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Peer-reviewedCC BY 4.0Mathematics

On the Frequencies of Scattering on Traps with Micro-Inhomogeneous Structures—Helmholtz Resonators

  • Gregory A. Chechkin1
  • Rustem R. Gadyl’shin2
  • Andrey L. Piatnitski3,4
4 affiliations
  1. 1Department of Differential Equations, Faculty of Mechanics and Mathematics, M.V. Lomonosov Moscow State University, Leninskie Gory, 1, Moscow 119991, Russia
  2. 2Department of Mathematical Analysis, Faculty of Physics and Mathematics, Akmulla Bashkir State Pedagogical University, Ufa 450077, Russia
  3. 3Faculty of Engineering Science and Technology, UiT The Arctic University of Norway, P.O. Box 385, 8505 Narvik, Norway
  4. 4Higher School of Modern Mathematics, Moscow Institute of Physics and Technology, 1 Klimentovski per., Moscow 115184, Russia
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Abstract

This paper studies the vibration of a body with a hole (trap), which is caused by a plane acoustic wave. We assume that the body contains a periodic microstructure of small inclusions, which can partially reflect acoustic waves, and consider the scattering problem for the acoustic waves, which is modeled by an appropriate boundary value problem for the Helmholtz equation. Our goal is to describe the asymptotic behavior of the spectrum of this boundary value problem as the period of the microstructure and the diameter of the hole tend to zero. The scattering frequencies are characterized by the poles of the analytic continuation of the resolvent of the corresponding boundary value problem.

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Cite

BibTeXRIS
BibTeX
@article{chechkin2026frequencies,
  title = {{On the Frequencies of Scattering on Traps with Micro-Inhomogeneous Structures—Helmholtz Resonators}},
  author = {Gregory A. Chechkin and Rustem R. Gadyl’shin and Andrey L. Piatnitski},
  journal = {Mathematics},
  year = {2026},
  month = oct,
  doi = {10.3390/math14203803},
  url = {https://www.mdpi.com/2227-7390/14/20/3803},
  note = {CC BY 4.0},
}
RIS
TY  - JOUR
TI  - On the Frequencies of Scattering on Traps with Micro-Inhomogeneous Structures—Helmholtz Resonators
AU  - Gregory A. Chechkin
AU  - Rustem R. Gadyl’shin
AU  - Andrey L. Piatnitski
PY  - 2026
DA  - 2026/10/20
JO  - Mathematics
DO  - 10.3390/math14203803
UR  - https://www.mdpi.com/2227-7390/14/20/3803
AB  - This paper studies the vibration of a body with a hole (trap), which is caused by a plane acoustic wave. We assume that the body contains a periodic microstructure of small inclusions, which can partially reflect acoustic waves, and consider the scattering problem for the acoustic waves, which is modeled by an appropriate boundary value problem for the Helmholtz equation. Our goal is to describe the asymptotic behavior of the spectrum of this boundary value problem as the period of the microstructure and the diameter of the hole tend to zero. The scattering frequencies are characterized by the poles of the analytic continuation of the resolvent of the corresponding boundary value problem.
SN  - 2227-7390
ER  -