Estimation of a class of quasi-resonances generated by multiple small particles with high surface impedances

摘要

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 lambda of the form lambda: = beta a(-1) - alpha beta a(-1 + h) where a is the maximum radi of the particles, with a < <1, and beta 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 alpha 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(ah,a1-h) for h is an element of (0,1) as a < <1.