= Solution
Take the light to travel along positive x, while the water moves with signed <velocity> $-v$. In the water's rest frame its light <phase velocity> is $c/n$. The <relativistic velocity addition> law from <5/a> gives the laboratory <phase velocity>
$$
w=\frac{c/n-v}{1-v/(nc)},\qquad
n'=\frac cw=n\frac{1-v/(nc)}{1-nv/c}.
$$
Expanding to first order in $v/c$, with $nv/c\ll1$, yields
$$
\boxed{n'\simeq n\left(1+\frac{nv}{c}-\frac{v}{nc}\right).}
$$
Equivalently $w\simeq c/n-v(1-1/n^2)$: the counterflow reduces the forward light <speed>. This is the nondispersive <Fresnel drag coefficient>. The <refractive index> is taken as constant over the relevant frequencies; <dispersion> and its frequency-shift corrections are outside this approximation.
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