We consider high-frequency homogenization in periodic media for travelling waves of several different equations: the wave equation for scalar-valued waves such as acoustics; the wave equation for vector-valued waves such as electromagnetism and elasticity; and a system that encompasses the Schrödinger equation. This homogenization applies when the wavelength is of the order of the size of the medium periodicity cell. The travelling wave is assumed to be the sum of two waves: a modulated Bloch carrier wave having crystal wavevector >k and frequency ω1 plus a modulated Bloch carrier wave having crystal wavevector >m and frequency ω2. We derive effective equations for the modulating functions, and then prove that there is no coupling in the effective equations between the two different waves both in the scalar and the system cases. To be precise, we prove that there is no coupling unless ω1=ω2 and (>k − >m) ⊙ Λ ∈ 2πℤd, where Λ=(λ1λ2…λd) is the periodicity cell of the medium and for any two vectors a = (a1, a2, …, ad), b = (b1, b2, …, bd) ∈ ℝd, the product a⊙b is defined to be the vector (a1b1,a2b2,…,adbd). This last condition forces the carrier waves to be equivalent Bloch waves meaning that the coupling constants in the system of effective equations vanish. We use two-scale analysis and some new weak-convergence type lemmas. The analysis is not at the same level of rigour as that of Allaire and co-workers who use two-scale convergence theory to treat the problem, but has the advantage of simplicity which will allow it to be easily extended to the case where there is degeneracy of the Bloch eigenvalue.
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机译:我们考虑周期性介质中高频均质化的几个不同方程的行波:标量值波的波方程,例如声学;矢量值波(例如电磁和弹性)的波动方程;以及包含薛定ding方程的系统。当波长为中等周期性单元的大小的数量级时,将应用这种均质化。假定行波是两个波的总和:具有晶体波矢量> k strong>和频率ω1的调制Bloch载波加上具有晶体波矢量> m strong>的调制Bloch载波和频率ω2。我们推导了调制函数的有效方程,然后证明了在标量和系统情况下两个不同波之间的有效方程没有耦合。确切地说,我们证明除非ω1=ω2和(> k strong> − > m strong>)⊙Λ∈ε2πℤ d sup>否则不存在耦合,其中Λ =(λ1λ2...λd)是介质的周期性像元,对于任何两个向量a =(a1,a2,...,ad),b =(b1, b em> 2,..., b em> d em>)∈∈ d em> sup>,乘积 a em>⊙ b em >定义为向量( a em> 1 b em> 1, a em> 2 b em> 2 sub>,…, a em> d em> sub> b em> d em> sub >)。最后一个条件迫使载波成为等效的布洛赫波,这意味着有效方程组中的耦合常数消失。我们使用两尺度分析和一些新的弱收敛型引理。该分析与使用两级收敛理论来解决问题的Allaire及其同事的严谨程度不同,但具有简单性的优势,可以轻松地将其扩展到存在以下情况的情况Bloch特征值的简并性。
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