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.

中文翻译：

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基于分层介质的高铁临界速度证明

众所周知，高铁的安全行驶速度必须保持在通常称为临界速度的明确限制以下。该上限反过来又取决于波在地面中传播的速度。但随着运动列车模型的复杂性增加，多年来临界速度的概念变得越来越模糊。直到 20 世纪后期才意识到临界速度以某种方式与整个系统的所谓色散谱有关，但直到现在，其在实践中的应用的理由仍然是经验性的，缺乏严谨的数学解释。这个任务在这篇文章中，