Doppler Effect Calculator

Calculate observed frequency when source or observer is moving using f' = f(v±v_o)/(v∓v_s).

🔊 Waves📐 Doppler Effect🚑 Frequency Shift

Sign convention: use + when observer moves toward source; − when moving away. For source: − when moving toward observer, + when moving away.

Source frequency (f) Hz
Wave speed (v) m/s
Observer velocity (v_o) m/s
Source velocity (v_s) m/s
Observer direction
Source direction
⚠️ Wave speed must be greater than source speed.

What Is Doppler Effect?

The Doppler effect describes the change in observed frequency of a wave when the source or observer (or both) are moving relative to the medium. The formula is f' = f × (v + v_o)/(v − v_s), where f is the emitted frequency, v is the wave speed, v_o is the observer's speed (+ if moving toward source), and v_s is the source speed (+ if moving away from observer). When source approaches: f' > f (blueshift). When receding: f' < f (redshift).

The Doppler effect applies to all waves: sound (ambulance siren pitch changes), light (redshift of galaxies), radar (speed guns), ultrasound (blood flow measurement), and sonar. For sound in air (v ≈ 343 m/s), the effect is easily audible. For light, the fractional shift Δf/f = v_r/c where v_r is radial velocity, so massive speeds are needed for measurable light Doppler shifts.

When a source moves faster than the wave speed (supersonic for sound), a shock wave (Mach cone) forms — the Doppler equation breaks down and a sonic boom occurs. The Mach number M = v_source/v_wave quantifies this. At M = 1 (source speed = wave speed), f' → ∞ as the formula denominator approaches zero.

Edwin Hubble used the light Doppler effect in 1929 to show galaxies are receding — their spectra are redshifted. The recession speed is proportional to distance (Hubble's Law: v = H₀d), providing evidence for the expanding universe and the Big Bang. Cosmic microwave background radiation is an extreme redshift of the early universe's visible light.

Formula Reference Table

Solve ForFormulaNotes
Observed frequencyf' = f(v + v_o)/(v − v_s)v_o: + toward, v_s: + away
Source approaching, obs stationaryf' = f·v/(v − v_s)f' > f — higher pitch
Source receding, obs stationaryf' = f·v/(v + v_s)f' < f — lower pitch
Beat frequencyΔf = |f₁ − f₂|Audible beats at low Δf
Light (radial velocity)Δλ/λ = v_r/cRedshift z = v_r/c
Mach numberM = v_source/v_waveM > 1: supersonic; sonic boom

3 Worked Examples

Example 1
Ambulance Approaching at 30 m/s

Ambulance siren at 800 Hz, v_s = 30 m/s toward observer, v = 343 m/s.

  • f' = 800 × (343 + 0)/(343 − 30)
  • f' = 800 × 343/313 = 876.7 Hz
  • After passing: f' = 800 × 343/(343+30) = 735 Hz
✓ Approaching: 877 Hz; Receding: 735 Hz (pitch drops by 142 Hz)
Example 2
Speed Gun — Police Radar

Police radar gun (f = 24.15 GHz, v_sound → c) measures reflected signal from car at 30 m/s.

  • Doppler shift: Δf = 2f·v_car/c (double — outgoing and return)
  • Δf = 2 × 24.15e9 × 30/(3e8) = 4,830 Hz
  • Beat frequency of 4,830 Hz → car speed calculated
✓ Doppler shift = 4,830 Hz → car speed 30 m/s detected
Example 3
Galaxy Redshift

A galaxy emits Hα at 656.3 nm but we observe 672 nm. Find recession speed.

  • z = (λ_obs − λ_emit)/λ_emit = (672 − 656.3)/656.3 = 0.0239
  • v_r = z×c = 0.0239 × 3×10⁸ = 7.17×10⁶ m/s = 7,170 km/s
  • Distance: v = H₀d → d = 7170/70 = 102 Mpc ≈ 333 million light-years
✓ Galaxy recession speed = 7,170 km/s; distance ≈ 333 Mly

Real-World Applications

🚑
Emergency Vehicles
Ambulance sirens use specifically pitched tones chosen to be audible and alarming. The Doppler shift as they approach/pass is immediately noticeable — designed into the warning system.
🌌
Cosmology
Galaxy recession speeds measured via redshift gave Hubble's constant H₀ ≈ 70 km/s/Mpc. This measurement established the age and expansion rate of the universe.
📡
Radar & Sonar
Speed cameras, weather radar (measuring rain velocity), and sonar (submarine detection) all use Doppler frequency shifts to measure velocities without physical contact.
⚕️
Medical Ultrasound
Doppler ultrasound measures blood flow velocity by comparing transmitted (2–15 MHz) and reflected frequencies from moving red blood cells. Color Doppler displays flow direction and speed.
✈️
Weather Forecasting
Doppler weather radar measures precipitation velocity by frequency shift. Rotating storm patterns (tornadoes) produce simultaneous Doppler blueshift (approaching rain) and redshift (receding) on opposite sides of the rotation.

Common Mistakes to Avoid

⚠️
Using wrong sign convention

The formula signs depend on direction. Check your convention carefully. This calculator uses: v_o positive = toward source (increases f'); v_s positive = away from observer (also decreases f').

⚠️
Using wave speed incorrectly

v in the formula is the wave speed in the medium (not relative to source). For sound at 20°C: 343 m/s. Use the actual medium speed, not c unless calculating light Doppler.

⚠️
Applying formula to supersonic sources

When v_s > v (supersonic), the denominator becomes negative, giving negative frequency — the formula is invalid. A shock wave (Mach cone) forms instead.

⚠️
Forgetting the 2x for radar/sonar

Radar measures reflected waves. The frequency shift doubles because it's Doppler-shifted twice: once when the wave reaches the moving target, and once when the reflected wave returns. Δf_total = 2f·v_target/c.

⚠️
Confusing f (emitted) with f' (observed)

f is the frequency the source actually emits. f' is what the observer measures. They differ when there is relative motion.

Frequently Asked Questions

Why does an ambulance siren change pitch? ▾
As the ambulance approaches, sound waves are compressed (shorter wavelength, higher frequency — you hear a higher pitch). After passing, waves are stretched (lower pitch). The speed of sound (343 m/s) stays constant — only wavelength and frequency change.
Does the Doppler effect apply to light? ▾
Yes. For electromagnetic waves: Δλ/λ ≈ v_r/c (for v << c), or exact relativistic formula: λ_obs/λ_emit = √((1+β)/(1-β)) where β = v/c. Recession → redshift; approach → blueshift. This is how astronomers measure stellar and galactic velocities.
What is a sonic boom? ▾
When a source moves at or above the speed of sound (Mach ≥ 1), wavefronts pile up, forming a conical shock wave. When this cone sweeps over an observer, the concentrated pressure gives a loud boom. The cone half-angle: sin(θ) = v_wave/v_source = 1/M.
How does Doppler radar detect tornadoes? ▾
Doppler weather radar sends out microwave pulses and measures the frequency shift of reflections. Rain moving toward the radar is blueshifted; rain moving away is redshifted. A tornado shows adjacent blueshift and redshift due to rotation — the Doppler velocity couplet is a diagnostic signature.
What is the cosmological redshift? ▾
Galaxy recession redshift: z = (λ_observed − λ_emitted)/λ_emitted = v_recession/c (for v << c). The most distant visible galaxies have z > 10. The cosmic microwave background (relic radiation from 380,000 years after Big Bang) has z ≈ 1100 — visible light shifted to microwave by universe expansion.
How does speed radar work? ▾
Police radar emits a continuous microwave beam (24–35 GHz range). Reflected waves from a moving car are Doppler-shifted by Δf = 2f·v/c. The radar measures beat frequency between emitted and received signals, calculates v = c·Δf/(2f). High-precision radar guns are accurate to ±1 km/h.
Can the Doppler effect measure blood flow? ▾
Yes — duplex ultrasound uses Doppler shift of 2–8 MHz ultrasound to measure blood cell velocity. Δf = 2f·v·cos(θ)/c_tissue (θ = beam angle). Continuous-wave Doppler measures maximum velocities; pulsed Doppler gives velocity at specific depths. Used to detect arterial blockages and measure cardiac output.
What is the transverse Doppler effect? ▾
When a source moves perpendicular to the line of sight, classical Doppler predicts no shift (f' = f). But special relativity predicts a blueshift: f' = f·γ (where γ = 1/√(1-v²/c²)) for the source moving transversely. This purely relativistic effect was confirmed experimentally in 1938 by Ives and Stilwell.

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Formula Explorer connections

Interpretation: This relationship connects frequency, wavelength, speed, phase, intensity or resonance in an oscillating system. Assumption: Identify the medium, boundary conditions and reference frame. Linear waves, small amplitudes, nondispersive media or ideal resonance may be assumed.

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