Calculation of the Kirchhoff coefficients for the Helmholtz resonator

A Helmholtz resonator is a shell Ω_shell separating a compact cavity Ω_int from a noncompact outer domain Ω_out. A small opening Ω ^δ in the shell connects the cavity with the outer domain, causing the transformation of real eigenfrequencies of the Neumann Laplacian in the cavity into the complex scattering frequencies of the full spectral problem for the Neumann Laplacian on Ω = ℝ³Ω_shell. The Kirchhoff model of 1882, see [21], gives a convenient ansatz

((1))

for the approximate calculation of the outer component of the scattered wave of the full spectral problem on Ω in terms of the scattered wave Ψ_out ^N (x, ν, λ) and the Green function G _out ^N (x, a, λ) of the Neumann Laplacian on the outer domain, with a pole at the pointwise opening Ω ^δ ≈ a. In this paper, we suggest an explicit formula for the Kirchhoff coefficient A _out, based on the construction of a fitted solvable model for the Helmholtz resonator with a narrow short channel Ω ^δ connecting the cavity with the outer domain. The correcting term of the scattering matrix of the model serves as a rational approximation, on a certain spectral interval, for the correcting term of the full scattering matrix of the Helmholtz resonator. Dedicated to the memory of Vladimir Andreevich Borovikov, who often chose a problem as an engineer and solved it by creating new and surprising mathematics