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The method of matched asymptotic expansions in the problem of the Helmholtz acoustic resonator. (English. Russian original) Zbl 0786.76075
J. Appl. Math. Mech. 56, No. 3, 340-345 (1992); translation from Prikl. Mat. Mekh. 56, No. 3, 412-418 (1992).
Summary: A Helmholtz resonator of fairly arbitrary form is considered. The asymptotic form with respect to a small parameter (the linear dimensions of the aperture) is constructed for the scattered field.

76Q05 Hydro- and aero-acoustics
35C20 Asymptotic expansions of solutions to PDEs
Full Text: DOI
[1] Rayleigh, O.M., The theory of Helmholtz’s resonator, (), 265-275 · JFM 46.1273.03
[2] Shestopalov, V.P., Summator equations in thĂ© modem theory of diffraction, (1983), Nauk. Dumka Kiev · Zbl 0571.73019
[3] Arsen’yev, A.A., Features of the analytic extension and resonance properties of the solution of the scattering problem for the Helmholtz equation, Zh. vychisl. mat. mat. fiz., 12, 112-138, (1972)
[4] Gadyl’shin, R.R., The amplitude of the oscillations for the Helmholtz resonator, Dokl. akad. nauk SSSR, 310, 1094-1097, (1990)
[5] Popov, I.Yu., The theory of extensions and localization of resonances for collector-type regions, Mat. sbornik, 181, 1366-1390, (1990) · Zbl 0718.35072
[6] Van Dyke, M., Perturbation methods in fluid mechanics, (1967), Mir Moscow · Zbl 0158.43905
[7] Il’in, A.M., Matching of asymptotic expansions of the solutions of boundary-value problems, (1989), Nauka Moscow · Zbl 0671.35002
[8] Maz’ya, V.G.; Nazarov, S.A.; Plamenevskii, B.A., Asymptotic expansions of the eigenvalues of boundary-value problems for the Laplace operator in regions with small openings, (), 347-371
[9] Gadyl’shin, R.R., The asymptotic form of the eigenvalue of a singularly perturbed self-conjugate elliptic problem with a small parameter in the boundary conditions, Diff. uravneniya, 22, 640-652, (1986) · Zbl 0616.35022
[10] Polya, G.; Seget, G., Isoperimetric inequalities in mathematical physics, (1962), Fizmatgiz Moscow
[11] Colton, D.; Cress, R., Methods of integral equations in scattering theory, (1987), Mir Moscow
[12] Babich, V.M., Analytical extension of the resolvent of external problems for the Laplace operator to the second sheet, () · Zbl 0292.35064
[13] Ramm, A.G., External diffraction problems, Radiotekhnika i elektronika, 17, 1362-1365, (1972)
[14] Eskin, G.I., Boundary-value problems for elliptic pseudodifferential equations, (1973), Nauka Moscow · Zbl 0292.35068
[15] Il’in, A.M.; Suleimanov, B.I., The asymptotic form of Green’s function for second-order elliptic equation in the vicinity of the boundary of a region, (), 1322-1339
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