Chapter 38, Problem 46AP

The motion of a transparent medium influences the speed of light. This effect was first observed by Fizeau in 1851. Consider a light beam in water. The water moves with speed v in a horizontal pipe. Assume the light travels in the same direction as the water moves. The speed of light with respect to the water is c/n. where n = 1.33 is the index of refraction of water. (a) Use the velocity transformation equation to show that the speed of the light measured in the laboratory frame is

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

(b) Show that for v << c, the expression from part (a) becomes, to a good approximation,

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

(c) Argue for or against the view that we should expect the result to be u = (c/n) + v according to the Galilean transformation and that the presence of the term −v/n² represents a relativistic effect appearing even at “nonrelativistic’’ speeds. (d) Evaluate u in the limit as the speed of the water approaches c.