Geodesic
Barrier composed of perforated resonators and boundary conditions
Eurasian mathematical journal, Tome 15 (2024) no. 3, pp. 68-76.

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We consider the Laplace operator with the Neumann boundary condition in a two-dimensional domain divided by a barrier composed of many small Helmholtz resonators coupled with the both parts of the domain through small windows of diameter $2a$. The main terms of the asymptotic expansions in a of the eigenvalues and eigenfunctions are considered in the case in which the number of the Helmholtz resonators tends to innity. It is shown that such a homogenization procedure leads to some energy-dependent boundary condition in the limit. We use the method of matching the asymptotic expansions of boundary value problem solutions. 
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I. Y. Popov; E. S. Trifanova; A. S. Bagmutov; I. V. Blinova. Barrier composed of perforated resonators and boundary conditions. Eurasian mathematical journal, Tome 15 (2024) no. 3, pp. 68-76. http://geodesic.mathdoc.fr/item/EMJ_2024_15_3_a6/

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