Spectrum of the laplacianon a domain with the boundary and barrier, composed of small resonators

In the present work, we consider a particular perturbation of the domain’s boundary for a Laplacian. Particularly, the perturbation consists of adding a number of Helholtz resonators, connected to the main domain through apertures, and we are interested in a limiting case, when number of attached resonators goes to infinity. Such perturbation refered to as “corrugated boundary” in literature. We will consider two different geometries: with the perturbation occuring on the boundary of a rectangular domain and on a barrier, appearing in the middle of rectangular domain. In the case of the barrier, domain is separated into two parts and each Helmholtz resonator has two apertures on the opposite walls, connecting it to each of the separated parts. For the calculation, we employ the model of zero-width apertures, which assumes apertures to be infinitely small (although their throughput can be controlled through a parameter), and allows one to use operator extensions theory and to describe the eigenfunctions explicitly. Let us denote original domain by Ω0, and Ω𝑁 are a family of domains, which coincide with Ω0 everywhere, except for a one-dimensional part of boundary (or barrier) Γ, where the perturbation takes place. Each domain Ω𝑁 has attached resonators. Let’s consider Laplacians -Δ0 and -Δ𝑁 on Ω0, and Ω𝑁 respectively, with Neumann boundary conditions on all boundaries, and denote eigenvalues of -Δ𝑁 by ψ𝑁𝑛 (𝑥). Let the dimensions of resonators be and ℎ, and δ are widths of all apertures. These parameters are dependent on number of resonators 𝑁. In our case, distance between holes is equal to height of resonators ℎ and = |Γ| /ℎ. As we increase the number of resonators 𝑁, their height ℎ decreases linearly, and the “length” of resonators should remain constant. Now we can state our main results. Theorem 1. In the case of boundary perturbations by Helmholtz resonators, when → ∞, ψ𝑁𝑛 (𝑥) converge to eigenvalues of the following boundary problem: ⎪⎪⎩ Δ𝑢 + 𝑘2𝑢 = 0, 𝜕𝑛|Γ = -𝑘 tan(𝑘𝑤)𝑢|Γ, 𝜕𝑛|𝜕Ω0∖Γ = 0. (17) Theorem 2. In the case of a perturbation by a barrier made of Helmholtz resonators, when → ∞, ψ𝑁𝑛 (𝑥) converge to eigenvalues of the following boundary problem: ⎪⎪⎪⎪⎩ Δ𝑢 + 𝑘2𝑢 = 0, 𝜕𝑛𝑅 |Γ𝑅 + 𝜕𝑛𝐿 |Γ𝐿 = -𝑟+(𝑘)(𝑢𝑅 + 𝑢𝐿)|Γ, 𝜕𝑛𝑅 |Γ𝑅 - 𝜕𝑛𝐿 |Γ𝐿 = -𝑟-(𝑘)(𝑢𝑅 - 𝑢𝐿)ˇ𝑢|Γ, 𝜕𝑛|𝜕Ω0∖Γ = 0. 𝑟-(𝑘) =lim δ→0 𝑟δ- = sin cos - 1 , 𝑟+(𝑘) =lim δ→0 𝑟δ+ = sin cos + 1 . We also provide results of numerical computations, corresponding to these problems, which confirm analytical results.