On Calculation of Kirchhoff Coefficients for Helmholtz Resonator

Helmholtz resonator is a shell Ωshell separating a compact domainthe cavity Ωint ⊂ R3 from the non-compact domain Ωout = R3\[Ωint∪Ωshell]. It is assumed that a small opening in the shell connects Ωint with Ωout, causing transformation of real eigen-frequencies of the Neumann Laplacian on Ωint into complex scattering frequencies of the corresponding stationary acoustic problem −∆u = λu, in Ω = Ωint ∪ Ωout, with Neumann boundary condition on the C2-smooth boundary ∂u ∂n ∣∣∣ ∂Ω = 0. The Kirchhoff model [1] gives a convenient Ansatz Ψout(x, ν, λ) = Ψ N out(x, ν, λ) + AoutG N out(x, a, , λ), x ∈ Ωout, Ψint(x, ν, λ) = AintG N int(x, a, λ), x ∈ Ωint, (1) for calculation of components of the scattered wave of the acoustic problem in Ωint, Ωout in terms of the scattered wave Ψ N out(x, ν, λ) and the Green functions Gint,out(x, a, λ) of the Neumann Laplacians in Ωint,out. In this paper we suggest an explicit formula for the Kirchhoff coefficients Aint,out, based on construction of a fitted solvable model for the Helmholtz resonator with a narrow short channel connecting Ωint,Ωout. The scattering matrix of the model serves an approximation of the scattering matrix of the Helmholtz resonator on a certain essential spectral interval. Calculation of the scattering frequencies of the model is reduced to solution of an algebraic equation. AMS Subject Classification: 35P25, 47A40,47F05 PACS numbers: 73.63.Hs,73.23.Ad,85.35.-p,85.35.Be