Abstract

The fundamental solution describing non-stationary elastic wave scattering on an isotopic defect in a one-dimensional harmonic chain is obtained in an asymptotic form. The chain is subjected to unit impulse point loading applied to a particle far enough from the defect. The solution is a large-time asymptotics at a moving point of observation, and it is in excellent agreement with the corresponding numerical calculations. At the next step, we assume that the applied point impulse excitation has random amplitude. This allows one to model the heat transport in the chain and across the defect as the transport of the mathematical expectation for the kinetic energy and to use the conception of the kinetic temperature. To provide a simplified continuum description for this process, we separate the slow in time component of the kinetic temperature. This quantity can be calculated using the asymptotics of the fundamental solution for the deterministic problem. We demonstrate that there is a thermal shadow behind the defect: the order of vanishing for the slow temperature is larger for the particles behind the defect than for the particles between the loading and the defect. The presence of the thermal shadow is related to a non-stationary wave phenomenon, which we call the anti-localization of non-stationary waves. Due to the presence of the shadow, the continuum slow kinetic temperature has a jump discontinuity at the defect. Thus, the system under consideration can be a simple model for the non-stationary phenomenon, analogous to one characterized by the Kapitza thermal resistance. Finally, we analytically calculate the non-stationary transmission function, which describes the distortion (caused by the defect) of the slow kinetic temperature profile at a far zone behind the defect.

中文翻译：

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具有同位素缺陷的一维谐波链中的非平稳弹性波散射和能量传输

摘要

以渐近形式获得了描述一维谐波链中同位素缺陷上非平稳弹性波散射的基本解。该链受到施加到距离缺陷足够远的粒子上的单位脉冲点载荷的作用。该解是移动观测点处的大时间渐近方程，与相应的数值计算非常吻合。在下一步中，我们假设所施加的点脉冲激励具有随机幅度。这使得人们能够将链中和跨缺陷的热传输建模为动能的数学期望的传输，并使用动温度的概念。为了提供该过程的简化连续统描述，我们分离了动力学温度的慢时间分量。该量可以使用确定性问题的基本解的渐进性来计算。我们证明了缺陷后面存在热阴影：缺陷后面的粒子的慢速温度消失的阶数大于加载和缺陷之间的粒子的消失阶数。热阴影的存在与非平稳波现象有关，我们称之为非平稳波的反局域化。由于阴影的存在，连续体慢动力学温度在缺陷处存在跳跃不连续性。因此，所考虑的系统可以是非平稳现象的简单模型，类似于以卡皮查热阻为特征的模型。最后，我们分析计算了非平稳传输函数，它描述了缺陷后面较远区域的慢速动态温度分布的畸变（由缺陷引起）。