= Fizeau experiment worked example
{c}
Choose $\ell=1.50\,\mathrm m$, $n=4/3$, $\lambda_0=550\,\mathrm{nm}$ and $u=6.00\,\mathrm{m\,s^{-1}}$, with $c=299792458\,\mathrm{m\,s^{-1}}$. These are illustrative choices, not measured values or a reconstruction of <Hippolyte Fizeau>'s data. Each beam traverses $L=3.00\,\mathrm m$ of <water>. The nondispersive prediction is
$$
N=\frac{4(1.50)(6.00)(7/9)}{(299792458)(550\times10^{-9})}\simeq0.16981,\qquad
2N\simeq0.33963,\qquad
\Delta t\simeq3.1154\times10^{-16}\,\mathrm s.
$$
The first number is the offset from rest to one flow setting; the second compares equal opposite settings. The very short equivalent delay explains why <interference> is useful: a measurable fraction of a <wavelength> replaces an unattainable direct timing measurement.
\Image[/fizeau-fringe-shift-versus-flow.png]
{title=Calculated fringe offsets for chosen parameters: partial drag gives 0.16981 fringe at 6 m/s, whereas equal opposite flows differ by 0.33963 fringe}
{id=image-fizeau-fringe-shift-versus-flow}
{description=Original model curves for tubes of length $\ell=1.50\,\mathrm m$, <refractive index> $n=4/3$ and vacuum <wavelength> $\lambda_0=550\,\mathrm{nm}$. Solid curves show the first-order rest-to-flow offset and the offset between reversed settings; the dashed curve is the rest-to-flow prediction from ordinary addition of $c/n$ and $u$. No points represent experimental measurements, and <optical dispersion> is omitted.}
The <Galilean transformation> would give <phase velocity> magnitudes $c/n\pm u$, hence $N_{\rm full}\simeq4\ell u n^2/(c\lambda_0)$. At the chosen values this is $0.38815$ fringe from rest, larger than the <special relativity> prediction by $n^2/(n^2-1)=16/7$. A model with no flow-induced change of <phase velocity> instead gives $N=0$. The three alternatives are experimentally distinguishable even when $u/c$ is tiny.
<Hippolyte Fizeau>'s positive displacement favoured <Augustin Jean Fresnel>'s partial-drag law over zero or full drag. He explicitly distinguished support for that law from establishing its underlying <luminiferous aether> hypothesis.<The Hypotheses Relating to the Luminous Aether by Hippolyte Fizeau>[Comparison of the three predictions and final paragraph qualifying the inference about Fresnel's hypothesis; scanned translation pp. 572–573]{ref} The <Fizeau experiment> therefore supplies a nonzero relative-motion measurement, not a null test proving an <luminiferous aether>'s absence. The derivation above exhibits the same first-order law within <special relativity>, showing why agreement with the coefficient alone cannot select an physical interpretation.
Two limitations matter for precision. The <velocity profile> sampled by the beam need not equal the cross-sectional average inferred from <volumetric flow rate>. Also, the <Doppler effect> gives opposite rest-frame <frequencies> for the counterpropagating beams. With <optical dispersion>, the coefficient $n^2-1$ in the <interference fringe> formula becomes
$$
n^2-1+n\omega\frac{dn}{d\omega}
=n^2-1-n\lambda_0\frac{dn}{d\lambda_0},
$$
where the <refractive index> and its derivative are evaluated for stationary <water> at the laboratory <angular frequency> $\omega=2\pi c/\lambda_0$. A wavelength-independent $n$ therefore defines an approximation, not a precision fit to real <water>.<Fizeau’s “Aether-Drag” Experiment in the Undergraduate Laboratory by Thierry Lahaye and others>[Section III, p. 499, equation (7), dispersion from opposite Doppler shifts; section IV.B, pp. 499–500, mean flow versus velocity at the beam]{ref} Flow reversal removes a fixed <phase difference>, but a mechanical or <temperature> change that reverses with the flow can still imitate the signal and must be controlled.
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