# Doppler Effect Simulator - Sound, Mach Cone, Redshift & Radar

> Doppler effect calculator with animation: moving source and observer, Mach cone and sonic boom, relativistic redshift, radar speed guns and pass-by audio.

- URL: https://qilingtools.com/en/math/doppler
- Category: Physics Tools (https://qilingtools.com/en/physics)
- Tags: [Physics](https://qilingtools.com/en/tags/physics), [Education](https://qilingtools.com/en/tags/education)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Watch a moving source send out circular wavefronts that bunch up ahead and spread out behind, and drag the listener anywhere to see the frequency it hears right now. Push past the speed of sound and the wavefronts pile into a Mach cone with its angle marked. The tool covers classical sound (moving source and observer, speed of sound from air temperature), the relativistic Doppler effect for light (redshift z, shifted spectral lines and colours, a Hubble-law distance) and the two-way shift of a police radar — and it can play the falling pitch of an ambulance driving past.

## How to use

1. Pick a preset (ambulance, train horn, supersonic jet, galaxy redshift, radar gun) or open the Sound, Light or Radar panel
2. Enter the source frequency, air temperature and the speeds and directions of source and observer — or a spectral line with a radial velocity or redshift
3. Drag the observer in the animation and read the heard frequency, wavelengths ahead and behind, Mach number, or the shifted wavelength and colour
4. Press the pass-by button to hear the pitch drop as the source goes by

## Features

- Animated wavefronts from a moving source, with a draggable (and optionally moving) observer and a live heard frequency
- Mach cone and Mach angle above the speed of sound, plus the delay before the sonic boom arrives
- General formula f′ = f(c + vₒ)/(c − vₛ) with the numbers plugged in, speed of sound from temperature, and the change in percent and semitones
- Exact pass-by curve f(t) using retarded time, playable with Web Audio including the loudness swell
- Relativistic Doppler: β, γ, z, observed wavelength, transverse Doppler, shifted Balmer lines and colour swatches
- Radar speed gun: speed ↔ frequency shift with cosine-error correction

## FAQ

### Why does an ambulance siren drop in pitch as it passes?

While the source approaches, each crest is emitted a little closer to you than the last, so the waves ahead are squeezed: shorter wavelength, higher frequency. Moving away, they’re stretched and the pitch falls. With f′ = f·c/(c ∓ vₛ), a 960 Hz siren at 90 km/h is heard at about 1035 Hz approaching and 895 Hz receding — more than two semitones apart. At the exact moment of passing, when the motion is perpendicular to the line of sight, you hear the true frequency.

### Is a moving source the same as a moving observer?

Not for sound, because sound travels through air. A source approaching at speed v gives f′ = f·c/(c − v); an observer approaching a still source at the same speed gets f′ = f·(c + v)/c. For small v they agree, but near the speed of sound they differ hugely: a source at Mach 1 produces a shock (the formula blows up), whereas an observer at Mach 1 just hears double the frequency. Light has no medium, so only the relative velocity matters and the relativistic formula applies.

### What are the Mach cone and the sonic boom?

When a source moves faster than sound (Mach number M = v/c > 1) it outruns its own wavefronts, and their envelope forms a cone with half-angle sinθ = 1/M — 30° at Mach 2. The pressure jumps sharply across that cone, and when it sweeps over the ground you hear a boom. Before the cone arrives there is silence, which is why a supersonic jet passes overhead before you hear it.

### How is redshift related to velocity?

Redshift is z = (λobserved − λ₀)/λ₀. For motion along the line of sight, the relativistic Doppler formula gives 1 + z = √((1 + β)/(1 − β)) with β = v/c, and for small speeds z ≈ v/c. At z = 0.05 the 656.28 nm Hα line moves to about 689 nm. The redshift of distant galaxies is mostly due to the expansion of space rather than motion through it; for small z, Hubble’s law d ≈ cz/H₀ estimates the distance. Andromeda is blueshifted because it approaches us at about 300 km/s.

### How does a radar speed gun measure speed?

The gun sends out microwaves; the car receives them as a moving observer and reflects them as a moving source, so the echo is shifted twice: Δf ≈ 2vf/c. For a 24.15 GHz K-band gun, a car at 100 km/h shifts the echo by about 4.5 kHz — roughly 44.7 Hz per km/h. If the beam isn’t aimed straight along the car’s path, the gun measures only the component v·cosφ and reads low (the cosine error).

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