najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2024-15-6-736-741najo-113Research ArticleMATHEMATICSМАТЕМАТИКАBoundary composed of small Helmholtz resonators: asymptotic approachГраница, состоящая из малых резонаторов Гельмгольца: асимптотический подходhttps://orcid.org/0000-0002-5251-5327ПоповИ. Ю.PopovI. Y.

Igor Y. Popov – Center of Mathematics

Kroverkskiy, 49, St. Petersburg, 197101

popov1955@gmail.comТрифановаЕ. С.TrifanovaE. S.

Ekaterina S. Trifanova – Center of Mathematics

Kroverkskiy, 49, St. Petersburg, 197101

etrifanova@gmail.comБагмутовА. С.BagmutovA. S.

Alexander S. Bagmutov – Center of Mathematics

Kroverkskiy, 49, St. Petersburg, 197101

bagmutov94@mail.ruЛытаевА. А.LytaevA. A.

Alexander A. Lytaev – Center of Mathematics

Kroverkskiy, 49, St. Petersburg, 197101

Vasilievskiy island, Bolshoi av., 61, St. Petersburg, 199178

sas-lyt@ya.ruITMO UniversityRussian FederationITMO University; Institute for Problems in Mechanical Engineering of the Russian Academy of SciencesRussian Federation202405062025156736741Copyright © Popov I.Y., Trifanova E.S., Bagmutov A.S., Lytaev A.A., 20252025Попов И.Ю., Трифанова Е.С., Багмутов А.С., Лытаев А.А.Popov I.Y., Trifanova E.S., Bagmutov A.S., Lytaev A.A.Данная работа распространяется под лицензией Creative Commons Attribution 4.0.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/113

We consider the solution of the two-dimensional Neumann problem for the Helmholtz equation in a complex region composed of a square resonator with large number of smaller square resonators connected to it through small apertures along one side. The sizes of the apertures and distances between the neighbour apertures tend to zero. We use the method of matching of asymptotic expansions of solutions. By directing the number of attached small resonators to infinity, we obtain a problem for the Laplacian in the main square with energy-dependent boundary condition.

Мы рассматриваем решение двумерной задачи Неймана для уравнения Гельмгольца в сложной области, состоящей из квадратного резонатора с большим количеством меньших квадратных резонаторов, присоединенных к нему через малые отверстия вдоль одной стороны. Размеры отверстий и расстояния между соседними отверстиями стремятся к нулю. Мы используем метод согласования асимптотических разложений решений. Устремляя число присоединенных малых резонаторов к бесконечности, мы получаем задачу для оператораЛапласа в главном квадрате с граничным условием, зависящим от энергии.

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The authors declare that there are no conflicts of interest present.