The low-frequency spectrum of small Helmholtz resonators

We analyse the spectrum of the Laplace operator in a complex geometry, representing a small Helmholtz resonator. The domain is obtained from a bounded set Ω⊂Rn by removing a small obstacle Σϵ⊂Ω of size ϵ>0. The set Σϵ essentially separates an interior domain Ωεinn (the resonator volume) from an exterior domain Ωεout, but the two domains are connected by a thin channel. For an appropriate choice of the geometry, we identify the spectrum of the Laplace operator: it coincides with the spectrum of the Laplace operator on Ω, but contains an additional eigenvalue με−1. We prove that this eigenvalue has the behaviour μϵ≈V ϵLϵ/Aϵ, where V ϵ is the volume of the resonator, Lϵ is the length of the channel and Aϵ is the area of the cross section of the channel. This justifies the well-known frequency formula ωHR=c0A/(LV) for Helmholtz resonators, where c0 is the speed of sound.