Relativistic Doppler Shift Calculator for Light

Doppler effect: f' = f·(v_sound ± v_obs)/(v_sound ∓ v_source)

When a source and an observer move relative to one another, the frequency you actually hear shifts. That is the Doppler effect, which Christian Doppler first described in 1842. For sound, you work it out with f' = f·(v_sound ± v_obs)/(v_sound ∓ v_source), taking the upper signs whenever something is moving toward the other party and the lower signs when it pulls away. Think of an ambulance siren. On the approach the wavefronts bunch up and the pitch climbs; once it has gone by, they spread out and the pitch sags. A 1 kHz siren coming at you at 30 m/s, with v_sound = 343 m/s, lands near 1096 Hz, and once it passes you it falls to roughly 920 Hz. Light follows the relativistic Doppler factor √((1+β)/(1-β)) instead, which is how astronomers read galaxy redshifts and how Hubble confirmed cosmic expansion back in 1929.

Applications

Police speed guns lean on it, reading the shift in the reflected wave. So does obstetric and cardiac ultrasound, where Doppler imaging shows blood flow, and weather radar, which picks up how fast precipitation is moving. Sonar uses it. So does astronomy: Doppler spectroscopy turned up the first exoplanet around a Sun-like star (51 Pegasi b, 1995), and the radial velocities of stars let us map how the Milky Way rotates.

FAQ

Why does the pitch drop suddenly as the ambulance passes? Because the radial velocity relative to you flips its sign almost instantly. Within a fraction of a second, approaching becomes receding.

Does the formula work for light too? Only at low velocities, as a rough approximation. Once v gets close to c you need the relativistic Doppler formula, which folds in time dilation as well.

What is redshift? It is a Doppler shift toward longer wavelengths that happens when a source is moving away from you. With cosmological redshift, though, the real driver is the expansion of space itself rather than ordinary motion.