= Classical Doppler effect for sound
{title2=$f_o=f_s(c-u_o)/(c-u_s)$}
Work in the material's rest <reference frame>, with constant <speed of sound> $c>0$. For one-dimensional propagation, choose the positive $x$ direction along the travelling <wave> from its emission point toward the receiver. Define $u_s=dx_s/dt$ and $u_o=dx_o/dt$ as the signed source and receiver <velocities> along that same axis. Both use the same sign convention: positive $u_s$ moves the source toward the receiver ahead, whereas positive $u_o$ moves that receiver away from the incoming <wave>. Assume $|u_s|<c$ and $|u_o|<c$, constant <velocities>, and no source-receiver crossing during the interval considered. Then
$$
f_o=f_s\frac{c-u_o}{c-u_s}.
$$
To derive this, let $T_s=1/f_s$ be the emission <wave period>. During $T_s$, one <wave crest> travels $cT_s$ while the source advances $u_sT_s$. Successive <wavefronts> along the chosen direction therefore have <wavelength>
$$
\lambda=(c-u_s)T_s=\frac{c-u_s}{f_s}.
$$
They overtake the receiver at relative speed $c-u_o$, so counting <wave crests> gives $f_o=(c-u_o)/\lambda$. This separates the source's effect on <wavelength> from the receiver's effect on crossing rate. Even with a moving source, a stationary receiver obeys $f_o\lambda=c$ because the <wave speed> through the material remains $c$.
For a stationary receiver and a source approaching at speed $v$, set $u_o=0$ and $u_s=v$: $f_o=f_sc/(c-v)>f_s$. For recession choose the positive axis along the departing <wave> toward the receiver again, so the source has $u_s=-v$: $f_o=f_sc/(c+v)<f_s$. For the deliberately chosen values $c=343\,\mathrm{m\,s^{-1}}$, $v=20\,\mathrm{m\,s^{-1}}$ and $f_s=500\,\mathrm{Hz}$, the two predictions are
$$
f_{\mathrm{approach}}=500\frac{343}{323}\,\mathrm{Hz}\simeq530.96\,\mathrm{Hz},\qquad f_{\mathrm{recede}}=500\frac{343}{363}\,\mathrm{Hz}\simeq472.45\,\mathrm{Hz}.
$$
The corresponding <wavelengths> are $0.646\,\mathrm m$ and $0.726\,\mathrm m$, compared with $c/f_s=0.686\,\mathrm m$ for a stationary source. These are calculations for chosen parameters, not reported measurements.
\Image[/doppler-source-wavefront-spacing.png]
{title=Moving-source <wavefront> spacing in the <classical Doppler effect for sound>}
{id=doppler-source-wavefront-spacing}
{description=Original <linear acoustics> model at $t=0$, with a deliberately enlarged speed ratio $v/c=0.4$ for readability. Lengths are measured in $\lambda_0=c/f_s$. Successive <circles> are sections of spherical <wavefronts> emitted one <wave period> apart; their centres stay at the old emission positions. The forward spacing is $0.6\lambda_0$ and the rear spacing $1.4\lambda_0$. These chosen parameters differ from the $20\,\mathrm{m\,s^{-1}}$ worked example.}
A receiver approaching a stationary source at $20\,\mathrm{m\,s^{-1}}$ instead has $u_s=0$, $u_o=-20\,\mathrm{m\,s^{-1}}$, giving $f_o=500(363/343)\,\mathrm{Hz}\simeq529.15\,\mathrm{Hz}$. Moving away gives $f_o=500(323/343)\,\mathrm{Hz}\simeq470.85\,\mathrm{Hz}$. The equal closing speeds do not make these equal to the moving-source values. If both source and receiver move together with $u_s=u_o$, however, the numerator and denominator cancel and $f_o=f_s$. For speeds small compared with $c$, the first-order shift is $(f_o-f_s)/f_s\simeq(u_s-u_o)/c$; the difference between moving-source and moving-receiver experiments appears beyond this first order.
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