一维准周期介质中波的传播
Wave propagation in one-dimensional quasiperiodic media
摘要：
本文研究了Helmholtz方程$-(μu')' - ρω^2 u = f$在一维无界介质中的解。我们假设这个方程的系数是准周期函数的局部扰动，即沿着高维周期函数的特定线的迹。利用拟周期性的定义，将问题提升到一个具有周期系数的高维问题。增强问题的周期性，使我们能够扩展的想法，基于DtN的方法开发的椭圆形的情况下。然而，相关的数学和数值分析的方法是更微妙的，因为增广偏微分方程是退化的，在这个意义上，其运营商的主部不再是椭圆形的。我们还研究了这个偏微分方程的数值分辨率，它依赖于Dirichlet细胞问题的解决方案，以及约束Riccati方程。
英文摘要：
This work is devoted to the resolution of the Helmholtz equation $-(μu')' - ρω^2 u = f$ in a one-dimensional unbounded medium. We assume the coefficients of this equation to be local perturbations of quasiperiodic functions, namely the traces along a particular line of higher-dimensional periodic functions. Using the definition of quasiperiodicity, the problem is lifted onto a higher-dimensional problem with periodic coefficients. The periodicity of the augmented problem allows us to extend the ideas of the DtN-based method developed for the elliptic case. However, the associated mathematical and numerical analysis of the method are more delicate because the augmented PDE is degenerate, in the sense that the principal part of its operator is no longer elliptic. We also study the numerical resolution of this PDE, which relies on the resolution of Dirichlet cell problems as well as a constrained Riccati equation.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
