An ambulance goes past and the siren drops in pitch, though it plays one note the whole time.

Head on#

The source makes a crest, and that crest at once loses all connection with the source. It spreads as a circle centred on the spot where it was made, at the speed of the air. Meanwhile the source moves on and makes the next one somewhere else.

Where each crest is born#

Each circle is centred where the source was when it was emitted, and grows at the speed of sound $v$. In one period the source advances $v_s T_0$ while its last crest advances $v T_0$, so the gap ahead is short by the distance the source covered:

$$\lambda_+ = \frac{v - v_s}{f_0} \qquad\text{and behind it}\qquad \lambda_- = \frac{v + v_s}{f_0}$$

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The frequency an observer hears#

Crests reach the observer $\lambda_\pm / v$ seconds apart, so the heard frequency is $v/\lambda_\pm$. For a source and an observer both moving along the line joining them,

$$f = f_0\,\frac{\vel{v} + \vel{v_o}}{\vel{v} - \vel{v_s}}$$

with $v_s$ positive when the source moves toward the observer and $v_o$ positive when the observer moves toward the source.

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Passing beside you#

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When the source outruns its own sound#

Push past Mach 1 and the source reaches each new emission point before its last crest does. The circles still grow at $v$ about fixed centres, so they pile up along a cone trailing the source, whose half-angle satisfies

$$\sin\theta = \frac{v}{v_s} = \frac{1}{M}$$

where $M$ is the Mach number.

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What comes next#

You heard a horn hold one pitch head on and sweep beside you, and you set a source fast enough to outrun its sound. Every step of this argument used two things: a wave that travels at a speed set by its medium, and a source that keeps moving between emissions. Take away the medium and most of it survives. Light from a receding galaxy arrives stretched toward the red, and the shift is read as a speed with a closely related formula. The details differ because light needs no medium and its speed is the same for every observer, but the picture of crests laid down one at a time by something that keeps moving carries straight over.