The mean round-trip speed, as in simple harmonic motion, is Galilean invariant. There are two senses in which this is the case: (1) the time is the same in both directions, and (2) the distance covered is the same in both directions. In the first case, the mean round-trip speed equals the arithmetic mean of the two speeds. In the second case, the mean round-trip speed equals the harmonic mean of the two speeds. In both cases, the mean round-trip speed is Galilean invariant.

The mean round-trip speed in the first case is the arithmetic mean:

In the second case the mean round-trip speed is the harmonic mean:

In order to apply the Galilean transformation, start with the standard configuration in which motion is along the *X* and *X′* axes in *S* and *S′* respectively with velocity *v* in *S′* relative to *S*.

In the first case, the speed in *S′* relative to the forward direction is:

And the speed relative to the return direction is:

Then the mean speed is:

Thus the mean round-trip speed is Galilean invariant.

In the second case, the speed in *S′* relative to the forward direction is (with the overall inverse left to the end):

And the speed relative to the return direction is:

Then the harmonic mean speed is:

This applies to the speed of light, *c*, which is Galilean invariant since only the mean round-trip speed can be measured. For a light clock with a nominal speed of *c* in each direction and an observer with longitudinal speed *v*, the mean speed of light is

An observer with transverse speed has no effect on the measurement of the speed of light *c*.

*Revised 2023-07-29.*