  The geometrical-optics expansion reduces the problem of solving wave
equations to one of solving transport equations along rays. Here we consider
scalar, electromagnetic and gravitational waves propagating on a curved
spacetime in general relativity. We show that each is governed by a wave
equation with the same principal part. It follows that: each wave propagates at
the speed of light along rays (null generators of hypersurfaces of constant
phase); the square of the wave amplitude varies in inverse proportion to the
cross section of the beam; and the polarization is parallel-propagated along
the ray (the Skrotskii/Rytov effect). We show that the optical scalars for a
beam, and various Newman-Penrose scalars describing a parallel-propagated null
tetrad, can be found by solving transport equations in a second-order
formulation. Unlike the Sachs equations, this formulation makes it
straightforward to find such scalars beyond the first conjugate point of a
congruence, where neighbouring rays cross, and the scalars diverge. We discuss
differential precession across the beam which leads to a modified phase in the
geometrical-optics expansion.
