AC-07Acoustic Fundamentals

Doppler Shift & Pass-by

Frequency shift for a moving source or observer - with an audible pass-by simulation.

Inputs

Source frequency f₀

Source speed v_s

m/s

108 km/h · 67 mph

Observer speed v_o (towards source)

m/s

Closest pass-by distance

Approaching

482.2Hz

Receding

404.6Hz

Pitch drop at pass

3.04semitones

Mach number

0.087

subsonic

Wavefront patternlive

Pass-by frequencystationary observer, d = 10 m

§ Theory of operation

The Doppler effect

When a source moves towards a listener, each successive wavefront is emitted closer than the previous one, so crests arrive more often: the perceived frequency rises. Moving away, wavefronts are stretched and pitch drops. For motion along the line between source and observer in still air:

\begin{aligned} f' &= f_0 \cdot \dfrac{c + v_o}{c - v_s} \qquad \text{approaching} \\ f' &= f_0 \cdot \dfrac{c - v_o}{c + v_s} \qquad \text{receding} \end{aligned}

with $v_s$ the source speed and $v_o$ the observer speed (positive towards each other in the first line). The effect is asymmetric: a source moving at Mach 0.5 raises pitch by an octave approaching but lowers it by only a fifth receding.

Pass-by signature

For a vehicle passing at closest distance d, only the velocity component along the line of sight matters, so the frequency glides smoothly through f₀ at the moment of closest approach:

f(t) = f_0 \cdot \dfrac{c}{c - v_s \cdot \cos\theta(t)}, \qquad \cos\theta = \dfrac{-x}{\sqrt{x^2 + d^2}}

The closer the pass, the more abrupt the glide. This pass-by curve is used in railway and road noise to separate tonal components and estimate vehicle speed. (The simulation ignores the small propagation-delay correction.)

Sonic boom

When $v_s \ge c$ the source outruns its own wavefronts; they pile up into a conical shock (the Mach cone, half-angle $\sin\alpha = 1/M$ ). Ahead of the source nothing is heard until the cone sweeps past. Raise the source speed above 343 m/s to see it.