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Estimation of a class of quasi‐resonances generated by multiple small particles with high surface impedances
Author(s) -
Challa Durga Prasad,
Mouffouk Fouzi,
Sini Mourad
Publication year - 2019
Publication title -
mathematical methods in the applied sciences
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.719
H-Index - 65
eISSN - 1099-1476
pISSN - 0170-4214
DOI - 10.1002/mma.5598
Subject(s) - eigenfunction , eigenvalues and eigenvectors , constant (computer programming) , mathematics , matrix (chemical analysis) , resonance (particle physics) , mathematical analysis , real line , cluster (spacecraft) , line (geometry) , surface (topology) , complex plane , physics , quantum mechanics , geometry , chemistry , chromatography , computer science , programming language
We deal with the stationary acoustic waves propagating in a cluster of small particles enjoying high contrasts. Such contrasts allow the appearance of (complex valued) resonances that are close to the real line as the size of the particles becomes small. For single (but not necessarily small) particles, we derive the characteristic equation that generates a class of these resonances (the ones for which the corresponding eigenfunctions are uniformly constant). For multiple and small particles, we provide sufficient conditions on the contrasts that generates quasi‐resonances for which the corresponding eigenfunctions are uniformly constant. Precisely, we show that, if we distribute the particles on a uniform line, then the existence of such quasi‐resonances is related to the eigenvalues of the Harary matrix. To show these results, we take, as the small contrasted particles, small obstacles with high surface impedances λ of the form λ : = β a −1 − α β a −1 + h where a is the maximum radi of the particles, with a < <1, and β is a universal and positive constant depending only on the shape of the particles (but not on their size). In this case, if the relative constant α is an eigenvalue of the Harary matrix, then the used frequency is a quasi resonance of the cluster of the small particles where the error of approximation is of the order max ( a h ,a 1 − h ) for h ∈ (0,1) as a < <1.