= Fresnel drag coefficient
{c}
{title2=$f=1-n^{-2}$}
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 $v'=c/n$. Dividing the <Lorentz transformation> differentials along that surface gives the same longitudinal <velocity-addition formula> as for a particle:
$$
v_{\rm lab}=\frac{c/n+u}{1+u/(nc)}
=\left(\frac cn+u\right)\left(1-\frac{u}{nc}+O\left(\frac{u^2}{c^2}\right)\right)
=\frac cn+u\left(1-\frac1{n^2}\right)+O\left(\frac{u^2}{c}\right).
$$
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.<Fizeau’s “Aether-Drag” Experiment in the Undergraduate Laboratory by Thierry Lahaye and others>[Section III, pp. 498–499, equation (1) and discussion of the collinear wave-four-vector derivation]{ref}
For $n=4/3$, the <Fresnel drag coefficient> is $f=7/16$. A chosen flow <speed> of $6\,\mathrm{m\,s^{-1}}$ therefore changes the forward <phase velocity> by approximately $2.625\,\mathrm{m\,s^{-1}}$, even though the stationary-medium <phase velocity> is about $2.25\times10^8\,\mathrm{m\,s^{-1}}$. These are model values. The invariant vacuum <speed of light> remains $c$; $c/n$ describes propagation through matter in its own <rest frame>.
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