Asymptotic Behavior of Solutions to the Problem of Diffraction of an Acoustic Wave on a Set of Small Obstacles
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Journal of Mathematical Sciences, Vol. 250, No. 2, October, 2020
ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE PROBLEM OF DIFFRACTION OF AN ACOUSTIC WAVE ON A SET OF SMALL OBSTACLES T. N. Bobyleva National Research Moscow State University of Civil Engineering 26, Yaroslavskoye Shosse, Moscow 129337, Russia [email protected]
A. S. Shamaev
∗
Ishlinsky Institute for Problems in Mechanics RAS 101-1, pr. Vernadskogo, Moscow 119526, Russia [email protected]
UDC 517.95
We consider the problem of diffraction of an acoustic wave on a set of small obstacles (cavities) with the boundary condition of the third kind on the obstacle boundary and the radiation condition at infinity. It is assumed that the obstacle diameters converge and for each obstacle the distance to the nearest obstacle is much greater than the obstacle diameter. We establish the closeness between the solutions to the boundary value problems in a domain with obstacles and the solution to the homogenized problem and prove the convergence of scattering frequencies to the scattering frequencies of the limiting problem with potential. Bibliography: 4 titles.
In this paper, we study diffraction of an acoustic wave on a “cloud” of a large number of “dust particles.” It is assumed that the diameter of each “dust particle” is much less than the distance to the nearest neighbor, i.e., the closest obstacle. Such problems were studied, for example, in [1, 2]. The distinctive feature of the problem studied in this paper is that, first, we consider the boundary conditions of the third kind on the obstacle surfaces (the Fourier condition) and, second, we study the asymptotic behavior of the so-called scattering frequencies (the definition, physical interpretation, and the main results can be found in [3]). As established in [1], a potential or the so-called strange term appears in the limit or homogenized model. Its appearance is caused by a certain rate of convergence to zero for the ratio of the obstacle diameter to the distance to the nearest neighbor. Such conditions were earlier considered in [1] in the case of the Dirichlet condition and in [2] in the case of the third boundary condition. The corresponding boundary value problems were considered in bounded domains. In this paper, ∗
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Translated from Problemy Matematicheskogo Analiza 105, 2020, pp. 59-66. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2502-0250
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we consider the diffraction problem in an unbounded domain (the complement of the set of all obstacles) with the radiation condition at infinity. To take into account the radiation condition, we apply the method [3] for reducting the diffraction problem in an unbounded domain to a problem in a bounded domain, but for a nonlocal operator. Then we perform the spectral analysis of the operator and obtain the asymptotics of the spectrum with respect to a small parameter ε characterizing the distance between obstacles. Here, we use the results of [4].
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Statement of the Problem
We assume that there a
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