What is the Doppler effect formula?

The general Doppler formula for sound is f' = f × (v + v_o) / (v − v_s), where f is source frequency, v is the speed of sound, v_o is observer velocity (positive toward source), and v_s is source velocity (positive toward observer). Direction signs vary; always confirm convention.

Why does a siren change pitch?

As an ambulance approaches, sound waves are compressed (higher frequency/pitch). As it moves away, waves are stretched (lower frequency/pitch). The total shift can be 100+ Hz for a fast-moving vehicle — clearly audible to anyone listening.

Does the Doppler effect work for light?

Yes, but with the relativistic formula: f' = f × √((1−β)/(1+β)), where β = v/c. Light has no medium, so the effect depends only on relative velocity. This is what causes cosmological redshift — galaxies receding from us appear redder.

How do radar guns measure speed?

They emit a microwave signal (typically 10-35 GHz). The signal bounces off a moving car, returning with a Doppler shift. Mixing the return with the transmitted signal produces a beat frequency proportional to vehicle speed. Modern guns achieve ~0.5 mph accuracy.

What is a sonic boom?

When a source moves faster than the wave speed (M > 1), wavefronts pile up into a shock wave that travels outward as a cone. When the shock reaches a listener, it's heard as a loud boom. Supersonic jets create continuous sonic booms along their path on the ground.

How does Doppler ultrasound work?

A transducer emits high-frequency ultrasound (2-10 MHz). Sound reflects off moving red blood cells with a Doppler shift. The shift is proportional to flow velocity. Used for echocardiography, vascular imaging, fetal monitoring. Color Doppler maps flow direction.

Does the Doppler effect depend on distance?

No, only on relative velocity. Distance affects intensity (1/r²) but not frequency shift. The shift is the same for a siren 10 m away or 1 km away — provided you know its velocity.

What did Doppler effect prove about the universe?

Edwin Hubble in 1929 measured redshifts of distant galaxies and found they recede faster the farther they are. This established cosmic expansion. The CMB redshift (z ≈ 1100) is direct evidence of the Big Bang and gave us the universe's age (~13.8 billion years).

Doppler Effect Calculator

Calculate the observed frequency due to the Doppler effect for sound waves. Accounts for both source and observer motion relative to the medium. Uses f′ = f × (v ± v_o) / (v ∓ v_s).

Last updated: May 29, 2026

About this calculator

The Doppler effect is the change in observed frequency of a wave when source and observer move relative to each other. Discovered in 1842 by Christian Doppler and famously verified with trumpet players riding trains, it explains everything from why ambulance sirens change pitch as they pass to how radar guns measure car speeds, how Hubble proved the universe is expanding, and how Doppler ultrasound images blood flow inside arteries.

The intuition is geometric: when a source moves toward you, successive wavefronts are emitted closer together — higher frequency. When it moves away, wavefronts spread out — lower frequency. The same happens if you move toward or away from a stationary source. Mathematically, the observed frequency is the source frequency times a ratio that depends on the speeds of source, observer, and wave medium.

For sound, the formula treats source and observer motion asymmetrically because sound propagates through a medium (air). Light has no such medium — the special-relativistic Doppler effect depends only on the relative velocity of source and observer, not their motion through space. This calculator handles the classical sound case; for light at high speeds, a relativistic formula is needed.

Common applications: speed enforcement (police radar/lidar), weather radar (rotation of severe storms), medical imaging (blood flow), navigation (GPS Doppler), astronomy (galactic redshifts confirming expansion), and any physics problem involving moving sources or observers.

Formula

**General Doppler formula for sound:** f' = f × (v ± v_o) / (v ∓ v_s) Where: - f' = observed frequency (Hz) - f = source frequency (Hz) - v = speed of sound in medium (m/s) - v_o = observer speed (positive when moving *toward* source) - v_s = source speed (positive when moving *toward* observer) **Sign convention** (common: motion toward each other increases frequency): - Use + for v_o when observer approaches source. - Use − for v_s when source approaches observer (gives smaller denominator, larger f'). **Worked example: car approaching at 30 m/s** A car horn at 440 Hz approaches you at 30 m/s. Sound speed 343 m/s. You are stationary. f' = 440 × 343 / (343 − 30) = 440 × 343 / 313 = 482 Hz **Pitch rises by 42 Hz** (about a third of a semitone — clearly noticeable). As the car passes and moves away: f' = 440 × 343 / (343 + 30) = 440 × 343 / 373 = 405 Hz **Pitch drops by 35 Hz** (down a quarter-tone). Total shift across the pass: 77 Hz. **Four cases:** | Case | Formula | Direction | |---|---|---| | Observer toward stationary source | f' = f × (v + v_o)/v | f' > f | | Observer away from stationary source | f' = f × (v − v_o)/v | f' < f | | Source toward stationary observer | f' = f × v/(v − v_s) | f' > f | | Source away from stationary observer | f' = f × v/(v + v_s) | f' < f | **Both moving:** Use full formula. Numerator: v + v_o (toward) or v − v_o (away). Denominator: v − v_s (toward) or v + v_s (away). **Asymmetry of source vs observer motion:** - Source moving = wavefronts compressed/stretched. - Observer moving = you encounter wavefronts faster/slower. They give slightly different formulas. At low speeds (v_o, v_s << v) both reduce to: Δf/f ≈ Δv/v Where Δv is the closing speed. **Beat frequency for nearly equal sources:** When two sources have slightly different frequencies, you hear beats at: f_beat = |f₁ − f₂| Useful in tuning musical instruments. **Mach cone (source faster than wave speed):** If v_s > v, sound piles up into a shock wave. Sonic boom from supersonic aircraft. sin θ = v / v_s Where θ is half-angle of the Mach cone. Mach number: M = v_s / v. **Relativistic Doppler (for light or fast sources):** f' = f × √((1 − β)/(1 + β)) (recession) Where β = v/c. At low speeds reduces to classical f' ≈ f(1 − v/c). **Doppler in radar:** Radar bounces signal off moving target. Round trip doubles the shift: Δf = 2 × f × v / c Police radar at 24.15 GHz with car at 30 m/s: Δf = 2 × 24.15e9 × 30 / 3e8 = 4,830 Hz This shift is measured by mixing returned signal with original — gives target speed to ~0.5 mph accuracy. **Doppler ultrasound (blood flow):** Same formula. Transducer emits ~2-10 MHz, detects shift from moving red blood cells. Δf proportional to flow velocity. Used in cardiac, vascular, obstetric imaging. **Astronomical redshift:** z = Δλ/λ = v/c (low z) For z > 0.1, relativistic formula needed. Hubble's law: v = H₀ × d → distant galaxies recede faster, confirming cosmic expansion.

How to use this calculator

Enter source frequency in Hz (e.g., 440 for A4, 1000 for typical voice).
Enter source velocity (positive = approaching observer).
Enter observer velocity (positive = approaching source).
Speed of sound default 343 m/s (air at 20°C). Use 1,500 for water.
Calculator returns observed frequency f'.
For two passes (approach + recede), calculate both for total pitch range.

Worked examples

Ambulance siren

**Scenario:** Ambulance siren at 700 Hz drives toward a stationary observer at 25 m/s (≈56 mph). Then passes. **Calculation:** Approaching: f' = 700 × 343 / (343 − 25) = 700 × 343 / 318 ≈ 755 Hz. Receding: f' = 700 × 343 / (343 + 25) = 700 × 343 / 368 ≈ 653 Hz. **Result:** Pitch jumps from 755 to 653 Hz as it passes — about 100 Hz total drop, clearly audible. The familiar "neeeeooowww" pattern.

Police radar speed measurement

**Scenario:** Police radar at 24.15 GHz measures a car at 30 m/s. **Calculation:** Δf = 2 × f × v / c = 2 × 24.15e9 × 30 / 3e8 = 4,830 Hz. **Result:** ~4.83 kHz Doppler shift in the returned signal. Each kHz of shift corresponds to ~6.21 m/s (~13.9 mph). Modern radar guns measure shift to within ~0.5 Hz, giving speed to ~0.1 mph precision.

Galactic redshift

**Scenario:** Hydrogen alpha line normally at 656.3 nm is observed at 700.1 nm from a distant galaxy. Recession velocity? **Calculation:** z = Δλ/λ = (700.1 − 656.3) / 656.3 = 0.0667. At z < 0.1, v ≈ zc = 0.0667 × 3e5 km/s ≈ 20,000 km/s. **Result:** Galaxy receding at ~20,000 km/s. By Hubble's law (H₀ ≈ 70 km/s/Mpc), distance ≈ 290 Mpc ≈ 950 million light-years. Such observations of countless galaxies established cosmic expansion.

When to use this calculator

**Use Doppler calculations for:**

- **Sound problems**: sirens, vehicle pass-by frequency analysis. - **Speed enforcement**: police radar, lidar, traffic monitoring. - **Weather radar**: storm rotation, tornado detection, wind speed. - **Doppler ultrasound**: cardiac, vascular, obstetric medicine. - **Aircraft Doppler**: ground speed, weather avoidance. - **Astronomy**: galactic redshifts, stellar radial velocities, exoplanet detection. - **Sonar**: underwater speed measurement. - **GPS**: receiver location refinement via Doppler.

**Classical vs relativistic:**

- **Sound** (or any wave in a medium): classical formula with v of medium. - **Light**: must use relativistic formula, depends only on relative velocity. - **Low-speed approximation**: at v << c (or v << v_sound), both reduce to Δf/f ≈ Δv/c (or Δv/v).

**Sign conventions vary by textbook:**

Most common: motion toward gives + (frequency increases).

Some use vector notation: v positive in one direction, all velocities measured against that.

This calculator uses: positive velocities mean approaching motion (intuitive).

**Sound speed reference:**

| Medium | Speed (m/s) | |---|---| | Air (0°C) | 331 | | Air (20°C) | 343 | | Air (40°C) | 355 | | Helium (20°C) | 1,007 | | Hydrogen | 1,310 | | Fresh water | 1,481 | | Sea water | 1,531 | | Rubber | 60-1,500 | | Aluminum | 6,420 | | Steel | 5,960 | | Diamond | 12,000 |

**Common applications:**

- **Pitch perception**: musicians and physicists analyzing pass-by sounds. - **Vehicle classification**: train horns, helicopter rotors. - **Marine biology**: dolphin/whale vocalizations. - **Sports**: pitch tracking (radar gun on baseball). - **Wind energy**: turbine blade tip Doppler shift. - **Industrial flow meters**: ultrasonic flow measurement. - **Heart-rate fetal monitoring**: Doppler ultrasound detects fetal heartbeats.

**Doppler in special relativity:**

For light (no medium): f' = f × √((1 − β)/(1 + β)) (recession) f' = f × √((1 + β)/(1 − β)) (approach)

Where β = v/c. At β → 1, frequency goes to 0 (recession) or infinity (approach).

Includes "transverse Doppler" (purely sideways motion still affects frequency — relativistic time dilation).

**Mach number context:**

- M < 0.8: subsonic. - 0.8 ≤ M < 1.0: transonic. - M = 1.0: speed of sound. - 1.0 < M < 5.0: supersonic. - M ≥ 5.0: hypersonic.

Supersonic aircraft create cone-shaped shock waves heard as sonic booms when they reach the ground.

**Software:**

- **MATLAB**: signal processing for Doppler analysis. - **Python (scipy.signal)**: frequency analysis of recordings. - **Sonic visualizer**: free audio spectrogram tool. - **Radar simulation tools**: MathWorks Phased Array Toolbox.

**Pitfalls:**

- **Sign convention errors**: every textbook has its own — be explicit. - **Source vs observer motion**: different formulas in classical (sound). - **Speed of sound**: depends on temperature, humidity, altitude. - **Approximation breakdown**: at high speeds, full formula required. - **Direction**: only the component of velocity along the line connecting source and observer matters. - **Mach cone**: at v_s > v, classical formula breaks down. - **Forgetting double-shift in radar**: round trip doubles Δf.

Common mistakes to avoid

Mixing sign conventions for source vs observer motion.
Forgetting that only the velocity component along the line of sight matters.
Applying classical Doppler to light at relativistic speeds.
Using wrong speed of sound for the medium or temperature.
Forgetting the round-trip doubles the radar Doppler shift.
Confusing source velocity with observer velocity (they're asymmetric in sound Doppler).
Ignoring transverse component in 2D/3D motion.
Assuming pitch change is constant during a pass (it changes rapidly near closest approach).

Frequently Asked Questions

Sources & further reading

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Doppler Effect Calculator

Calculate the observed frequency due to the Doppler effect for sound waves. Accounts for both source and observer motion relative to the medium. Uses f′ = f × (v ± v_o) / (v ∓ v_s).

Inputs

Source Frequency (Hz)

Source Velocity (m/s)

Positive = approaching observer

Observer Velocity (m/s)

Positive = approaching source

Speed of Sound (m/s)

Results

Observed Freq

478.48 Hz

Shift

+38.48 Hz

Change

+8.7%

Doppler Effect Results

Parameter	Value
Source Frequency	440 Hz
Source Velocity	0 m/s (0.0 km/h)
Observer Velocity	30 m/s (108.0 km/h)
Speed of Sound	343 m/s
Observed Frequency	478.48 Hz
Frequency Shift	+38.48 Hz (+8.75%)
Source Wavelength	0.7795 m
Observed Wavelength	0.7168 m
Formula	f' = f(v + v_o)/(v - v_s)