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On a system of imbedded resonators. (English. Russian original) Zbl 0793.76079
Russ. Acad. Sci., Dokl., Math. 46, No. 2, 383-386 (1993); translation from Dokl. Akad. Nauk, Ross. Akad. Nauk 326, No. 6, 939-942 (1992).
From the text: We consider two resonators, one imbedded in the other. The given system is the “interior” resonator, surrounded by an “exterior” resonator of the type of a spherical layer. Suppose \(k^ 2_ 0\neq 0\) (\(k\) is the wave number) is a simple eigenvalue of the closed interior resonator but is not an eigenvalue of the closed resonator of the type of a spherical layer. It would seem that in this situation the types of singularities of the scattered field corresponding to a single resonator should be preserved. Actually, in the exterior of the system in a peak regime the difference of the scattered fields is of order \(O(1)\), but in the interior resonator the growth is already of order \(O(\varepsilon^{- 2})\) \((0< \varepsilon\ll 1\) is the linear dimension of the aperture). Thus, there is a certain type of roll-up by the nonresonance spherical layer (in the spherical layer of order \(O(\varepsilon^{-1})\)). This effect is connected with the fact that in the present case \(\text{Im }\tau_ \varepsilon\) (\(\tau_ \varepsilon\) is a pole of the Green function of the resonator) has order \(O(\varepsilon^ 4)\) and not \(O(\varepsilon^ 2)\) as in the case of a single resonator. We remark that for a chain of successively connected resonators this phenomenon may occur.
Reviewer’s remark: This is an abstract paper in the general area of acoustic scattering, and “only time will tell whether it will prove useful”.

76Q05 Hydro- and aero-acoustics
35Q35 PDEs in connection with fluid mechanics
35B40 Asymptotic behavior of solutions to PDEs