An Exercise in Relativity

Consider the following situation: Imagine a relativistic mathematical model where a train is heading straight to the station at a speed u, distance is measured along the tracks by counting and there is a clock at the station and one on the train. An initial distance is given.

The locomotive whistle blows and the sound propagates at v “the speed of sound”.

When this signal reaches the station it is reflected immediately at the same v, a bell that echoes back to the train. Where once again a signal is reflected back to the station and this is repeated to-and-fro over-and-over again over all the shorter distance.

The problem is to calculate the total distance of the signal travelled within special relativity.

It would be a good idea to do the calculations in the classical setting first there is an observation that simplifies things considerably. In relativity one would obviously need to use the Lorenz coordinate transformations in some form or another.

I don't plan providing a solution unless asked explicitly this is merely an elementary exercise in special relativity and is easy enough.