Playground
Interactive Doppler effect: a moving source emits circular wavefronts that compress ahead and stretch behind. Adjust source speed to see the frequency shift.
Variables
| Symbol | Name | SI | Dimension | Range |
|---|---|---|---|---|
| Observed frequencyoutput Frequency perceived by the observer | Hz | T^-1 | 1 – 2000 | |
| Source frequency Frequency emitted by the source at rest | Hz | T^-1 | 1 – 2000 | |
| Wave speed in medium Speed of the wave in the medium (e.g., speed of sound in air) | m/s | LT^-1 | 100 – 1500 | |
| Observer velocity Speed of the observer toward the source (positive = approaching) | m/s | LT^-1 | -100 – 100 | |
| Source velocity Speed of the source toward the observer (positive = approaching) | m/s | LT^-1 | -100 – 100 |
Deep dive
Derivation
A source emitting at frequency f produces wavefronts separated by λ = v/f. If the source moves at v_s toward the observer, each successive wavefront is emitted v_s/f closer, compressing the effective wavelength to λ' = (v - v_s)/f. The observer moving at v_o toward the source encounters wavefronts at rate f' = (v + v_o)/λ' = f·(v + v_o)/(v - v_s).
Experimental verification
Buys Ballot (1845) placed trumpeters on a train to verify pitch shifts. Modern: police radar guns use the electromagnetic Doppler effect, and Doppler ultrasound measures blood flow velocity in medical imaging.
Common misconceptions
- The Doppler effect changes the speed of sound — it changes the observed frequency and wavelength, not the wave speed in the medium.
- It only applies to sound — it applies to all waves, including light (redshift/blueshift of galaxies).
- Source and observer motion are equivalent — for mechanical waves they are not: the medium defines a preferred frame. Only for light (relativistic Doppler) are they symmetric.
Real-world applications
- Radar speed guns: police measure vehicle speed from the frequency shift of reflected microwaves.
- Medical ultrasound: Doppler imaging measures blood flow velocity in arteries.
- Astronomical redshift: recession velocity of galaxies reveals the expansion of the universe.
- Weather radar: Doppler radar detects wind speed and rotation in storm cells.
Worked examples
Approaching ambulance siren
Given:
- f:
- 700
- v:
- 343
- v_s:
- 30
- v_o:
- 0
Find: f_prime
Solution
f' = f × v/(v - v_s) = 700 × 343/(343 - 30) = 700 × 343/313 = 767.1 Hz
Receding ambulance siren
Given:
- f:
- 700
- v:
- 343
- v_s:
- -30
- v_o:
- 0
Find: f_prime
Solution
f' = 700 × 343/(343 + 30) = 700 × 343/373 = 643.7 Hz
Scenarios
What if…
- scenario:
- What if v_s = v (source at wave speed)?
- answer:
- Denominator → 0, f' → ∞. All wavefronts pile up into a shock wave — the sonic boom at Mach 1.
- scenario:
- What if both source and observer move toward each other?
- answer:
- Both effects compound: f' = f·(v + v_o)/(v - v_s). The frequency shift is larger than either motion alone.
- scenario:
- What if applied to light?
- answer:
- For electromagnetic waves, use the relativistic Doppler formula: f' = f·√((1+β)/(1-β)) where β = v/c. There is no medium, so only relative velocity matters.
Limiting cases
- condition:
- v_s → 0, v_o → 0
- result:
- f' = f
- explanation:
- No relative motion means no frequency shift.
- condition:
- v_s → v
- result:
- f' → ∞
- explanation:
- Source at wave speed: wavefronts pile up into a sonic boom (Mach 1).
- condition:
- v_s → -v (receding at wave speed)
- result:
- f' → f/2
- explanation:
- Source receding at wave speed halves the observed frequency.
Context
Christian Doppler · 1842
Doppler predicted the effect for light from binary stars. Buys Ballot confirmed it in 1845 using musicians on a moving train.
Hook
Why does an ambulance siren sound higher-pitched as it approaches and lower as it drives away?
An ambulance siren emits at 700 Hz, approaching at 30 m/s. Find the perceived frequency using f' = f * v/(v - v_s) with v = 343 m/s.
Dimensions:
- lhs:
- f' → [T⁻¹]
- rhs:
- f·(v + v_o)/(v - v_s) → [T⁻¹]·[LT⁻¹]/[LT⁻¹] = [T⁻¹]
- check:
- Both sides are [T⁻¹] = Hz. ✓
Validity: Valid for mechanical waves when source speed < wave speed (subsonic). For electromagnetic waves, use the relativistic Doppler formula instead.