Oscillations and waves

The Doppler effect

Relative motion between a wave source and observer shifts the received frequency; for sound in a stationary medium, approach raises the observed frequency.

An ambulance siren sounds higher as it approaches and lower as it recedes. Wavefronts are compressed ahead of the moving source and spread behind it; sound speed in the medium is still set by the medium. The effect is named for physicist Christian Doppler, who proposed a shift in the color of moving stars; for sound, the analogous observation is a frequency shift.

Moving source, stationary observer

For a source approaching a stationary listener, $f_{ m obs}=fv/(v-v_s)$; receding gives $f_{ m obs}=fv/(v+v_s)$. Emitted frequency stays fixed while spacing of wavefronts in the medium changes. A moving listener changes the numerator too.

Quick check

The acoustic Doppler shift depends on motion relative to the medium, not merely the instantaneous source–listener distance. An approaching source shortens the wavelength ahead; a moving listener in a stationary medium encounters more wavefronts each second. Resolve source and observer velocities along their line of sight separately, then apply the sign convention rather than combining speeds indiscriminately.

When applying a model, state which quantities are held fixed and choose the appropriate reference frame. Keeping units beside numerical values helps prevent confusion between quantities or between measured and predicted values. After calculating, check dimensions, signs, and limiting cases: the result should fit the assumptions, such as small oscillations, a uniform medium, or negligible resistance. If real conditions differ, explain the discrepancy instead of applying a formula mechanically.

A useful physical check is to substitute special cases into the relation: equilibrium, maximum displacement, one full cycle, or no relative motion. These cases often make a quantity vanish or reach an extreme, exposing sign errors and confusion between period and frequency. When comparing experiments, change one condition at a time so that the cause of a changed result is clear.