DOPPLER
Same speed, shorter spacing — pitch shift is geometry, not magic.
Use the simulation above to change the variables and play through the guided stages. The explanation below describes the default starting values; the simulation updates its explanation as you experiment.
Setup
A source hums one steady note and moves. An observer waits — or moves too. Everyone with an ear near a train track knows the answer changes; the question is by exactly how much, and why.
Waves
The medium enforces one law: every wavefront expands at c from wherever it was born, indifferent to the motion of its parent. A moving source simply births each crest somewhere new — crowding them ahead, stretching them behind. The rings tell the whole story before any algebra.
Solve
Squeeze the wavelength: λ = (c−v_s)/f ahead of the source. The observer meets those crests at speed c+v_o, and the ratio is the famous formula. When v_s reaches c the spacing hits zero — and past it, the equation politely declines to answer.
Propagate
Honest slow motion, every Nth crest drawn. Watch two things: the rings never change speed, and the source barely gains on its own sound — that is how fast sound is, and why outrunning it earns a named number. Each violet flash is a crest arriving at the ear.
Audit
The ledger: shifted pitch is unchanged speed over changed spacing — mechanism, not magic. Moving source and moving observer at the same speed disagree, because the air picks a reference frame; the fact that light refuses to show this asymmetry started twentieth-century physics.