Doppler Effect Calculator

The Doppler effect, discovered by Austrian physicist Christian Doppler in 1842, describes the perceived change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source. In classical acoustics, where sound propagates through a physical medium, the apparent frequency differs depending on whether it is the wave source, the observer, or both that are in motion relative to the stationary fluid.

For sound waves propagating with speed c through a still medium, the relationship between emitted frequency f_s and observed frequency f_o is given by the classical Doppler equation: f_o = f_s · (c + v_o) / (c - v_s), where velocities are defined as positive when approaching one another and negative when receding. When an ambulance siren approaches a stationary listener, wavefronts are compressed in the forward direction of motion, leading to a shorter wavelength and higher perceived pitch. Once the siren passes and moves away, wavefronts are stretched, reducing frequency and dropping perceived pitch.

In contrast to sound, electromagnetic radiation travels through the vacuum of space at the universal speed of light c without requiring a physical medium. According to Einstein's special theory of relativity, optical Doppler shifts must incorporate relativistic time dilation (γ Lorentz factor), yielding the relativistic longitudinal Doppler equation: f_o = f_s · √((1 - β) / (1 + β)), where β = v / c for a receding source. Astronomers use this redshift parameter z = (λ_o - λ_s) / λ_s to measure galaxy recession velocities and map the cosmic expansion of the universe via Hubble's law.

In high-speed acoustics, when a source velocity v_s equals or exceeds the medium's sound speed c (Mach number M ≥ 1), the Doppler equation approaches a mathematical singularity, representing the coalescence of wavefronts into a conical acoustic shock wave (Mach cone). The semi-apex angle of the shock cone is given by μ = arcsin(1 / M). This calculator provides seamless computation across both acoustic and relativistic physics regimes.