9.1 The Doppler shift — kinematic derivation

A stationary observer hears a sound. The source emits the same waveform but is moving — say, approaching the observer at velocity $v_s$. The sound the observer hears is shifted to a higher pitch. This is the Doppler shift, the most familiar everyday consequence of motion in acoustics.

The kinematics

The source emits successive wavecrests separated in time by the source’s period $T_s$. Between two emissions, the source itself moves a distance $v_s T_s$ closer to the observer. So the wavecrests in transit are compressed in space: their spacing is

The wave still travels at speed $c$ (the medium doesn’t care about the source), so the period at which crests arrive at the stationary observer is

The observed frequency is the reciprocal:

For a receding source the sign flips: $f_o = f_s / (1 + v_s/c)$. For $v_s \ll c$, both reduce to $f_o \approx f_s (1 \pm v_s/c)$ — fractional shift equal to Mach number. A car at 30 m/s has Mach 0.09, giving a 9% pitch shift between approach and recession.

When the observer moves

If the observer moves with velocity $v_o$ toward a stationary source, the wave reaches them faster:

(Note the linear form: when the observer moves, the wavelength doesn’t change — the observer just intercepts more crests per second.)

When both source and observer move (in the same direction, with the source ahead of and approaching the observer or whatever the geometry is), the combined formula is

with the sign conventions: $v_o$ positive if observer moves toward source, $v_s$ positive if source moves toward observer.

Asymmetric in source vs observer

Notice that the formula is not symmetric in source motion and observer motion. Source motion changes the wavelength; observer motion changes the rate of crest interception. The two differ at second order in $v/c$ — for small Mach numbers they’re indistinguishable, but for large Mach numbers (the source approaching the speed of sound), source motion is the dominant effect.

This asymmetry is what makes the Doppler shift in sound different from the relativistic Doppler shift in light — light has no preferred frame, so the formula is symmetric and depends only on relative velocity. Acoustic Doppler is fundamentally three-frame: there is a source, an observer, and a medium, each potentially moving in its own way.

What about a moving medium?

If the medium (the air) is itself moving — wind blowing from source toward observer — the effective speed of sound in the laboratory frame is $c + v_\text{wind}$. The Doppler formula in this case generalises but is still derivable from the same kinematic reasoning. The key is to be clear about which frame you are in.

The proper way to do all this is to derive the wave equation in a frame attached to the moving medium, then transform — which is the topic of the next lesson.

Examples

• Car passing. A car horn at 500 Hz approaching at 30 m/s: $f_o = 500 / (1 - 30/343) \approx 548$ Hz. Receding: $500/(1 + 0.087) \approx 460$ Hz. The drop from 548 to 460 happens over a small spatial interval near closest approach — about 1.7 semitones (a “minor third” drop).

• Police radar. Doppler is used to measure speed: $\Delta f / f = 2 v / c$ (the factor of 2 because the radar pulse reflects from the moving target). For visible light $c = 3 \times 10^8$ m/s, but the same formula form. Acoustic Doppler is used in medical ultrasound for blood-flow velocity measurements.

• Bat echolocation. Bats use Doppler shifts to determine the relative velocity of their prey. Their auditory systems are tuned to compensate for the Doppler shift caused by the bat’s own motion, so they can decode the residual shift from a moving target.

Doppler is the gentlest of the moving-media phenomena. The wave equation itself doesn’t change shape — it’s just being applied in a non-stationary frame. Next lesson, we look at what happens to the wave equation when the medium itself is in steady motion.