Why are very short times so long and very long times so short in elastic waves?

TL;DR

This paper clarifies the apparent paradox in thermoelastic wave behavior, showing that short periods are effectively isothermal and long periods are isentropic, based on mathematical analysis and experimental evidence.

Contribution

The authors resolve the contradiction between intuitive understanding and mathematical analysis of thermoelastic waves regarding period length and thermal effects.

Findings

Short periods are effectively isothermal waves.

Long periods are effectively isentropic waves.

Experimental evidence supports the mathematical analysis.

Abstract

In a first study of thermoelastic waves, such as on the textbook of Landau and Lifshitz, one might at first glance understand that when the given period is very short, waves are isentropic because heat conduction does not set in, while if the given period is very long waves are isothermal because there is enough time for thermalization to be thoroughly accomplished. When one pursues the study of these waves further, by the mathematical inspection of the complete thermoelastic wave equation he finds that if the period is very short, much shorter than a characteristic time of the material, the wave is isothermal, while if it is very long, much longer than the characteristic time, the wave is isentropic. One also learns that this fact is supported by experiments: at low frequencies the elastic waves are isentropic, while they are isothermal when the frequencies are so high that can be attained in few cases. The authors show that there is no contradiction between the first glance understanding and the mathematical treatment of the elastic wave equation: for thermal effects very long periods are so short and very short periods are so long.