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Choose $\ell =1.50\mathrm{m}$, $n=4/3$, ${\lambda }_0=550\mathrm{nm}$ and $u=6.00\mathrm{m}{\mathrm{s}}^{-1}$, with $c=299792458\mathrm{m}{\mathrm{s}}^{-1}$. 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 isThe 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.

The Galilean transformation would give phase velocity magnitudes $c/n\pm u$, hence $N_{\mathrm{full}}\simeq 4\ell u{n}^2/(c{\lambda }_0)$. 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.^[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 becomeswhere 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.^[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.

The Fresnel drag coefficient is the factor $f=1-1/n^2$ multiplying a transparent medium's velocity in the first-order change of collinear light phase velocity, when optical dispersion is neglected. Here $n$ is the medium's refractive index in its rest frame. The name recalls Augustin Jean Fresnel's partial-drag law; special relativity obtains the same coefficient from the velocity-addition formula without introducing a luminiferous aether.

Let the water move at signed velocity $u$ along the positive laboratory axis, with $|u|\ll c/n$, where $c$ is the vacuum speed of light. Assume a transparent medium with constant $n>1$, uniform flow and negligible optical dispersion. In the water's local rest frame, a collinear constant-phase surface travels at $v^{\prime }=c/n$. Dividing the Lorentz transformation differentials along that surface gives the same longitudinal velocity-addition formula as for a particle:The correction comes from the transformed time denominator. Simply adding $u$ to $c/n$ would omit it. This argument concerns collinear phase velocity; in a medium with optical dispersion, group velocity is a different quantity.^[ref]

For $n=4/3$, the Fresnel drag coefficient is $f=7/16$. A chosen flow speed of $6\mathrm{m}{\mathrm{s}}^{-1}$ therefore changes the forward phase velocity by approximately $2.625\mathrm{m}{\mathrm{s}}^{-1}$, even though the stationary-medium phase velocity is about $2.25\times 10^8\mathrm{m}{\mathrm{s}}^{-1}$. These are model values. The invariant vacuum speed of light remains $c$; $c/n$ describes propagation through matter in its own rest frame.

In 1913 Georges Sagnac mounted a lamp, collimator, beam splitter, four mirrors and photographic plate on a horizontal rotating platform. The mirrors defined a closed polygon; the two beams traversed it in opposite senses. A slight beam splitter adjustment produced visible interference fringes. A motor brought the platform to steady rotation, and switching the lamp exposed the plate. Comparing opposite senses of rotation revealed a reversing displacement. Changing the beam splitter adjustment supplied a further check against optical deformation. The report compared measured and calculated displacements. Georges Sagnac interpreted this as evidence for a luminiferous aether; the relativistic derivation below reproduces the delay without that hypothesis.^[ref]

The readout measures a relative wave phase, not two stopwatch readings separated by an optical transit time. Reversal isolates a contribution odd in angular velocity: if a signed interference fringe position is $q(\Omega )=q_0+k\Omega$ to leading order, then $[q(\Omega )-q(-\Omega )]/2=k\Omega$ cancels the fixed offset $q_0$. Changes of alignment or temperature correlated with reversal can survive this subtraction. Thus the experimental inference concerns a reproducible rotating-loop phase difference, with controls for disturbances; it is not by itself a unique test of an underlying propagation hypothesis.