The low-frequency spectrum of small Helmholtz resonators
Ben Schweizer · Proceedings of the Royal Society A Mathematical Physical and Engineering Sciences · 2014
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⊂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≈VεLε/Aε, where Vε 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.