Summary: As is well known, the safe travel velocity of high-speed rail must remain below a well-defined limit commonly referred to as the critical speed. This upper bound depends, in turn, on the speed at which waves propagate in the ground. But as models for trains in motion have increased in complexity, over the years the concept of critical speed has grown in obscurity. It was not until late in the twentieth century that it was realized that the critical speed was somehow connected to the so-called dispersion spectrum for the complete system, but until now, the justification for its application in practice has remained empirical and lacking in rigorous mathematical explanations. This task is taken up in this article, where for the first time a most general proof is provided for the problem at hand that is applicable to any layered soil configuration when one such system is subjected to one or more loads in motion.