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 ⊂ R n by removing a small obstacle Σ ε ⊂ Ω of size ε > 0. The set Σ ε essentially separates an interior domain Ω inn ε (the resonator volume) from an exterior domain Ω out ε , 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 μ −1 ε . We prove that this eigenvalue has the behaviour μ ε ≈ V ε L ε /A ε , where V ε is the volume of the resonator, L ε is the length of the channel and A ε is the area of the cross section of the channel. This justifies the well-known frequency formula ω HR = c 0 A/(LV) for Helmholtz resonators, where c 0 is the speed of sound.

Effective acoustic properties of a meta-material consisting of small Helmholtz resonators

“…(d) Disconnected subregions. In the work at hand we use a different construction, based on the Helmholtz resonator that has been studied mathematically in [30]. Again, a topological effect plays a role: Every cell has an interior part R Y (the resonator) and an exterior part Q Y .…”

Transmission Conditions for the Helmholtz-Equation in Perforated Domains

“…But even for the Helmholtz equation with a fixed frequency ω, order-1 effects are possible, namely in a Helmholtz resonator geometry. For a mathematical study of the Helmholtz resonator we refer to [11]. We emphasize that the lowest order effect of [11] is only possible by introducing three scales: The macroscopic scale (order 1, size of Ω), the microscopic scale ε (size of the resonator), and a sub-micro-scale which is small compared to ε (the diameter of a channel connecting the interior of the resonator to the exterior).…”

“…For a mathematical study of the Helmholtz resonator we refer to [11]. We emphasize that the lowest order effect of [11] is only possible by introducing three scales: The macroscopic scale (order 1, size of Ω), the microscopic scale ε (size of the resonator), and a sub-micro-scale which is small compared to ε (the diameter of a channel connecting the interior of the resonator to the exterior). Effects of highest order by introducing small structures are also known from a related equation, namely the time homogeneous Maxwell equation (of which the Helmholtz equation is a special case): Using split-ring microscopic geometries, the effective behavior of solutions to Maxwell equations can be changed dramatically: Negative index materials with negative index of refraction can occur as homogenized materials, see [1,5].…”

Outgoing wave conditions in photonic crystals and transmission properties at interfaces

“…One example is the bounded domain Ω = (0, 1) ⊂ R 1 with the solution u(x) = sin(πx) for ω = π. A more relevant example in higher dimension (2 or 3) is the Helmholtz resonator: When ω coincides with the resonance frequency, there is a nontrivial solution to homogeneous boundary conditions, see [36]. For regular exterior domains, the Sommerfeld condition implies uniqueness: The Helmholtz operator has only a continuous spectrum and no point spectrum.…”