Geodesic
Splitting of the poles of a Helmholtz resonator
Izvestiya. Mathematics , Tome 43 (1994) no. 2, pp. 233-260

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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. 
@article{IM2_1994_43_2_a2,
     author = {R. R. Gadyl'shin},
     title = {Splitting of the poles of a {Helmholtz} resonator},
     journal = {Izvestiya. Mathematics },
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     number = {2},
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     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a2/}
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R. R. Gadyl'shin. Splitting of the poles of a Helmholtz resonator. Izvestiya. Mathematics , Tome 43 (1994) no. 2, pp. 233-260. http://geodesic.mathdoc.fr/item/IM2_1994_43_2_a2/