Abstract A closed system of differential equations has been derived to describe thermal processes in a one-dimensional harmonic crystal on an elastic foundation. It is shown that the evolution of thermal perturbation in such a crystal is described by a discrete unsteady-state equation, a special case of which is the hyperbolic equation of ballistic heat conduction. This equation remains valid with negative stiffness of bonds between particles of the crystal in its entire stability range. The thermal perturbation front propagates with the maximum group velocity of mechanical waves. The propagation of a short-term thermal perturbation in the crystal on the elastic foundation is determined by the equation of ballistic thermal conductivity of the same type as in the crystal without an elastic foundation. The only parameter of this equation is the maximum group velocity (in absolute value), i.e., the maximum rate of energy propagation in the crystal on the elastic foundation. This quantity is proportional to the absolute value of the half-difference of the upper and lower cutoff frequencies. The rate of heat wave propagation in the crystal on the elastic foundation with positive stiffness is always lower than that in the crystal without an elastic foundation. The obtained equation is found to be valid both for positive stiffness values and for negative ones, for which the chain stability condition is satisfied. As an example, a dynamic problem of heat distribution is solved exactly for a parabolic initial temperature profile to model heating of a one-dimensional crystal on a foundation by a short laser pulse. Due to the dispersion of mechanical waves in the chain on the foundation, their group velocity depends on the wave number and the ratio of bond stiffnesses in the chain and the elastic foundation. The thermal front propagates with the maximum possible group velocity in the system, which depends only on this ratio.

中文翻译：

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弹性地基上一维谐波晶体中的热传播

摘要 推导出了一个封闭的微分方程组来描述弹性基础上的一维谐波晶体中的热过程。结果表明，这种晶体中热扰动的演化是由离散的非稳态方程描述的，它的一个特例是弹道热传导的双曲方程。该方程在其整个稳定性范围内晶体颗粒之间的键刚度为负时仍然有效。热扰动前沿以机械波的最大群速度传播。在弹性基础上的晶体中的短期热扰动的传播由与没有弹性基础的晶体中相同类型的弹道热导率方程确定。该方程的唯一参数是最大群速度（绝对值），即弹性基础上晶体中能量传播的最大速率。该数量与上限和下限截止频率的半差的绝对值成正比。热波在具有正刚度的弹性基础上的晶体中的传播速率总是低于没有弹性基础的晶体中的热波传播速率。发现获得的方程对于正刚度值和负刚度值都是有效的，对于链稳定性条件满足。例如，精确求解抛物线初始温度分布的热分布动态问题，以模拟短激光脉冲对基础上一维晶体的加热。由于机械波在基础上的链条中的弥散，其群速度取决于波数以及链条与弹性基础中的键刚度比。热锋在系统中以最大可能的群速度传播，这仅取决于该比率。