论文标题
在局部共振频率下缓慢变化的波导中的Helmholtz问题
The Helmholtz problem in slowly varying waveguides at locally resonant frequencies
论文作者
论文摘要
本文旨在在缓慢变化的波导中介绍对Helmholtz问题的一般研究。这项工作在局部谐振频率上特别令人感兴趣，在局部谐振频率中，可以观察到量子力学中schrödinger方程接近隧道效应的现象。在这种情况下，局部共振模式以通风函数的形式在波导中传播。使用先前在Schrödinger方程式上的数学结果，我们证明了在此类波导中具有外向条件的Helmholtz源问题的独特解决方案。我们提供了该解决方案的显式模态近似，以及对H1LOC中近似误差的控制。在波导的情况下，主要定理证明了单调变化的轮廓，然后使用匹配策略进行了概括。我们最终通过将模态近似与基于有限元方法的数值解决方案进行比较来验证。
This article aims to present a general study of the Helmholtz problem in slowly varying waveguides. This work is of particular interest at locally resonant frequencies, where a phenomenon close to the tunnel effect for Schrödinger equation in quantum mechanics can be observed. In this situation, locally resonant modes propagate in the waveguide under the form of Airy functions. Using previous mathematical results on the Schrödinger equation, we prove the existence of a unique solution to the Helmholtz source problem with outgoing conditions in such waveguides. We provide an explicit modal approximation of this solution, as well as a control of the approximation error in H1loc. The main theorem is proved in the case of a waveguide with a monotonously varying profile and then generalized using a matching strategy. We finally validate the modal approximation by comparing it to numerical solutions based on the finite element method.
