Doppler Effect Calculator
Calculate the observed frequency shift from the Doppler effect for sound and light — solving the complete Doppler equation f'=f(v+v_o)/(v+v_s) for any variable with full step-by-step working, sign convention explanation, and an animated wavefront simulation showing compressed and stretched waves.
Drag the slider to change source speed. Watch wavefronts compress ahead (higher frequency) and stretch behind (lower frequency). At M≥1 a Mach cone forms.
Observed Frequency (f')
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Doppler equation: f' = f × (v + vo) / (v + vs)
I want to find...
Result
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Relativistic Doppler Formula
Approaching: f' = f × √((1+β)/(1−β))
Receding: f' = f × √((1−β)/(1+β))
β = v/c | Redshift z = f_emit/f_obs − 1
REDSHIFT
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| Situation | v_o sign | v_s sign | Effect on f' |
|---|---|---|---|
| Observer → source | + | — | f' increases ↑ |
| Observer ← source | − | — | f' decreases ↓ |
| Source → observer | — | − | f' increases ↑ |
| Source ← observer | — | + | f' decreases ↓ |
| Both → each other | + | − | f' greatly increases ↑↑ |
| Both ← each other | − | + | f' greatly decreases ↓↓ |
| Case | Doppler Formula |
|---|---|
| General (sound) | f' = f(v+v_o)/(v+v_s) |
| Source approaching, obs stationary | f' = f·v/(v−v_s) |
| Source receding, obs stationary | f' = f·v/(v+v_s) |
| Observer approaching, source stationary | f' = f·(v+v_o)/v |
| Observer receding, source stationary | f' = f·(v−v_o)/v |
| Relativistic (approaching) | f' = f·√((1+β)/(1−β)) |
| Relativistic (receding) | f' = f·√((1−β)/(1+β)) |
| Wavelength shift (non-rel.) | Δλ/λ ≈ v_r/c |
| Ultrasound Doppler shift | Δf = 2f₀v_blood·cosθ/c_tissue |
| Medium | Speed (m/s) | Notes |
|---|---|---|
| Air (0°C) | 331 | Standard temperature |
| Air (20°C) | 343 | Room temperature |
| Air (100°C) | 387 | Boiling point of water |
| Water (20°C) | 1481 | Much faster than air |
| Sea water (20°C) | 1522 | Salt increases speed |
| Steel | 5960 | Very fast in solids |
| Aluminum | 6320 | Fastest common metal |
| Glass | 5640 | |
| Wood (oak) | 3850 | |
| Concrete | 3200 | |
| Rubber | 100 | Very slow |
| Application | Medium | Frequency Range |
|---|---|---|
| Police speed radar | Electromagnetic | 10–35 GHz |
| Weather Doppler radar | Electromagnetic | 2–10 GHz |
| Medical ultrasound | Sound (tissue) | 1–20 MHz |
| Bat echolocation | Sound (air) | 20–200 kHz |
| Sonar | Sound (water) | 1–100 kHz |
| Astronomical redshift | Light | Visible/UV/IR |
| Ambulance siren | Sound (air) | 700–1000 Hz |
The Doppler Effect Formula — f' = f(v+vo)/(v+vs)
The Doppler effect formula for sound is the complete Doppler equation: f' = f × (v + v_o) / (v + v_s). This single Doppler formula covers all cases of relative motion between source and observer. The formula for the Doppler effect defines five variables:
- f' — observed (apparent) frequency in Hz
- f — source frequency (emitted) in Hz
- v — speed of sound in the medium (always positive, m/s)
- v_o — observer speed (m/s, with sign convention)
- v_s — source speed (m/s, with sign convention)
The physical meaning of the Doppler effect equation: when source and observer move toward each other, wavefronts bunch up — more arrive per second at the observer's ear — increasing the observed frequency. When moving apart, wavefronts stretch, decreasing frequency. This is exactly what you hear when an ambulance siren rises in pitch as it approaches and drops as it recedes.
The equation of Doppler effect is the foundation of this calculator. All four single-motion Doppler effect equations follow directly from this one general formula by setting the appropriate variable to zero or applying the sign convention.
Doppler Sign Convention — The Most Confusing Part
The Doppler sign convention is the #1 source of student errors. Two different textbook conventions exist. This calculator uses: v_o positive when observer moves TOWARD source; v_s positive when source moves AWAY from observer.
Memory aid: Signs are chosen so that motion TOWARD always increases the fraction (and therefore f'). Observer toward → numerator gets bigger. Source toward → denominator gets smaller. Both effects raise f'.
The four single-motion cases using the Doppler frequency equation:
| Case | Doppler Shift Formula | f' vs f |
|---|---|---|
| Source → observer (stationary obs) | f' = f·v/(v−v_s) | f' > f ↑ |
| Source ← observer (stationary obs) | f' = f·v/(v+v_s) | f' < f ↓ |
| Observer → source (stationary src) | f' = f·(v+v_o)/v | f' > f ↑ |
| Observer ← source (stationary src) | f' = f·(v−v_o)/v | f' < f ↓ |
Notice how the Doppler shift equation for a source approaching uses v−v_s in the denominator (v_s is negative in the general formula, so the denominator shrinks, making f' larger). The equation for Doppler shift for a receding source uses v+v_s (larger denominator → smaller f').
How to Calculate the Doppler Effect — Step-by-Step
Use this five-step method to calculate the Doppler effect reliably every time:
- Step 1 — Identify all values: Source frequency f, speed of sound v in the medium, source speed v_s, observer speed v_o
- Step 2 — Determine directions and apply sign convention: Is the observer moving toward or away? Is the source moving toward or away? Apply the signs: v_o positive toward, v_s positive away
- Step 3 — Substitute into f'=f(v+v_o)/(v+v_s): Write the full Doppler formula with all numerical values substituted
- Step 4 — Calculate: Compute numerator = v + v_o, denominator = v + v_s, then multiply by f
- Step 5 — Find wavelength shift: λ = v/f (source wavelength), λ' = v/f' (observed wavelength), Δλ = λ' − λ
Example 1 — Ambulance Approaching (Source toward observer)
- Given: f = 700 Hz, v = 343 m/s, v_s = 30 m/s toward observer, v_o = 0
- Sign convention: source toward → v_s_signed = −30; observer stationary → v_o_signed = 0
- Substitute: f' = 700 × (343 + 0)/(343 + (−30)) = 700 × 343/313
- Calculate: 700 × 1.09585 = 767.1 Hz
- Shift: Δf = +67.1 Hz (+9.59%), λ_obs = 343/767.1 = 0.447 m (compressed)
Example 2 — Observer Approaching Stationary Source (440 Hz, v_o = 5 m/s)
- Given: f = 440 Hz, v = 343 m/s, v_s = 0, v_o = 5 m/s toward source
- Sign: v_o_signed = +5, v_s_signed = 0
- f' = 440 × (343+5)/(343+0) = 440 × 348/343 = 440 × 1.01458 = 446.4 Hz
- Δf = +6.4 Hz (+1.46%)
Example 3 — Both Moving Toward Each Other (f=500 Hz, v_s=20 m/s, v_o=5 m/s)
- v_o_signed = +5, v_s_signed = −20
- f' = 500 × (343+5)/(343+(−20)) = 500 × 348/323 = 500 × 1.0774 = 538.7 Hz
- Maximum shift — both contributions compound
The Ambulance Problem — Approaching and Receding Frequencies
The classic textbook problem for the Doppler effect formula. An ambulance siren emits 700 Hz and travels at 30 m/s. The Doppler equation gives two frequencies:
The Doppler shift total pitch drop as the ambulance passes: 767.1 − 644.5 = 122.6 Hz. This sudden drop happens because the Doppler formula switches from using (v−v_s) to (v+v_s) in the denominator the instant the source passes the observer. The ratio of approaching to receding frequency:
f'_approach / f'_recede = (v+v_s)/(v−v_s) = 373/313 = 1.192 — the ambulance sounds 19.2% higher in pitch before passing than after
Doppler Effect for Light — Redshift and Blueshift
For electromagnetic waves at everyday speeds (v ≪ c), the Doppler formula approximates to f' ≈ f(1 ± v/c) — identical in form to the sound formula. For significant speeds, the relativistic Doppler formula must be used:
- Redshift (z > 0): Galaxy moving away → wavelength increases (toward red) → frequency decreases. Hubble's Law: all distant galaxies show redshift proportional to distance — evidence of expanding universe
- Blueshift (z < 0): Andromeda galaxy approaching Earth at ~110 km/s → slight blueshift, λ decreases, f increases
- Police radar: Uses EM Doppler effect at 24 GHz. Car at 30 m/s → Δf ≈ 4.8 kHz (tiny but precisely measurable)
- Weather radar: Measures Doppler shift of rain droplets to map wind velocity inside storms
Doppler Effect in Ultrasound — Medical Applications
Medical ultrasound uses the Doppler shift formula to measure blood flow velocity non-invasively. The ultrasound Doppler equation is:
The Doppler shift equation for ultrasound gives: blood moving toward transducer → higher frequency return (blueshift); blood moving away → lower frequency (redshift). Typical values: f₀ = 5 MHz, blood velocity 0.1–1.5 m/s, θ = 60°. Δf ≈ 2 × 5×10⁶ × 1.0 × 0.5 / 1540 ≈ 3247 Hz — easily measured.
Sonic Boom — When Source Speed Exceeds Sound Speed
When v_s ≥ v_sound, the Doppler formula breaks down completely. The denominator (v − v_s) becomes zero (infinite frequency at M=1) or negative (meaningless). The source outruns its own wavefronts — all wavefronts pile up into a conical shock wave:
- Mach number: M = v_s / v_sound (dimensionless)
- Mach cone half-angle: sin(α) = v_sound / v_source = 1/M
- At M=1 (sonic speed): cone angle = 90° (flat wavefront perpendicular to motion)
- At M=2: sin(α) = 0.5, α = 30° (narrow Mach cone)
The Doppler effect formula is only valid for subsonic sources (v_s < v_sound). The animation above shows the Mach cone forming when you drag the slider past M=1.
Common Mistakes in Doppler Effect Calculations
Mistake 1 — Wrong sign convention (most common)
- ❌ Wrong: Source approaching → use +v_s in denominator → f' = 700×343/373 = 644.5 Hz (this is actually the receding frequency!)
- ✅ Correct: Source approaching → v_s_signed = −v_s → denominator = v − v_s = 313 → f' = 767.1 Hz
- Check: moving toward should always give f' > f
Mistake 2 — Swapping numerator/denominator variables
- ❌ Wrong: Putting v_s in numerator and v_o in denominator
- ✅ Correct: v_o (observer speed) always goes in the NUMERATOR; v_s (source speed) always goes in the DENOMINATOR. Memory: Observer/Numerator both start related — the observer receives the waves so goes on top.
Mistake 3 — Using relative speed instead of v_sound
- ❌ Wrong: Replacing v in the Doppler formula with v_sound − v_source
- ✅ Correct: v in the Doppler equation is ALWAYS the speed of sound in the medium — a fixed physical property, not a relative speed
Mistake 4 — Using classical formula for light at high speed
- ❌ Wrong: f' = f(v+v_o)/(v+v_s) where v=c for fast-moving astronomical objects
- ✅ Correct: Use f' = f√((1±β)/(1∓β)) — the relativistic Doppler formula
Mistake 5 — Thinking Doppler changes wave speed
- ❌ Wrong: "The approaching ambulance's waves travel faster toward me"
- ✅ Correct: Wave speed depends only on the MEDIUM (v_sound = 343 m/s in air regardless of source motion). Only frequency and wavelength change — not wave speed.
Worked Examples — 8 Complete Problems
1. Ambulance (700 Hz, 30 m/s) approaching stationary observer
- f' = 700 × (343+0)/(343+(−30)) = 700 × 343/313
- f' = 767.1 Hz | Δf = +67.1 Hz | +9.59%
2. Same ambulance receding from observer
- f' = 700 × (343+0)/(343+30) = 700 × 343/373
- f' = 644.5 Hz | Δf = −55.5 Hz | −7.93%
3. Observer walking (1.5 m/s) toward stationary source (440 Hz)
- f' = 440 × (343+1.5)/(343+0) = 440 × 344.5/343
- f' = 441.9 Hz | Δf = +1.9 Hz | +0.44%
4. Both moving toward each other: source 20 m/s, observer 5 m/s, f=500 Hz
- v_o_signed = +5, v_s_signed = −20
- f' = 500 × (343+5)/(343−20) = 500 × 348/323
- f' = 538.7 Hz | Δf = +38.7 Hz | +7.74%
5. Find source speed: observed f'=580 Hz, emitted f=500 Hz, v=343 m/s, observer stationary
- 580 = 500 × 343/(343−v_s) [source approaching]
- 343−v_s = 500×343/580 = 295.7
- v_s = 343−295.7 = 47.3 m/s (170.3 km/h)
6. Speed radar: car at 120 km/h (33.33 m/s), radar f=24 GHz
- β = 33.33/c = 1.111×10⁻⁷; non-relativistic fine
- Δf ≈ 2×f×v/c = 2×24×10⁹×33.33/2.998×10⁸ = 5,333 Hz ≈ 5.33 kHz
- (Factor of 2: wave travels to car and reflects back)
7. Galaxy receding at β=0.05c — redshift
- f' = f × √((1−0.05)/(1+0.05)) = f × √(0.95/1.05) = f × 0.9513
- z = 1/0.9513 − 1 = z = 0.0513 (redshift)
- λ_obs/λ_emit = 1.0513 — wavelength stretched by 5.13%
8. Ultrasound blood velocity: f₀=3 MHz, Δf=975 Hz, θ=60°, c_tissue=1540 m/s
- Δf = 2f₀v_blood cosθ/c → v_blood = Δf×c/(2f₀cosθ)
- v_blood = 975×1540/(2×3×10⁶×0.5) = 1,501,500/3,000,000
- v_blood = 0.500 m/s (50 cm/s — typical arterial flow)
Frequently Asked Questions — Doppler Effect
Related Calculators
v_o positive (observer toward)
v_s negative (source toward)
v_o negative (observer away)
v_s positive (source away)
Memory: motion TOWARD always increases f'