# Invariance of round-trip speed

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.