Splitting of the poles of a Helmholtz resonator

R. R. Gadyl'shin

Izvestiya. Mathematics , Tome 43 (1994) no. 2, pp. 233-260.

Résumé

It is shown that in a neighborhood of a two-fold eigenvalue of the Neumann problem there are two poles of the Green's function of the Helmholtz resonator. Asymptotic expressions for them with respect to a small parameter $\varepsilon$ (the linear size of the hole) are constructed, and the principal terms of the asymptotics are written out for the corresponding scattering and radiation problems.

Bibliographie

[1] Rayleigh O. M., “The Theory of Helmholtz Resonator”, Proceeding of Royal Society. London. A., 92, no. 638, 1916, 265–275 | Zbl

[2] Miles J. W., “Scattering by a Spherical cap”, Pt. 2, Jornal Acoustical Society of America, 50:3 (1971), 892–903 | DOI | MR

[3] Shestopalov V. P., Summatornye uravneniya v sovremennoi teorii difraktsii, Nauk. dumka, Kiev, 1983 | MR

[4] Arsenev A. A., “Ob osobennostyakh analiticheskogo prodolzheniya i rezonansnykh svoistvakh resheniya zadachi rasseyaniya dlya uravneniya Gelmgoltsa”, ZhVM i MF, 12:1 (1972), 112–138 | MR

[5] Beale J. T., “Scattering Frequencies of Resonators”, Communications on Pure and Applied Mathematics, 26:4 (1973), 549–564 | DOI | MR

[6] Petras S. V., “O rasscheplenii serii rezonansov na “nefizicheskom liste””, Zapiski nauch. seminarov LOMI, 51, 1975, 155–169 | MR | Zbl

[7] Gotlib V. Yu., “Rasseyanie na kruglom rezonatore s uzkoi schelyu kak zadacha teorii vozmuschenii”, DAN SSSR, 287:5, 1109–1113 | MR

[8] Gadylshin P. P., “Ob amplitude kolebanii dlya rezonatora Gelmgoltsa”, DAN SSSR, 310:5 (1990), 1094–1097 | MR

[9] Ilin A. M., Soglasovanie asimptoticheskikh razlozhenii reshenii kraevykh zadach, Nauka, M., 1989 | MR

[10] Kolton D., Kress R., Metody integralnykh uravnenii v teorii rasseyaniya, Mir, M., 1987 | MR

[11] Agranovich M. S., “Spektralnye svoistva zadach difraktsii”, Dopolnenie k knige: Voitovich N. N., Katsenelenbaum B. Z., Sivov A. N., Obobschennyi metod sobstvennykh kolebanii v teorii difraktsii, Nauka, M., 1977 | MR | Zbl

[12] Babich V. M., “Ob analiticheskom prodolzhenii rezolventy vneshnikh zadach dlya operatora Laplasa na vtoroi list”, Teoriya funktsii, funktsionalnyi analiz i ikh prilozheniya, no. 3, Kharkovsk. un-t, Kharkov, 1966, 151–157

[13] Ramm A. G., “O vneshnikh zadachakh difraktsii”, Radiotekhnika i elektronika, 17:7 (1972), 1362–1365 | MR

[14] Eskin G. I., Kraevye zadachi dlya ellipticheskikh psevdodifferentsialnykh uravnenii, Nauka, M., 1973 | MR

[15] Kato T., Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | MR | Zbl

[16] Shabat B. V., Vvedenie v kompleksnyi analiz. Ch. I, Nauka, M., 1985 | MR

[17] Polia G., Sege G., Izoperimetricheskie neravenstva v matematicheskoi fizike, Fizmatgiz, M., 1962

[18] Landkof N. S., Osnovy sovremennoi teorii potentsiala, Nauka, M., 1966 | MR | Zbl

[19] Pavlov B. S., Faddeev M. D., “O rasseyanii na polom rezonatore s malym otverstiem”, Zapiski nauch. seminarov LOMI, 126, 1983, 159–169 | MR | Zbl

[20] Popov I. Yu., “Teoriya rasshirenii i lokalizatsiya rezonansov dlya oblastei lovushechnogo tipa”, Matem. sb., 181:10 (1990), 1366–1390

[21] Popov I. Yu., “Rezonator Gelmgoltsa i teoriya rasshirenii operatorov v prostranstve s indefinitnoi metrikoi”, Matem. sb., 183:3 (1992), 3–37 | Zbl

[22] Gadylshin P. P., “Metod sraschivaemykh asimptoticheskikh razlozhenii v zadache ob akusticheskom rezonatore Gelmgoltsa”, Prikladnaya matem. i mekhanika, 56:3 (1992), 412–418 | MR

[23] Gadylshin P. P., “O vliyanii vybora mesta otverstiya i ego formy na svoistva akusticheskogo rezonatora Gelmgoltsa”, TMF, 93:1 (1992), 107–118 | MR

[24] Gadylshin P. P., “O kvazistatsionarnom rezhime rezonatora Gelmgoltsa”, Prikladnaya matem. i mekhanika, 57 (1993) | MR