Chapter 1, Problem 22P

Speed of light in a moving medium. The motion of a medium such as water influences the speed of light. This effect was first observed by Fizeau in 1851. Consider a light beam passing through a horizontal column of water moving with a speed v. (a) Show that if the beam travels in the same direction as the flow of water, the speed of light measured in the laboratory frame is given by

$u=\frac{c}{n}\left(\frac{1+nv/c}{1+v/nc}\right)$

where n is the index of refraction of the water. (Hint: Use the inverse Lorentz velocity transformation and note that the speed of light with respect to the moving frame is given by c/n.) (b) Show that for v << c, the preceding expression is in good agreement with Fizeau’s experimental result:

$u\approx \frac{c}{n}+v-\frac{v}{n^2}$

This proves that the Lorentz velocity transformation and not the Galilean velocity transformation is correct for light.