The low-frequency spectrum of small Helmholtz resonators

Abstract

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 ⊂Ωof 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 ≈ VεLε/Aε, where Vε is the volume of the resonator, Lε is the length of the channel and Aεis the area of the cross section of the channel. This justifies the well-known frequency formula ωHR = c0 √A/(LV) for Helmholtz resonators, where c0 is the speed of sound.