Doppler Effect of Sound

The Doppler effect is the change in the frequency of a wave as perceived by an observer, occurring whenever there is relative motion between the source of the wave and the observer along the line joining them (or, more generally, whenever there is relative motion between source, observer, and the medium). Qualitatively, the perceived (apparent) frequency rises above the source's own (true) frequency whenever source and observer are approaching each other, and falls below it whenever they are receding from each other -- the everyday example being the familiar rise, then sudden drop, in the pitch of a vehicle's horn or a train's whistle as it passes a stationary listener. Physically, motion changes the rate at which successive wavefronts (crests of compression) are received, even though the source itself keeps emitting at its own fixed, unchanged frequency in its own reference frame: a source moving towards a listener effectively crowds its successive wavefronts closer together ahead of it (compressing the wavelength on that side), while a listener moving towards a source intercepts wavefronts at a faster rate than a stationary listener would. Taking $v$ as the speed of sound in the (stationary) medium, $f_0$ as the source's true frequency, $v_s$ as the speed of the source along the line joining source and observer (taken positive when the source moves TOWARD the observer), and $v_o$ as the speed of the observer along the same line (taken positive when the observer moves TOWARD the source), the general formula f …