frequency by only about 1 %, but the effect becomes steep close to the wave speed.
Worked example with the values above
- Denominator: c − vₛ = 343 − 30 = 313 m/s.
- Ratio: c/(c − vₛ) = 1.0958.
- f′ = 1,000 × 1.0958 = 1,096 Hz.
Assumptions and limits
- Source speed less than the wave speed (vₛ < c).
- The source moves directly towards a stationary observer, in a still medium.
Common mistakes
- Using the formula for a moving observer. That has the observer’s speed in the numerator instead.
- Using it for radar or light, where the relativistic and two-way forms apply.
Statistics and uncertainty
Near the defaults f′ moves by only about 0.1 × the relative error in vₛ, but the sensitivity to vₛ grows as vₛ/(c − vₛ), so at 90 % of the wave speed a 1 % error in vₛ gives about a 9 % error in f′. Temperature changes c by about 0.6 m/s per °C, which matters when the shift is used to measure speed.
Where it is used
- Measuring vehicle or aircraft speed from the change in pitch.
- The basis of Doppler sonar and the sound of a passing siren.
Maths you need
- Wave speed: c = fλ
- Ratios: a fraction c/(c − v)