论文标题
二维正方形晶格中瞬时波的传播
The propagation of transient waves in two-dimensional square lattices
论文作者
论文摘要
本文的目的是研究平面和抗平台问题中2D晶格中瞬时低频波的衰减。本文的主要思想是，可以通过在无限长波的准芬太西附近的渐近反转方法来获得离散周期介质问题的分析解决方案；此外，在这种方法中，可以考虑短波的贡献。使用此方法，我们在局部瞬态载荷下获得平面和抗平台公式中晶格中扰动的渐近分数。此外，我们表明，描述正方形晶格的2D平面运动的方程式可以以两个线性独立的波程式的形式表示，每个方程仅包含一个未知函数。通过与弹性理论类似，一个方程描述了晶格中剪切波的传播，而另一个方程式描述了纵向波的传播。结果，可以表明，在同质的无限晶格中，可以以这种方式指定载荷，使得形成主要的纵向或主要是剪切波。研究的问题还通过有限的差异方法解决。显示了渐近和数值解的定性和定量对应关系。
The aim of this article is to study the attenuation of transient low-frequency waves in 2D lattices in both plane and antiplane problems. The main idea of this article is that analytical solutions to problems of mechanics of discrete periodic media can be obtained by a method of asymptotic inversion of the Laplace and Fourier transforms in the vicinity of the quasi-front of infinitely long waves; moreover, in this method it is possible to take into account the contribution of short waves. Using this method, we obtain asymptotics of perturbations in lattices in plane and antiplane formulations under a local transient load. Besides, we show that equations describing 2D plane motion of a square lattice can be represented in the form of two linearly independent wave equations, each of which contains one unknown function only. By analogy with the theory of elasticity, one equation describes the propagation of shear waves in the lattice, while the other equation describes the propagation of longitudinal waves. As a result, it is shown that, in a homogeneous infinite lattice, a load can be specified in such a manner that either predominantly longitudinal or predominantly shear waves are formed. The problems under study are also solved by a finite difference method. The qualitative and quantitative correspondence of asymptotic and numerical solutions is shown.
