×
Gadyl’shin, R. R.

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.

Cited in 6 Documents

MSC:

76Q05 Hydro- and aero-acoustics
35C20 Asymptotic expansions of solutions to PDEs

Keywords:

Sommerfeld radiation condition; Meixner condition; small parameter; scattered field
PDFBibTeX XMLCite
\textit{R. R. Gadyl'shin}, J. Appl. Math. Mech. 56, No. 3, 1 (1992; Zbl 0786.76075); translation from Prikl. Mat. Mekh. 56, No. 3, 412--418 (1992)
Full Text: DOI

References:

[1] Rayleigh, O. M., The theory of Helmholtz’s resonator, (Proc. Roy. Soc. London Ser. A, 92 (1916)), 265-275 · JFM 46.1273.03
[2] Shestopalov, V. P., Summator Equations in thé Modem Theory of Diffraction (1983), Nauk. Dumka: 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: Mir Moscow · Zbl 0158.43905
[7] Il’in, A. M., Matching of Asymptotic Expansions of the Solutions of Boundary-value Problems (1989), Nauka: 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, (Ser. Mat., 48 (1984), Izv. Akad. Nauk SSSR), 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: Fizmatgiz Moscow · Zbl 0101.41203
[11] Colton, D.; Cress, R., Methods of Integral Equations in Scattering Theory (1987), Mir: Mir Moscow · Zbl 0621.35002
[12] Babich, V. M., Analytical extension of the resolvent of external problems for the Laplace operator to the second sheet, (Theory of Functions, Functional Analysis and their Applications (1966), Khar’kov Univ. Press: Khar’kov Univ. Press Khar’kov) · 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: 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, (Ser. Mat., 47 (1983), Izv. Akad. Nauk SSSR), 1322-1339
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.