A mathematical framework for tunable metasurfaces. Part I

Abstract

In this paper, a mathematical modeling of the physical mechanism underlying the concept of tunable metasurfaces is provided. The scattering properties of two metasurfaces are analyzed. The considered metasurfaces are designed by placing either a Helmholtz resonator or a pair of Helmholtz resonators in a periodic lattice. The subwavelength resonant properties and the concept of hybridization of Helmholtz resonators are exploited in order to shape the scattered waves in unusual way by such metasurfaces.

Keywords

Subwavelength resonance Helmholtz resonator hybridization metasurface

1. Introduction

Controlling waves in cavities, which are used in numerous domains of applied and fundamental physics, has become a major topic of interest [10,11,16,18]. The wave fields established in cavities are fixed by their geometry. They are usually modified by using mechanical parts. Nevertheless, tailoring the cavity boundaries permits one to design at will the wave fields they support. Here, we present a way to modify the frontier, with means of tunable metasurfaces, which allows switching between Dirichlet and Neumann conditions. The concept of metasurfaces is a powerful tool to shape waves by governing precisely the phase response of each constituting element through its subwavelength resonance properties [2,7,12,19]. Subwavelength resonators have been also used as the building block of super-resolution imaging [1,8,9].

A metasurface is a thin sheet with patterned subwavelength structures, which nevertheless has a macroscopic effect on wave propagation. Based on the concept of hybridized resonators, a tunable metasurface can be designed. Hence, it can be transformed into a tunable component that allows shaping waves dynamically in unprecedented ways [13,17,18]. The mechanism is based on the very general concept of hybridized coupled resonant elements whose resonant frequencies can be tuned by adjusting the coupling strength. The idea from [17] is to design a metasurface that can be either resonant or not resonant at a given operating frequency. In the first case, the collective resonant behaviour of the subwavelength resonators provides a change of the boundary condition while in the second case, the metasurface is transparent to the incident wave. To that aim, one can take as unit cell of the metasurface a system made out of two individual subwavelength resonators: one static resonator (referred to as the main resonator), whose frequency is fixed to the operating frequency and one tunable parasitic resonator (referred to as the parasitic resonator) whose frequency can be wisely adjusted by a given tunable mechanism. In the first case, the resonance frequency of the parasitic resonator is different enough from the resonance frequency of the main resonator, so that the two resonant elements do not couple. At the operating frequency, the metasurface is then resonant. In the second case however, one sets the resonance frequency of the parasitic resonator to match that of the static resonator. In that case, the subwavelength resonators hybridize to create a dimer whose eigenfrequencies are respectively under and above the initial resonance frequency.

A new and simple procedure for maximizing the Green’s function between two points at a chosen frequency in terms of the boundary conditions is developed in [5]. Its algorithm is a one shot optimization algorithm and can then be used in real-time to focalize the wave on a given spot by maximizing the transmission between an emitter and a receiver through specific eigenmodes of the cavity or on the contrary, to minimize the field on a receiver. To this end, the above discussed tunable metasurfaces are essential.

In this paper, we mathematically and numerically model the physical mechanism underlying the concept of tunable metasurfaces. We consider Helmholtz resonators. We show that an array of Helmholtz resonators behaves as an equivalent surface with Neumann boundary condition at a resonant frequency which corresponds to a wavelength much greater than the size of the Helmholtz resonators. Analytical formulas for the hybridized resonances of coupled Helmholtz resonators are also derived. Numerical simulations confirm their accuracy. We also propose an efficient approach to characterize the Green’s function of a cavity with mixed (Dirichlet and Neumann) boundary conditions. The use of tunable metasurfaces allows us to find a criterion ensuring that modifying parts of a cavity’s boundaries turn it into a completely different one.

The paper is organized as follows. Section 2 is devoted to give preliminary results on the so-called Neumann functions, which play a key role in proving the results in Sections 3 and 4. We first introduce the quasi-periodic fundamental solutions to the Helmholtz equation and recall in Lemmas 2.2–2.4 some key results from [2]. Then, we consider the Neumann functions, which depends crucially on certain remainder functions, for which we provide exact formulas in Lemmas 2.9–2.12.

In Section 3, we look at one periodically repeated Helmholtz resonator above a ground plate. After treating them with an incident wave, we obtain a scattered wave, whose resonant values are discussed and its behavior at the far-field are examined. We show in Theorem 3.2 that the structure behaves as an equivalent surface with Neumann boundary condition at the resonant frequencies characterized in Theorem 3.1. The proof uses a combined technique of [9] and [2].

Section 4 has the same objective as the previous section, but this time we have two periodically repeated Helmholtz resonator above a ground plate. As shown in Theorem 4.1, the strong coupling between the periodically repeated pair of resonators leads us to hybridized resonances. It is shown in Theorem 4.2 that at only these hybridized frequencies the structure behaves as an equivalent surface with Neumann boundary condition.

2. Preliminaries

In this section, we introduce the quasi-periodic fundamental solutions to the Helmholtz equation. The explicit formula derived in [2] will be helpful for us. Then we consider the Neumann functions and their remainders.

2.1. Quasi-periodic fundamental solution to the Helmholtz equation with Dirichlet boundary conditions

Let $k : = {(k_{1}, k_{2})}^{T} \in R^{2}$ and $k : = | k | : = {(k_{1}^{2} + k_{2}^{2})}^{1 / 2} \in [0, \infty)$ , where T denotes the transpose. Then for $k \in (0, \infty)$ we define $Γ^{k} : R^{2} ∖ {0} \to C$ as the fundamental solution to $(△ + k^{2}) Γ^{k} (x) = δ_{0} (x)$ satisfying the Sommerfeld radiation condition. For $k = 0$ we define $Γ^{0} : R^{2} ∖ {0} \to C$ as the fundamental solution to $△ Γ^{0} (x) = δ_{0} (x)$ . For $k \in [0, \infty)$ , we define $Γ^{k} (z, x) : = Γ^{k} (z - x)$ . For $x \in R^{2} ∖ {0}$ we have that $\begin{matrix} Γ^{0} (x) = \frac{1}{2 π} ln | x |, Γ^{k} (x) = - \frac{i}{4} H_{0}^{(1)} (k | x |), \end{matrix}$ where $H_{0}^{(1)}$ is the Hankel function of the first kind, we refer to [6, Chapter 2].

The quasi-periodic fundamental solutions $Γ_{♯}^{k} : R^{2} ∖ {{(n p, 0)}^{T} ∣ n \in Z} \to C$ and $Γ_{*}^{k} : R^{2} ∖ {{(n p, 0)}^{T} ∣ n \in Z} \to C$ are defined by $\begin{matrix} Γ_{♯}^{k} (x) = \sum_{n \in Z} Γ^{k} (x + (\begin{matrix} n p \\ 0 \end{matrix})) e^{- i k_{1} n p}, Γ_{*}^{k} (x) = \sum_{n \in Z} Γ^{k} (x + (\begin{matrix} n p \\ 0 \end{matrix})) e^{i k_{1} n p} . \end{matrix}$ $Γ_{♯}^{k}$ and $Γ_{*}^{k}$ are solutions to $\begin{array}{l} (△ + k^{2}) Γ_{♯}^{k} (x) = \sum_{n \in Z} e^{- i k_{1} n p} δ_{{(n p, 0)}^{T}} (x), (△ + k^{2}) Γ_{*}^{k} (x) = \sum_{n \in Z} e^{+ i k_{1} n p} δ_{{(n p, 0)}^{T}} (x) . \end{array}$ Using Poisson summation formula, this leads us to the following formulas [2, Lemma 3.2]:

Lemma 2.1.
For the case $k = 0$ we have $\begin{matrix} (2.1) & Γ_{♯}^{0} (x) = Γ_{}^{0} (x) = \frac{| x_{2} |}{2 p} - \sum_{\begin{array}{c} n \in Z ∖ {0} \\ l : = 2 π n / p \end{array}} \frac{1}{2 p | l |} e^{- | l | | x_{2} |} e^{i (l x_{1})} . \end{matrix}$ If k satisfies* $k^{2} < inf {| l - k_{1} |^{2} ∣ l : = 2 π n / p, n \in Z ∖ {0}}$ , we have $\begin{array}{l} Γ_{♯}^{k} (x) = \frac{e^{- i k_{1} x_{1}} e^{i k_{2} | x_{2} |}}{2 i k_{2} p} - \sum_{\begin{array}{c} n \in Z ∖ {0} \\ l : = 2 π n / p \end{array}} \frac{e^{- i k_{1} x_{1}}}{2 p \sqrt{| l - k_{1} |^{2} - k^{2}}} e^{- \sqrt{| l - k_{1} |^{2} - k^{2}} | x_{2} |} e^{i (l x_{1})}, \\ Γ_{}^{k} (x) = \frac{e^{i k_{1} x_{1}} e^{i k_{2} | x_{2} |}}{2 i k_{2} p} - \sum_{\begin{array}{c} n \in Z ∖ {0} \\ l : = 2 π n / p \end{array}} \frac{e^{i k_{1} x_{1}}}{2 p \sqrt{| l + k_{1} |^{2} - k^{2}}} e^{- \sqrt{| l + k_{1} |^{2} - k^{2}} | x_{2} |} e^{i (l x_{1})} . \end{array}$

Using Lemma 2.1 we can expand $Γ_{♯}^{δ k}$ and $Γ_{}^{δ k}$ with respect to δ near 0 and obtain $\begin{array}{l} (2.2) & Γ_{♯}^{δ k} (x) = \frac{1}{2 i δ k_{2} p} + Γ_{0, ♯}^{k} (x) + \sum_{n = 1}^{\infty} δ^{n} Γ_{n, ♯}^{k}, \\ (2.3) & Γ_{}^{δ k} (x) = \frac{1}{2 i δ k_{2} p} + Γ_{0, }^{k} (x) + \sum_{n = 1}^{\infty} δ^{n} Γ_{n, }^{k}, \end{array}$ where the kernels $Γ_{0, ♯}^{k}$ and $Γ_{0, }^{k}$ for $n ⩾ 1$ can be computed explicitly and shown to be smoother than $Γ_{0, ♯}^{k}$ and $Γ_{0, }^{k}$ , see [6, Chapters 7.3, 7.4]. For the zeroth kernel we have with Equation (2.1), for $k \neq 0$ , that $\begin{matrix} (2.4) & Γ_{0, ♯}^{k} (x) = Γ_{♯}^{0} (x) - \frac{k_{1} x_{1}}{2 k_{2} p}, Γ_{0, }^{k} (x) = Γ_{}^{0} (x) + \frac{k_{1} x_{1}}{2 k_{2} p} . \end{matrix}$ We define $Γ_{♯}^{k} (z, x) : = Γ_{♯}^{k} (z - x)$ and $Γ_{}^{k} (z, x) : = Γ_{}^{k} (z - x)$ . Consider that $Γ_{♯}^{k}$ and $Γ_{}^{k}$ are not symmetric for $k \neq 0$ in general, that is $Γ_{♯}^{k} (z, x) \neq Γ_{♯}^{k} (x, z)$ .

We define the parity operator $P : R^{2} \to R^{2}$ as $\begin{matrix} (2.5) & P (x_{1}, x_{2}) = (\begin{matrix} x_{1} \\ - x_{2} \end{matrix}) . \end{matrix}$

We introduce the quasi-periodic fundamental solutions to the Helmholtz equation with Dirichlet boundary condition $Γ_{+}^{k}, Γ_{\times}^{k} : {(z, x) \in R^{2} \times R^{2} ∣ \neg (\exists n \in Z : | z_{1} - x_{1} | = n p \land z_{2} = x_{2})} \to C$ by $\begin{matrix} (2.6) & Γ_{+}^{k} (z, x) : = Γ_{♯}^{k} (z, x) - Γ_{♯}^{k} (z, P (x)), Γ_{\times}^{k} (z, x) : = Γ_{}^{k} (z, x) - Γ_{}^{k} (z, P (x)) . \end{matrix}$ From Lemma 2.1, we see that for $z \in \partial R_{+}^{2}$ or $x \in \partial R_{+}^{2}$ , we have that $Γ_{+}^{k} (z, x) = 0$ and $Γ_{\times}^{k} (z, x) = 0$ . Moreover, $Γ_{+}^{k}$ and $Γ_{\times}^{k}$ are not symmetric, in general, and $Γ_{+}^{k}$ and $Γ_{\times}^{k}$ are not translation invariant in general, that is $Γ_{+}^{k} (z, x) \neq Γ_{+}^{k} (z - x, 0)$ . The following results hold from [7]. Lemma 2.2.
$Γ_{+}^{δ k}$ and $Γ_{\times}^{δ k}$ admit the expansions of the form $\begin{matrix} Γ_{+}^{δ k} (z, x) = Γ_{+, 0}^{k} (z, x) + \sum_{n = 1}^{\infty} δ^{n} Γ_{+, n}^{k} (z, x), Γ_{\times}^{δ k} (z, x) = Γ_{\times, n}^{k} (z, x) + \sum_{n = 1}^{\infty} δ^{n} Γ_{\times, n}^{k} (z, x), \end{matrix}$ where $Γ_{+}^{0} (z, x) = Γ_{+, 0}^{k} (z, x) = Γ_{\times, 0}^{k} (z, x)$ and $\begin{array}{l} Γ_{+}^{0} (z, x) & = \frac{1}{4 π} [log (sinh {(\frac{π}{p} (z_{2} - x_{2}))}^{2} + sin {(\frac{π}{p} (z_{1} - x_{1}))}^{2}) \\ (2.7) & - log (sinh {(\frac{π}{p} (z_{2} + x_{2}))}^{2} + sin {(\frac{π}{p} (z_{1} - x_{1}))}^{2})] . \end{array}$
Lemma 2.3.
Let $z, x \in R^{2}$ , such that $x_{2} > z_{2} > 0$ , and let k be small enough, then $Γ_{+}^{k} (z, x) = Γ_{+, p}^{k} (z, x) + Γ_{+, e}^{k} (z, x)$ , where explicit formulas can be found in [ 4 , Lemma 2.2]. Especially, $\begin{matrix} Γ_{+, p}^{k} (z, x) : = - \frac{sin (k_{2} z_{2})}{k_{2} p} e^{i (k_{2} x_{2} - k_{1} (z_{1} - x_{1}))}, \nabla_{x} Γ_{+, p}^{k} (z, x) = i k Γ_{+, p}^{k} (z, x) . \end{matrix}$
Lemma 2.4.
Let $z, x \in R^{2}$ , such that $z_{2} > x_{2} > 0$ , and let k be small enough, then $Γ_{+}^{k} (z, x) = Γ_{+, p}^{k} (z, x) + Γ_{+, e}^{k} (z, x)$ , where explicit formulas can be found in [ 4 , Lemma 2.3]. Especially, $\begin{matrix} Γ_{+, p}^{k} (z, x) : = \frac{sin (k_{2} x_{2})}{- k_{2} p} e^{i (k_{2} z_{2} - k_{1} (z_{1} - x_{1}))}, \nabla_{x} Γ_{+, p}^{k} (z, x) = (\begin{matrix} \frac{- i k_{1}}{k_{2} p} sin (k_{2} x_{2}) \\ \frac{- 1}{p} cos (k_{2} x_{2}) \end{matrix}) e^{i (k_{2} z_{2} - k_{1} (z_{1} - x_{1}))} . \end{matrix}$

2.2. Layer potentials

Let $E \subset R^{2}$ be an open, bounded, simply connected Lipschitz domain. Then we define the double-layer potential $S_{\partial E}^{k} : H^{- 1 / 2} (\partial E) \to H^{1 / 2} (R^{2})$ and the Neumann-Poincaré operator $K_{\partial E}^{k} : H^{1 / 2} (\partial E) \to H^{1 / 2} (\partial E)$ as $\begin{matrix} S_{\partial E}^{k} [φ] (x) : = \int_{\partial E} Γ^{k} (x, y) φ (y) d σ_{y}, K_{\partial E}^{k} [φ] (x) : = p . v . \int_{\partial E} \partial_{ν_{y}} Γ^{k} (x, y) φ (y) d σ_{y}, \end{matrix}$ where ‘p.v.’ denotes the principal value. $K_{\partial E}^{k}$ is a bounded function and if E has a $C^{2}$ -boundary, $K_{\partial E}^{k}$ is a compact operator. Moreover, we have the following result [6, Propositions 2.5 and 2.6]:

Lemma 2.5.
The operator $(- \frac{1}{2}) I + K_{\partial E}^{k}$ is invertible if and only if k is not a Neumann eigenvalue of the operator $- △$ . Then, ${((- \frac{1}{2}) I + K_{\partial E}^{k})}^{- 1}$ has a continuation to an operator-valued meromorphic function in $C$ . Also, all Neumann eigenvalues of the operator $- △$ are characteristic values of the operator $(- \frac{1}{2}) I + K_{\partial E}^{k}$ .

We define now the double-layer potential and the Neumann-Poincaré operator on a periodic structure. Let $q_{1}, q_{2} \in R^{2}$ and let $E ⋐ {y \in R^{2} ∣ | y_{1} | < q_{1} / 2 and 0 < y_{2} < q_{2}}$ be a open, bounded, simply connected domain. We define $Ω : = ⋃_{n \in Z} E + {(n q_{1}, 0)}^{T}$ and with that let $S_{\partial Ω, \times}^{k} : H^{- 1 / 2} (\partial E) \to H^{1 / 2} (R^{2})$ and $K_{\partial Ω, \times}^{k} : H^{1 / 2} (\partial E) \to H^{1 / 2} (\partial E)$ be defined as $\begin{matrix} S_{\partial Ω, \times}^{k} [φ] (x) : = \int_{\partial Ω} Γ_{\times}^{k} (x, y) φ (y) d σ_{y}, K_{\partial Ω, \times}^{k} [φ] (x) : = p . v . \int_{\partial Ω} \partial_{ν_{y}} Γ_{\times}^{k} (x, y) φ (y) d σ_{y} . \end{matrix}$

We know that if E has a $C^{2}$ -boundary, then $K_{\partial Ω, \times}^{k}$ is compact. We also have the following standard result (see [6, Lemma 7.4]): Lemma 2.6.
The operator $\frac{1}{2} I + K_{\partial Ω, \times}^{k} : H^{1 / 2} (\partial E) \to H^{1 / 2} (\partial E)$ is invertible.

2.3. Neumann functions and their remainders

Let $k_{E, min, △}$ be the first non-zero eigenvalue of the operator $- △$ with Neumann conditions on the boundary $\partial E$ of an open bounded domain $E \subset R^{2}$ . We now can define the following functions:

Definition 2.7.
Let $E \subset R^{2}$ be a open, bounded, simply connected $C^{2}$ -domain. Let $0 \neq | k | < k_{E, min, △} / 2$ . Then we define $N_{E}^{k} : E \times \overline{E} ∖ {(z, x) \in R^{2} \times R^{2} ∣ z = x} \to C$ and $N_{\partial E}^{k} : \partial E \times \overline{E} ∖ {(z, x) \in R^{2} \times R^{2} ∣ z = x} \to C$ as $\begin{array}{l} N_{E}^{k} (z, x) : = \frac{1}{2 π} log | z - x | + \frac{1 / | E |}{k^{2}} + R_{E}^{k} (z, x), \\ N_{\partial E}^{k} (z, x) : = \frac{1}{π} log | z - x | + \frac{1 / | E |}{k^{2}} + R_{\partial E}^{k} (z, x), \end{array}$ where $| E |$ denotes the area of E and $N_{E}^{k}$ and $N_{\partial E}^{k}$ are solutions to $\begin{matrix} \{\begin{matrix} (△_{x} + k^{2}) N_{E}^{k} (z, x) = δ_{0} (z - x) & for x \in E, \\ \partial_{ν_{x}} N_{E}^{k} (z, x) = 0 & for x \in \partial E, \end{matrix} \end{matrix}$ where $z \in E$ is fixed and where $ν_{x}$ denotes the outside normal on $\partial E$ with respect to x, and $\begin{matrix} \{\begin{matrix} (△_{x} + k^{2}) N_{\partial E}^{k} (z, x) = δ_{0} (z - x) & for x \in E, \\ \partial_{ν_{x}} N_{\partial E}^{k} (z, x) = 0 & for x \in \partial E, \end{matrix} \end{matrix}$ where $z \in \partial E$ is fixed.

With this definition the function $N_{E}^{k}$ is a solution to the Helmholtz equation with $δ_{0} (z - x)$ on the right-hand side and has a vanishing normal derivative on the boundary. The same is valid for the function $N_{\partial E}^{k}$ , although the source point z is on the boundary. The remainders $R_{E}^{k} (z, \cdot)$ and $R_{\partial E}^{k} (z, \cdot)$ are smooth enough. We refer to [6, Chapter 2.3.5].
Definition 2.8.
Let $q_{1}, q_{2} \in R^{2}$ and let $E ⋐ {y \in R^{2} ∣ | y_{1} | < q_{1} / 2 and 0 < y_{2} < q_{2}}$ be a open, bounded, simply connected $C^{2}$ -domain. Let $0 \neq | k | < k_{E, min, △} / 2$ . We define $Ω : = ⋃_{n \in Z} E + {(n q_{1}, 0)}^{T}$ and let ∧ and $∣ \neg$ respectively mean “and” and “not”. Then we define $\begin{array}{l} N_{Ω, +}^{k} : {(z, x) \in (R_{+}^{2} ∖ \overline{Ω}) \times (R_{+}^{2} ∖ Ω) ∣ \neg (\exists n \in Z : z_{1} - x_{1} = n q_{1} \land z_{2} = x_{2})} \to C, \\ N_{\partial Ω, +}^{k} : {(z, x) \in \partial Ω \times (R_{+}^{2} ∖ Ω) ∣ \neg (\exists n \in Z : z_{1} - x_{1} = n q_{1} \land z_{2} = x_{2})} \to C, \end{array}$ as $\begin{matrix} N_{Ω, +}^{k} (z, x) : = Γ_{+}^{k} (z, x) + R_{Ω, +}^{k} (z, x), N_{\partial Ω, +}^{k} (z, x) : = 2 Γ_{+}^{k} (z, x) + R_{\partial Ω, +}^{k} (z, x), \end{matrix}$ with $R_{Ω, +}^{k}$ , $R_{\partial Ω, +}^{k}$ being solutions to $\begin{matrix} \{\begin{matrix} (△_{x} + k^{2}) R_{Ω, +}^{k} (z, x) = 0 & for x \in R_{+}^{2} ∖ \overline{Ω}, \\ \partial_{ν_{x}} R_{Ω, +}^{k} (z, x) = - \partial_{ν_{x}} Γ_{+}^{k} (z, x) & for x \in \partial Ω, \\ R_{Ω, +}^{k} (z, x) = 0 & for x \in \partial R_{+}^{2}, \end{matrix} \end{matrix}$ where $z \in R_{+}^{2} ∖ \overline{Ω}$ is fixed, where $ν_{x}$ denotes the outside normal on $\partial Ω$ with respect to x, and $\begin{matrix} \{\begin{matrix} (△_{x} + k^{2}) R_{\partial Ω, +}^{k} (z, x) = 0 & for x \in R_{+}^{2} ∖ \overline{Ω}, \\ \partial_{ν_{x}} R_{\partial Ω, +}^{k} (z, x) = - 2 \partial_{ν_{x}} Γ_{+}^{k} (z, x) & for x \in \partial Ω, \\ R_{\partial Ω, +}^{k} (z, x) = 0 & for x \in \partial R_{+}^{2} . \end{matrix} \end{matrix}$ Here, $z \in \partial Ω$ is fixed, and $R_{Ω, +}^{k}$ , $R_{\partial Ω, +}^{k}$ satisfy the outgoing radiation condition (see [14]), thus in particular $\begin{matrix} | \partial_{x_{2}} R_{Ω, +}^{k} (z, x) - i k_{2} R_{Ω, +}^{k} (z, x) | \to 0 for x_{2} \to \infty . \end{matrix}$ Analogously, we define $N_{Ω, \times}^{k}$ , $N_{\partial Ω, \times}^{k}$ , $R_{Ω, \times}^{k}$ and $R_{\partial Ω, \times}^{k}$ .

For the remainder functions we have the following formulas: Lemma 2.9.
Let $z, x \in R^{2}$ , such that $z_{2} > q_{2}$ and $x_{2} \in \partial E$ then $\begin{matrix} R_{Ω, +}^{k} (z, x) = - \int_{\partial E} N_{\partial Ω, \times}^{k} (x, y) \partial_{ν_{y}} Γ_{+}^{k} (z, y) d σ_{y} . \end{matrix}$
Lemma 2.10.
Let $z, x \in \partial E$ and $k \neq 0$ , then $\begin{matrix} (\frac{1}{2} I + K_{\partial Ω, \times}^{k}) [R_{\partial Ω, +}^{k} (z, \cdot)] (x) = S_{\partial Ω, \times}^{k} [- 2 \partial_{ν_{\cdot}} Γ_{+}^{k} (z, \cdot)] (x) . \end{matrix}$
Lemma 2.11.
Let $z, x \in \partial E$ and $0 \neq k < k_{E, min, △} / 2$ then $\begin{matrix} (\frac{1}{2} I - K_{\partial E}^{k}) [R_{\partial E}^{k} (z, \cdot)] (x) = - \int_{E} (2 k^{2} Γ^{0} (z, y) + \frac{1}{| E |}) Γ^{k} (x, y) d y + S_{\partial E}^{k} [2 \partial_{ν_{x}} Γ^{0} (z, \cdot)] (x) . \end{matrix}$
Lemma 2.12.
For $z \in \partial E$ we have that $\begin{matrix} (2.8) & \int_{E} (\frac{1}{| E |}) \frac{1}{2 π} log (k) d y = 2 \int_{\partial E} \partial_{ν_{y}} Γ^{0} (z, y) \frac{1}{2 π} log (k) d σ_{y} . \end{matrix}$

3. One periodically arranged Helmholtz resonator

In this section, we look at a bounded, connected, domain D, which has height h. Additionally, D has a gap Λ at its boundary, which allows the incident wave $U_{0}^{k}$ to pass through. The incident wave rebounds inside D and leaves at the gap Λ, which then leads to the scattered wave $U_{s}^{k}$ . We repeat the geometry with periodicity p along the $x_{1}$ -axis and scale it by a factor δ.

We look for an accurate approximation of the resonance as well as the scattered wave in the far-field. We will see, that this approximation satisfies the Helmholtz equation with a Robin boundary condition at the $x_{1}$ -axis which approximates a Dirichlet boundary or a Neumann boundary depending on the magnitude of the incoming wave vector k and δ.

3.1. Mathematical description of the physical problem

3.1.1. Geometry

Before we consider the periodic and macroscopic problem, we first define the geometry of our Helmholtz resonator in the unit cell, shown in Fig. 1. Let $D ⋐ (- p / 2, p / 2) \times (0, 1)$ be a open, bounded, simply connected and connected domain, where $p \in R$ and p is close enough to 1. For sake of simplicity we assume that D is a $C^{2}$ -domain. We define $Λ \subset \partial D$ to be the gap of D, where Λ is a line segment parallel to the $x_{1}$ -axis. Λ is centered at ${(0, h)}^{T}$ , where $h \in (0, 1)$ is the height of Λ, and Λ has length $2 ε$ , where $ε \in (0, 1)$ and it is small enough. To facilitate future computations we assume that $h / p ⩾ 1 / 2$ .

Fig. 1.

The physical setup. In (a) we have the microscopic, non-periodic view. In (b) we have the macroscopic, periodic view.

Let us define the macroscopic view. We define the collection of periodically arranged Helmholtz resonators $Ω^{δ}$ , with period $δ p$ , and the collection of gaps of those Helmholtz resonators $Ξ^{δ}$ , where a single gap has length $2 δ ε$ , as $\begin{matrix} Ω^{δ} : = ⋃_{n \in Z} δ (D + n (\begin{matrix} p \\ 0 \end{matrix})), Ξ^{δ} : = ⋃_{n \in Z} δ (Λ + n (\begin{matrix} p \\ 0 \end{matrix})) . \end{matrix}$

3.1.2. Incident wave

Let $k : = {(k_{1}, k_{2})}^{T} \in R^{2}$ be the wave vector. We will fix the direction of the wave vector, that is, $k_{1} / k$ and $k_{2} / k$ , where $k : = | k | : = {(k_{1}^{2} + k_{2}^{2})}^{1 / 2} \in [0, \infty)$ , but let the magnitude k vary. With that, we define the function $U_{0}^{k} : R^{2} \to C$ as $\begin{matrix} U_{0}^{k} (x) : = a_{0} e^{- i k_{1} x_{1}} e^{- i k_{2} x_{2}}, \end{matrix}$ where $a_{0} \in R$ denotes the amplitude. $U_{0}^{k}$ will be our incident wave. From (2.5), it follows that $\begin{matrix} U_{0}^{k} \circ P (x) = a_{0} e^{- i k_{1} x_{1}} e^{i k_{2} x_{2}}, (U_{0}^{k} - U_{0}^{k} \circ P) (x) = - 2 i a_{0} e^{- i k_{1} x_{1}} sin (k_{2} x_{2}) . \end{matrix}$ Consider also that $U_{0}^{k}$ and $U_{0}^{k} \circ P$ are quasi-periodic with quasi-momentum $- k_{1} p$ , that is, $\begin{matrix} U_{0}^{k} (x + (\begin{matrix} p \\ 0 \end{matrix})) = e^{- i k_{1} p} U_{0}^{k} (x), U_{0}^{k} \circ P (x + (\begin{matrix} p \\ 0 \end{matrix})) = e^{- i k_{1} p} U_{0}^{k} \circ P (x) . \end{matrix}$

3.1.3. The resulting wave

With the geometry and the incident wave, we model the electromagnetic scattering problem and the resulting wave $U^{k} : R_{+}^{2} ∖ \partial Ω^{δ} \to C$ by the following system of equations: $\begin{matrix} (3.1) & \{\begin{matrix} (△ + k^{2}) U^{k} = 0 & in R_{+}^{2} ∖ \partial Ω^{δ}, \\ U^{k} |_{+} - U^{k} |_{-} = 0 & on Ξ^{δ}, \\ \partial_{ν} U^{k} |_{+} - \partial_{ν} U^{k} |_{-} = 0 & on Ξ^{δ}, \\ \partial_{ν} U^{k} |_{+} = 0 & on \partial Ω^{δ} ∖ Ξ^{δ}, \\ \partial_{ν} U^{k} |_{-} = 0 & on \partial Ω^{δ} ∖ Ξ^{δ}, \\ U^{k} = 0 & on \partial R_{+}^{2}, \end{matrix} \end{matrix}$ where $\cdot |_{+}$ denotes the limit from outside of $Ω^{δ}$ and $\cdot |_{-}$ denotes the limit from inside of $Ω^{δ}$ , and $\partial_{ν}$ denotes the normal derivative on $\partial Ω^{δ}$ . Similar to diffraction problems for gratings, the above system of equations is complemented by a certain outgoing radiation condition imposed on the scattered field $U_{s}^{k} : = U^{k} - (U_{0}^{k} - U_{0}^{k} \circ P)$ and quasi-periodicity on $U^{k}$ . More precisely, we are interested in the quasi-periodic solutions, that is, $\begin{matrix} U^{k} (x + (\begin{matrix} p \\ 0 \end{matrix})) = e^{- i k_{1} p} U^{k} (x) for x \in R_{+}^{2}, \end{matrix}$ and solutions satisfying the outgoing radiation condition, thus we have $\begin{matrix} | \partial_{x_{2}} U_{s}^{k} - i k_{2} U_{s}^{k} | \to 0 for x_{2} \to \infty . \end{matrix}$

As a remark, in the general case where $U_{0}^{k}$ is a superposition of plane-waves, we can decompose $U_{0}^{k}$ using Bloch–Flocquet theory [6,20]. We obtain a family of problems to solve, each one with its own outgoing radiation condition. The final solution is then the superposition of all these solutions.

Consider also that in absence of Helmholtz resonators the solution to (3.1) is given by $U_{0}^{k} - U_{0}^{k} \circ P$ .

3.1.4. The resulting wave in the microscopic view

Given the resulting wave $U^{k} (x)$ , the function $u^{δ k} (x) : R_{+}^{2} ∖ \partial Ω^{1} \to C$ , $u^{δ k} (x) : = U^{k} (δ x)$ represents the resulting wave, but where the Helmholtz resonators are scaled-back and thus are of height h and not $δ h$ . $u^{δ k}$ satisfies $\begin{matrix} (3.2) & \{\begin{matrix} (△ + {(δ k)}^{2}) u^{δ k} = 0 & in R_{+}^{2} ∖ \partial Ω^{1}, \\ u^{δ k} |_{+} - u^{δ k} |_{-} = 0 & on Ξ^{1}, \\ \partial_{ν} u^{δ k} |_{+} - \partial_{ν} u^{δ k} |_{-} = 0 & on Ξ^{1}, \\ \partial_{ν} u^{δ k} |_{+} = 0 & on \partial Ω^{1} ∖ Ξ^{1}, \\ \partial_{ν} u^{δ k} |_{-} = 0 & on \partial Ω^{1} ∖ Ξ^{1}, \\ u^{δ k} = 0 & on \partial R_{+}^{2}, \end{matrix} \end{matrix}$ where $\cdot |_{+}$ denotes the limit from outside of $Ω^{1}$ and $\cdot |_{-}$ denotes the limit from inside of $Ω^{1}$ , and $\partial_{ν}$ denotes the normal derivative on $\partial Ω^{1}$ .

We can adopt the quasi-periodicity from the macroscopic view and obtain $\begin{matrix} u^{δ k} (x + (\begin{matrix} p \\ 0 \end{matrix})) = e^{- i δ k_{1} p} u^{δ k} (x) for x \in R_{+}^{2} . \end{matrix}$ Defining $u_{s}^{δ k} : = u^{δ k} - (u_{0}^{δ k} - u_{0}^{δ k} \circ P)$ we also get that $| \partial_{x_{2}} u_{s}^{δ k} - i δ k_{2} u_{s}^{δ k} | \to 0$ for $x_{2} \to \infty$ . We see that $u^{δ k}$ solves the same partial differential equation like $U^{k}$ in the rescaled geometry, but with the scaled wave vector $δ k$ . We will see that we can express $u^{δ k}$ as an expansion in terms of δ and we will give an analytic expression for the first order term.

3.2. Main results

We assume that $δ k \in K : = {\hat{k} \in R ∣ 0 \neq | \hat{k} | < k_{D, min, △} / 2 and | \hat{k} |^{2} < inf {| l - \hat{k} e_{1} |^{2} ∣ l : = 2 π n / p, n \in Z ∖ {0}}}$ , where $k_{D, min, △}$ is defined as the first non-zero eigenvalue of the operator $- △$ with Neumann conditions on the boundary $\partial D$ . If we would extend the domain $K$ to $K_{C} : = {k_{*} \in C ∣ \sqrt{k_{*} {\bar{k}}_{*}} < k_{D, min, △} / 2}$ , we would obtain following resonance values for our physical problem:

Theorem 3.1.
For ε and δ small enough, we have exactly two resonance values in $K_{C}$ for $U^{k}$ . These are approximated by $\begin{array}{l} k_{+}^{δ, ε} = k_{0}^{δ, ε} + \frac{α_{0} | D |}{2 δ} c_{ε}^{3 / 2} + \frac{α_{1} | D |}{2 δ} c_{ε}^{2} + \frac{1}{δ} O (c_{ε}^{5 / 2}), \\ k_{-}^{δ, ε} = - k_{0}^{δ, ε} - \frac{α_{0} | D |}{2 δ} c_{ε}^{3 / 2} + \frac{α_{1} | D |}{2 δ} c_{ε}^{2} + \frac{1}{δ} O (c_{ε}^{5 / 2}), \\ δ k_{0}^{δ, ε} : = c_{ε}^{1 / 2} : = \sqrt{\frac{- π}{2 | D | log (ε / 2)}}, \end{array}$ where $α_{0}$ and $α_{1}$ are given by $\begin{array}{l} (3.3) & α_{0} : = R_{\partial D}^{0} ((\begin{matrix} 0 \\ h \end{matrix}), (\begin{matrix} 0 \\ h \end{matrix})) + R_{\partial Ω^{1}, +}^{0} ((\begin{matrix} 0 \\ h \end{matrix}), (\begin{matrix} 0 \\ h \end{matrix})) + \frac{1}{π} log (\frac{π}{p}) - \frac{1}{π} log (sinh (\frac{π}{p} 2 h)), \\ (3.4) & α_{1} : = \partial_{δ k} R_{\partial D}^{0} ((\begin{matrix} 0 \\ h \end{matrix}), (\begin{matrix} 0 \\ h \end{matrix})) + \partial_{δ k} R_{\partial Ω^{1}, +}^{0} ((\begin{matrix} 0 \\ h \end{matrix}), (\begin{matrix} 0 \\ h \end{matrix})) . \end{array}$

We have the following approximation for the resulting wave $U^{k}$ :
Theorem 3.2.
Let $V_{r} : = {z \in R_{+}^{2} ∣ z_{2} > r}$ . There exist constants $C_{(3.5)}, {\tilde{C}}_{(3.5)} > 0$ such that $\begin{array}{l} {‖ U_{s}^{k} - (U_{S_{p}^{⋆}}^{k} + U_{T_{p}^{⋆}}^{k} + U_{RHS, p}^{k}) ‖}_{L^{\infty} (V_{r})} + {‖ \nabla [U_{s}^{k} - (U_{S_{p}^{⋆}}^{k} + U_{T_{p}^{⋆}}^{k} + U_{RHS, p}^{k})] ‖}_{L^{\infty} (V_{r})} \\ (3.5) & ⩽ C_{(3.5)} (\frac{δ}{| log (ε) |^{3}} (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}}) + \frac{δ}{| log (ε) |^{2}} + δ ε + δ e^{- {\tilde{C}}_{(3.5)} r} + δ^{2}), \end{array}$ for δ, ε small enough and r large enough, where $\begin{array}{l} U_{S_{p}^{⋆}}^{k} (z) = - e^{i (k_{2} z_{2} - k_{1} z_{1})} \frac{h}{p} \int_{Λ} μ_{⋆}^{δ k} (y) d σ_{y}, \\ U_{T_{p}^{⋆}}^{k} (z) = e^{i (k_{2} z_{2} - k_{1} z_{1})} \frac{1}{p} \int_{Λ} \int_{\partial D} μ_{⋆}^{δ k} (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) d σ_{w} d σ_{y}, \\ \begin{matrix} U_{RHS, p}^{k} (z) & = e^{i (k_{2} z_{2} - k_{1} z_{1})} [\int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) \frac{sin (δ k_{2} y_{2}) e^{i δ k_{1} y_{1}}}{δ k_{2} p} d σ_{y} \\ - \int_{\partial D} \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} \frac{i k_{1}}{p k_{2}} sin (δ k_{2} w_{2}) \\ \frac{1}{p} cos (δ k_{2} w_{2}) \end{matrix}) e^{i δ k_{1} w_{1}} d σ_{w} d σ_{y}] . \end{matrix} \end{array}$ and where, for $y \in Λ$ , we have $\begin{matrix} μ_{⋆}^{δ k} (y) = \frac{1}{\sqrt{ε^{2} - y_{1}^{2}}} (\frac{- π}{4 | D | {(log (ε / 2))}^{2}} \frac{f^{δ k} (0)}{δ k_{+}^{δ, ε} - δ k_{-}^{δ, ε}} (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}}) + \frac{f^{δ k} (0)}{2 log (ε / 2)}), \end{matrix}$ with the constant $f^{δ k} (0) \in R$ being given by $\begin{matrix} f^{δ k} (0) = 2 i a_{0} sin (δ k_{2} h) - 2 a_{0} δ \int_{\partial D} ν_{y} \cdot (\begin{matrix} k_{1} e^{- i δ k_{1} y_{1}} sin (δ k_{2} y_{2}) \\ i k_{2} e^{- i δ k_{1} y_{1}} cos (δ k_{2} y_{2}) \end{matrix}) N_{\partial Ω^{1}, +}^{δ k} ((\begin{matrix} 0 \\ h \end{matrix}), y) d σ_{y} . \end{matrix}$

We see from Theorem 3.2 that the function $\begin{matrix} U_{app}^{k} (z) : = (U_{0}^{k} - U_{0}^{k} \circ P) (z) + U_{S_{p}^{⋆}}^{k} (z) + U_{T_{p}^{⋆}}^{k} (z) + U_{RHS, p}^{k} (z), \end{matrix}$ gives an accurate approximation of $U^{k}$ in the far-field. Moreover, it satisfies the Helmholtz equation in $R_{+}^{2}$ with the boundary condition $\begin{matrix} U_{app}^{k} (z) + δ c_{IBC} \partial_{z_{2}} U_{app}^{k} (z) = 0, for z \in \partial R_{+}^{2}, \end{matrix}$ and $U_{app}^{k} - U_{0}^{k}$ satisfies the outgoing radiation condition. The boundary condition is called ‘Impedance Boundary Condition’ and for $c_{IBC} = 0$ it yields a Dirichlet boundary condition while for $δ c_{IBC} ≫ 1$ it approximates a Neumann boundary condition for $U_{app}^{k}$ . Using Theorem 3.2, we can express $c_{IBC}$ as follows. Theorem 3.3.
The constant in the impedance boundary condition is given by $\begin{matrix} c_{IBC} = \frac{1}{2 i a_{0} δ k_{2}} C_{(3.7)}^{δ k} + O (δ), \end{matrix}$ where $C_{(3.7)}^{δ k} \in R$ is defined as $\begin{array}{l} C_{(3.7)}^{δ k} & = - \frac{h}{p} \int_{Λ} μ_{⋆}^{δ k} (y) d σ_{y} + \frac{1}{p} \int_{Λ} \int_{\partial D} μ_{⋆}^{δ k} (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) d σ_{w} d σ_{y} \\ + \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) \frac{sin (δ k_{2} y_{2}) e^{i δ k_{1} y_{1}}}{δ k_{2} p} d σ_{y} \\ (3.6) & - \int_{\partial D} \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} \frac{i k_{1}}{p k_{2}} sin (δ k_{2} w_{2}) \\ \frac{1}{p} cos (δ k_{2} w_{2}) \end{matrix}) e^{i δ k_{1} w_{1}} d σ_{w} d σ_{y}, \end{array}$ from which we see that $\begin{matrix} (3.7) & C_{(3.7)}^{δ k} = O (δ) (\frac{1}{log (ε / 2)} + \frac{1}{{(log (ε / 2))}^{2}} (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}})) . \end{matrix}$

3.3. Proof of the main results

We want to proof Theorems 3.1–3.3. First, we express the resulting wave outside the Helmholtz resonators and the resulting wave inside the Helmholtz resonators through operators acting on the resulting wave, but restricted on the gap. This leads us to a condition with the linear operator $A^{δ k, ε}$ , whose solution is the resulting wave on the gap up to a term of order $δ^{2}$ . We solve this linear system based on the procedure given in [9]. We will see that it is solvable for a complex wave vector near 0 except in three points, two of which are the resonances of our system. With this we obtain the resulting wave on the gap. Then we recover the resulting wave outside the resonators up to a term of order $δ^{2}$ . We will see, that we can split the resulting wave into a propagating wave and an evanescent one. The propagating one leads us to the impedance boundary condition constant $c_{IBC}$ .

3.3.1. Collapsing the wave-informations on the gap

Let us consider the resulting wave $u^{δ k}$ in the microscopic view described in Section 3.1.4. We will keep the microscopic view until Section 3.3.6. We define the main strip $Y : = {y \in R_{+}^{2} ∣ | y_{1} | < p / 2}$ . D is the Helmholtz resonator on that strip and Λ the gap on $\partial D$ . Furthermore, we fix $k_{1} / k = : e_{1}$ and $k_{2} / k = : e_{2}$ and assume that $δ k \in K : = {\hat{k} \in R ∣ 0 \neq | \hat{k} | < k_{D, min, △} / 2 and | \hat{k} |^{2} < inf {| l - \hat{k} e_{1} |^{2} ∣ l : = 2 π n / p, n \in Z ∖ {0}}}$ . Consider that $u^{δ k}$ is continuous on the gap Λ, thus $u^{δ k} (z)$ is well-defined for $z \in Λ$ .

Proposition 3.4.
Let $N_{D}^{k}$ be as in Definition 2.7 . Let $z \in D$ , then we have $\begin{matrix} u^{δ k} (z) = - \int_{Λ} \partial_{ν} u^{δ k} (y) N_{D}^{δ k} (z, y) d σ_{y} . \end{matrix}$ Let $z \in Λ$ . Then we have $\begin{matrix} (3.8) & u^{δ k} (z) = - \int_{Λ} \partial_{ν} u^{δ k} (y) N_{\partial D}^{δ k} (z, y) d σ_{y} . \end{matrix}$
Proposition 3.5.
Let $N_{Ω, +}^{k}$ , $N_{Ω, \times}^{k}$ be as in Definition 2.8 . Let $z \in Y ∖ \overline{D}$ . Then it follows that $\begin{matrix} u_{s}^{δ k} (z) = \int_{Λ} \partial_{ν} u^{δ k} (y) N_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} - \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} . \end{matrix}$ Let $z \in Λ$ . Then we have $\begin{matrix} (3.9) & u_{s}^{δ k} (z) = \int_{Λ} \partial_{ν} u^{δ k} (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} - \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} . \end{matrix}$

Using that $u^{δ k}$ is continuous on the gap we can deduce from the following proposition a necessary condition for $\partial_{ν} u^{δ k} |_{Λ}$ . Assume we can obtain a solution $\partial_{ν} u^{δ k}$ from that condition, then from Propositions 3.4 and 3.5 we can recover the resulting wave on Y.
Proposition 3.6 (Gap Formula).

Let $z \in Λ$ then $\begin{array}{l} \int_{Λ} \partial_{ν} u^{δ k} (y) (N_{\partial Ω^{1}, +}^{δ k} (z, y) + N_{\partial D}^{δ k} (z, y)) d σ_{y} \\ (3.10) & = \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} - (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (z) . \end{array}$

Consider that the right-hand side in (3.10) does not depend on $u^{δ k}$ and it is computable.

Proof of Proposition 3.4.
Let us look at (3.8) first. Let $z \in \partial D$ then using Green’s formula with $(△ + k^{2}) u^{δ k} = 0$ we have $\begin{array}{l} u^{δ k} (z) & = \int_{D} u^{δ k} (y) (△_{y} + k^{2}) N_{\partial D}^{δ k} (z, y) d y \\ = \int_{\partial D} u^{δ k} (y) \partial_{ν_{y}} N_{\partial D}^{δ k} (z, y) d σ_{y} - \int_{\partial D} \partial_{ν} u^{δ k} (y) N_{\partial D}^{δ k} (z, y) d σ_{y} . \end{array}$ Using that $\partial_{ν_{y}} N_{\partial D}^{δ k} (z, y) = 0$ on $\partial D$ and $\partial_{ν} u^{δ k} (y) = 0$ on $\partial D ∖ Λ$ we obtain the desired equation. We get the other equation analogously. □
Proof of Proposition 3.5.
Like in the previous proof, using Green’s formula on the complement of the Helmholtz resonator intersected with the unit strip, using the quasi-periodic Neumann function $N_{\partial Ω^{1}, +}^{δ k}$ , we readily obtain Proposition 3.5. For details consider [4, Proposition 3.5]. □
Proof of Proposition 3.6.
Using that $u^{δ k}$ is continuous at Λ we have that $u^{δ k} |_{+} (z) - u^{δ k} |_{-} (z) = u^{δ k} (z) - u^{δ k} (z) = 0$ , for $z \in Λ$ . From (3.9) and (3.8), we obtain (3.10). □

3.3.2. Expanding the gap formula in terms of δ

We define $f^{δ k} : Λ \to C$ as the right-hand side of the Gap Formula 3.6, that is, $\begin{matrix} (3.11) & f^{δ k} (z) : = \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} - (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (z) . \end{matrix}$ We identify $f^{δ k} (τ)$ with $f^{δ k} ({(τ, h)}^{T})$ , for $τ \in (- ε, ε)$ . Let us define the following operator spaces and their respective norms:

Definition 3.7.
Let $μ^{'}$ represent the distributional derivative of μ. We define $\begin{array}{l} X^{ε} : = {μ \in L^{2} ((- ε, ε)) | \int_{- ε}^{ε} \sqrt{ε^{2} - t^{2}} {| μ (t) |}^{2} d t < \infty}, \\ Y^{ε} : = {μ \in C^{0} ([- ε, ε]) ∣ μ^{'} \in X^{ε}}, \\ ‖ μ ‖_{X^{ε}} : = {(\int_{- ε}^{ε} \sqrt{ε^{2} - t^{2}} {| μ (t) |}^{2} d t)}^{1 / 2}, ‖ μ ‖_{Y^{ε}} : = {(‖ μ ‖_{X^{ε}}^{2} + {‖ μ^{'} ‖}_{X^{ε}}^{2})}^{1 / 2} . \end{array}$

Consider that for $μ \in X^{ε}$ $\begin{matrix} (3.12) & \int_{- ε}^{ε} μ (t) d t = \int_{- ε}^{ε} μ (t) \frac{\sqrt[4]{ε^{2} - t^{2}}}{\sqrt[4]{ε^{2} - t^{2}}} d t ⩽ \sqrt{π} ‖ μ ‖_{X^{ε}}, \end{matrix}$ because of the $L^{2}$ -Cauchy–Schwarz inequality.
Definition 3.8.
Let $μ \in X^{ε}$ and g be a real function and $α > 0$ . We say $μ = O_{X^{ε}} (g (α))$ for $α \to 0$ if $\frac{‖ μ ‖_{X^{ε}}}{g (α)}$ is bounded as $α \to 0$ .
Definition 3.9.
The following operators are defined as functions from $X^{ε}$ to $Y^{ε}$ . Let $n \in N$ , then $\begin{array}{l} L^{ε} [μ] (τ) : = \int_{- ε}^{ε} μ (t) log (| τ - t |) d t, K^{ε} [μ] (τ) : = \frac{1}{| D |} \int_{- ε}^{ε} μ (t) d t, \\ \begin{matrix} R^{δ k, ε} [μ] (τ) & : = \int_{- ε}^{ε} μ (t) [R_{\partial D}^{δ k} (\begin{matrix} (\begin{matrix} τ \\ h \end{matrix}), (\begin{matrix} t \\ h \end{matrix}) \end{matrix}) + R_{\partial Ω^{1}, +}^{δ k} (\begin{matrix} (\begin{matrix} τ \\ h \end{matrix}), (\begin{matrix} t \\ h \end{matrix}) \end{matrix}) \\ + \frac{1}{π} log (\frac{π}{p}) + \frac{1}{π} log (sinc | \frac{π}{p} (τ - t) |) \\ - \frac{1}{2 π} log (sinh {(\frac{π}{p} 2 h)}^{2} + sin {(\frac{π}{p} (τ - t))}^{2})] d t, \end{matrix} \\ {\hat{Γ}}_{+, n}^{k, ε} [μ] (τ) : = \int_{- ε}^{ε} μ (t) Γ_{+, n}^{k} (\begin{matrix} (\begin{matrix} τ \\ h \end{matrix}), (\begin{matrix} t \\ h \end{matrix}) \end{matrix}) d t, \\ A^{δ k, ε} [μ] (τ) : = \frac{2}{π} L^{ε} [μ] (τ) + \frac{K^{ε} [μ] (τ)}{δ^{2} k^{2}} + R^{δ k, ε} [μ] (τ), \end{array}$ where $Γ_{+, n}^{k}$ is given in Lemma 2.2.

Later, we will show that $\begin{matrix} \partial_{ν} u^{δ k} |_{Λ} = {(A^{δ k, ε})}^{- 1} [f^{δ k}] + O (δ^{2}) . \end{matrix}$
Proposition 3.10.
Let $2 ε < p$ , let $τ \in (- ε, ε)$ , then $\begin{matrix} A^{δ k, ε} [\partial_{ν} u^{δ k} |_{Λ}] (τ) + \sum_{n = 1}^{\infty} δ^{n} 2 {\hat{Γ}}_{+, n}^{k, ε} [\partial_{ν} u^{δ k} |_{Λ}] (τ) = f^{δ k} ({(τ, h)}^{T}) . \end{matrix}$
Proof.
Let $z : = {(τ, h)}^{T} \in Λ$ . From Proposition 3.6, it follows that $\begin{matrix} \int_{Λ} \partial_{ν} u^{δ k} (y) (2 Γ_{+}^{δ k} (z, y) + R_{\partial Ω^{1}, +}^{δ k} (z, y) + \frac{1}{π} log | z - y | + \frac{2 / | D |}{δ^{2} k^{2}} + R_{\partial D}^{δ k} (z, y)) d σ_{y} = f^{δ k} (z) . \end{matrix}$ Using Lemma 2.2, we can rearrange the last equation and obtain $\begin{array}{l} \int_{Λ} \partial_{ν} u^{δ k} (y) (2 Γ_{+}^{0} (z, y) + \frac{1}{π} log | z - y | + \frac{1 / | D |}{δ^{2} k^{2}} \\ (3.13) & + R_{\partial Ω^{1}, +}^{δ k} (z, y) + R_{\partial D}^{δ k} (z, y) + \sum_{n = 1}^{\infty} δ^{n} 2 Γ_{+, n}^{k} (z, y)) d σ_{y} = f^{δ k} (z) . \end{array}$ Using that Λ is a line segment parallel to the $x_{1}$ -axis, we have that $d σ_{y} = d t$ , by writing $y = {(t, h)}^{T}$ . Using (2.7) for $Γ_{+}^{0} (z, y)$ and using that the expansion in δ (see Lemma 2.2) is uniform, we can interchange the infinite sum and the integration. Let $μ (t) : = \partial_{ν} u^{δ k} (y)$ , we have that $\begin{array}{l} f^{δ k} (z) & = \int_{- ε}^{ε} μ (t) [\frac{1}{π} log | τ - t | + \frac{1}{2 π} log (sin {(\frac{π}{p} (τ - t))}^{2}) + R_{\partial Ω^{1}, +}^{δ k} (z, y) + R_{\partial D}^{δ k} (z, y) \\ - \frac{1}{2 π} log (sinh {(\frac{π}{p} 2 h)}^{2} + sin {(\frac{π}{p} (τ - t))}^{2})] d t + \frac{K^{ε} [μ] (τ)}{δ^{2} k^{2}} + \sum_{n = 1}^{\infty} δ^{n} 2 {\hat{Γ}}_{+, n}^{k, ε} [μ] (τ) . \end{array}$ Now consider that for $2 ε < p$ , we can conclude that $\begin{matrix} \frac{2}{π} L^{ε} [μ] (τ) + R^{δ k, ε} [μ] (τ) + \frac{K^{ε} [μ] (τ)}{δ^{2} k^{2}} + \sum_{n = 1}^{\infty} δ^{n} {\hat{Γ}}_{+, n}^{k, ε} [μ] (τ) = f^{δ k} (z) . \end{matrix}$ With that we have proven Proposition 3.10. □

Let us show that $L^{ε}$ is bijective and let us consider its inverse.
Proposition 3.11.
Let $0 < ε < 2$ . The operator $L^{ε} : X^{ε} \to Y^{ε}$ is linear, bounded and invertible and has the inverse $\begin{matrix} {(L^{ε})}^{- 1} [η] (t) = - \frac{1}{π^{2} \sqrt{ε^{2} - t^{2}}} p . v . \int_{- ε}^{ε} \frac{\sqrt{ε^{2} - τ^{2}} η^{'} (τ)}{t - τ} d τ + \frac{C_{L} [η]}{π log (ε / 2) \sqrt{ε^{2} - t^{2}}}, \end{matrix}$ where $\begin{matrix} (3.14) & C_{L} [η] : = η (τ) - L^{ε} [- \frac{1}{π^{2} \sqrt{ε^{2} - t^{2}}} p . v . \int_{- ε}^{ε} \frac{\sqrt{ε^{2} - s^{2}} η^{'} (s)}{t - s} d s] \end{matrix}$ is a constant depending on η and it is linear in η.
Proof.
The proof for invertiblity is given in [21, Chapter 11.5], the exact formula is derived in [6, Chapter 5.2.3]. □

With straightforward calculations we obtain the following lemma.
Lemma 3.12.
We have that $\begin{array}{l} L^{ε} [\frac{1}{\sqrt{ε^{2} - t^{2}}}] (τ) = π log (ε / 2), L^{ε} [\frac{t}{\sqrt{ε^{2} - t^{2}}}] (τ) = - π τ, \\ {(L^{ε})}^{- 1} [1] (t) = \frac{1}{π log (ε / 2) \sqrt{ε^{2} - t^{2}}} . \end{array}$

From Lemma 3.12, we also readily compute the following lemma:
Lemma 3.13.
We have that $\begin{matrix} {‖ {(L^{ε})}^{- 1} [1] ‖}_{X^{ε}} = \frac{1}{\sqrt{π} | log (ε / 2) |}, {‖ {(L^{ε})}^{- 1} [τ] ‖}_{X^{ε}} = \frac{ε}{\sqrt{2 π}} . \end{matrix}$

Since $R_{\partial D}^{δ k}$ and $R_{\partial Ω^{1}, +}^{δ k}$ are continuous, $R^{δ k, ε}$ is a compact operator. Thus we have that $\frac{2}{π} L^{ε} + R^{δ k, ε}$ is a Fredholm operator of index zero. Hence, for the operator $A^{δ k, ε}$ , extending the domain $K$ to the complex numbers in a disk-shaped form, we will see that $A^{δ k, ε}$ is invertible except for a finite amount of values of $δ k$ . Some of those values are the resonances of our physical system. To that end, we will need the following result. Lemma 3.14.
Let $R$ be the integral operator defined from $X^{ε}$ into $Y^{ε}$ by $\begin{matrix} R [μ] (τ) = \int_{- ε}^{ε} R (τ, t) μ (t) d t, \end{matrix}$ where $R (τ, t)$ is of class $C^{1, η}$ in τ and t, for $0 < η < 1$ . There exists a positive constant $C_{(3.15)}$ , independent of ε, such that $\begin{matrix} (3.15) & {‖ {(L^{ε})}^{- 1} R ‖}_{L (X^{ε}, X^{ε})} ⩽ \frac{C_{(3.15)}}{| log (ε) |}, \end{matrix}$ where $L (X^{ε}, X^{ε})$ denotes the space of bounded linear operators from $X^{ε}$ to $X^{ε}$ .
Proof.
The proof is given in [6, Lemma 5.4]. □

3.3.3. Characteristic values of $A^{δ k, ε}$ and the two resonance values

Let us first look at the characteristic values of $\begin{matrix} Q^{δ k, ε} [μ] : = \frac{2}{π} L^{ε} [μ] + \frac{K^{ε} [μ]}{δ^{2} k^{2}}, \end{matrix}$ where $μ \in X^{ε}$ and $δ k \in K_{C} : = {k_{*} \in C ∣ \sqrt{k_{*} {\bar{k}}_{*}} < k_{D, min, △} / 2}$ . Let ${(\cdot, \cdot)}_{ε}$ be defined as the $L^{2} ((- ε, ε))$ inner product.

Lemma 3.15.
$Q^{δ k, ε}$ has exactly the two characteristic values $\pm k_{0}^{δ, ε}$ where both have the characteristic function $μ_{0}^{δ, ε}$ , with $\begin{array}{l} k_{0}^{δ, ε} = \frac{1}{δ} {(- \frac{π}{2 | D |} {({(L^{ε})}^{- 1} [1], 1)}_{ε})}^{1 / 2} = \frac{1}{δ} {(- \frac{π}{2 | D | log (ε / 2)})}^{1 / 2}, \\ μ_{0}^{ε} = - \frac{π}{2 | D |} \frac{{(L^{ε})}^{- 1} [1]}{{(δ k_{0}^{δ, ε})}^{2}}, \end{array}$ after imposing ${(μ, 1)}_{ε} = 1$ .

Consider that $k_{0}^{δ, ε}$ is real and positive.
Proof.
We are looking for $μ \in X^{ε}$ such that $Q^{δ k, ε} [μ] = 0$ . Since ${(L^{ε})}^{- 1} [0] = 0$ , the last equation is equivalent to $\begin{matrix} (3.16) & \frac{2}{π} μ + \frac{1}{| D |} \frac{{(μ, 1)}_{ε} {(L^{ε})}^{- 1} [1]}{{(δ k)}^{2}} = 0 . \end{matrix}$ Applying ${(\cdot, 1)}_{ε}$ on both sides yields $\begin{matrix} (3.17) & {(μ, 1)}_{ε} (\frac{2}{π} + \frac{1}{| D |} \frac{{({(L^{ε})}^{- 1} [1], 1)}_{ε}}{{(δ k)}^{2}}) = 0 . \end{matrix}$ If ${(μ, 1)}_{ε} = 0$ , then $L^{ε} [μ] = 0$ , because of the condition $Q^{δ k, ε} [μ] = 0$ , and then $μ = 0$ , since $L^{ε}$ is invertible and linear. But 0 cannot be a characteristic function, by definition. Hence $(μ, 1) \neq 0$ . Thus the second factor in (3.17) has to be zero. This leads us to $\begin{matrix} δ^{2} k^{2} = - \frac{π}{2 | D |} {({(L^{ε})}^{- 1} [1], 1)}_{ε} . \end{matrix}$ Using Lemma 3.12, we can calculate that ${({(L^{ε})}^{- 1} [1], 1)}_{ε} = \frac{1}{log (ε / 2)}$ , and obtain the characteristic values.

As for the characteristic functions, we rewrite (3.16) as $\begin{matrix} \frac{μ}{(μ, 1)} = - \frac{π}{2 | D |} \frac{{(L^{ε})}^{- 1} [1]}{{(δ k)}^{2}} . \end{matrix}$ Imposing the normalization on μ we have proven our statement. □

To facilitate future expressions we define $\begin{matrix} c_{ε} : = - \frac{π}{2 | D | log (ε / 2)} = - \frac{π}{2 | D |} {({(L^{ε})}^{- 1} [1], 1)}_{ε} . \end{matrix}$ Next we will look at the characteristic values of $A^{δ k, ε}$ . Denote ${\tilde{L}}^{δ k, ε} : = L^{ε} + \frac{π}{2} R^{δ k, ε}$ and $S^{δ k, ε} : = \frac{π}{2} {(L^{ε})}^{- 1} R^{δ k, ε}$ , where we fixed the angles of the incoming wave vector, but let the magnitude be complex. Using that $R^{δ k, ε}$ is in $C^{1, η}$ , for $η \in (0, 1)$ , and $L^{ε}$ invertible and using Lemma 3.14, we can apply the Neumann series, whenever ε is small enough, and thus we have $\begin{matrix} (3.18) & {({\tilde{L}}^{δ k, ε})}^{- 1} = {(I + S^{δ k, ε})}^{- 1} {(L^{ε})}^{- 1} = \sum_{l = 0}^{\infty} {(- S^{δ k, ε})}^{l} {(L^{ε})}^{- 1}, \end{matrix}$ where $I$ denotes the identity function in $X^{ε}$ . We then define $\begin{matrix} (3.19) & A^{δ k, ε} : = - \frac{π}{2 | D |} {({({\tilde{L}}^{δ k, ε})}^{- 1} [1], 1)}_{ε} . \end{matrix}$
Lemma 3.16.
The characteristic values of $A^{δ k, ε}$ are zeros of the function $δ^{2} k^{2} - A^{δ k, ε}$ and we have the asymptotic formula $\begin{matrix} δ^{2} k^{2} = {(δ k_{0}^{δ, ε})}^{2} + δ^{2} O (c_{ε}^{2}), for ε \to 0 . \end{matrix}$
Proof.
We prove that the zeros of $δ^{2} k^{2} - A^{δ k, ε}$ are the characteristic values in the same way as in the proof of Lemma 3.15, but substitute $L^{ε}$ with ${\tilde{L}}^{δ k, ε}$ . In order to derive the asymptotic formulas, we use (3.18) and Lemma 3.14. □
Proposition 3.17.
There exist two characteristic values, counting multiplicity, for the operator $A^{δ k, ε}$ in $K_{C}$ . Moreover, they have the asymptotic expansions $\begin{matrix} δ k = \pm δ k_{0}^{δ, ε} + δ O (c_{ε}), for ε \to 0 . \end{matrix}$

The proof relies on the generalized Rouché’s theorem [6, Theorem 1.15]. For details we refer to [4, Proposition 3.17].

Now, let us give an asymptotic expression for those resonances. We recall that $α_{0}$ and $α_{1}$ , defined in (3.3) and (3.4), are used in the decomposition of $R^{δ k, ε}$ , that is, $R^{δ k, ε} = R_{1}^{δ k, ε} + R_{2}^{δ k, ε} + R_{3}^{δ k, ε}$ , with $\begin{array}{l} (3.20) & R_{1}^{δ k, ε} [μ] : = α_{0} {(μ, 1)}_{ε}, \\ (3.21) & R_{2}^{δ k, ε} [μ] : = δ k α_{1} {(μ, 1)}_{ε}, \\ (3.22) & R_{3}^{δ k, ε} [μ] (τ) : = \int_{- ε}^{ε} μ (t) (τ \tilde{\partial_{τ}} R^{δ k} (τ, t) + t \tilde{\partial_{t}} R^{δ k} (τ, t) + δ^{2} k^{2} {\tilde{\partial_{δ k}}}^{2} R^{δ k} (τ, t)) d t, \end{array}$ with $R^{δ k} (τ, t)$ denotes the kernel of $R^{δ k, ε}$ , see Definition 3.9, and $\tilde{\partial}$ denotes the derivative part of the remainder in Taylor’s theorem in the Peano form.
Lemma 3.18.
For all $δ k \in K_{C}$ , $R_{3}^{δ k, ε}$ satisfies the estimate $\begin{matrix} (3.23) & {‖ {(L^{ε})}^{- 1} R_{3}^{δ k, ε} ‖}_{L (X^{ε}, X^{ε})} = O (ε \cdot c_{ε} + c_{ε} \cdot δ^{2} k^{2}) . \end{matrix}$

The proof mimics the proof given in [6, Proof of Lemma 5.4]. For more details consider [4, Lemma 3.18].

Let $l ⩾ 1$ be an integer, we define $\begin{matrix} S_{(l)}^{δ k, ε} : = - \frac{π}{2 | D |} {({(- S^{δ k, ε})}^{l} {(L^{ε})}^{- 1} [1], 1)}_{ε} . \end{matrix}$ Because of (3.19), we can write $A^{δ k, ε} = c_{ε} + \sum_{l = 1}^{\infty} S_{(l)}^{δ k, ε}$ . We want to give a second order analytic expression for $S_{(1)}^{δ k, ε} = \frac{π}{2 | D |} {(\frac{π}{2} {(L^{ε})}^{- 1} R^{δ k, ε} {(L^{ε})}^{- 1} [1], 1)}_{ε}$ . To this end, we define $\begin{matrix} (3.24) & S_{(1, j)}^{δ k, ε} : = \frac{π}{2 | D |} {(\frac{π}{2} {(L^{ε})}^{- 1} R_{j}^{δ k, ε} {(L^{ε})}^{- 1} [1], 1)}_{ε} for j \in {1, 2, 3} . \end{matrix}$ We obtain that second order analytic expression with the following lemma:
Lemma 3.19.
For all $l \in N$ , we have that $| S_{(l)}^{δ k, ε} | = O (c_{ε}^{l + 1})$ , moreover $\begin{matrix} S_{(1, 1)}^{δ k, ε} = α_{0} c_{ε}^{2} | D |, S_{(1, 2)}^{δ k, ε} = α_{1} δ k c_{ε}^{2} | D |, S_{(1, 3)}^{δ k, ε} = O (ε c_{ε}^{2} + c_{ε}^{2} δ^{2} k^{2}) . \end{matrix}$
Proof.
The proof follows from straightforward calculation using Lemma 3.14 and the expressions in (3.23), (3.20)–(3.22) and (3.24). □

Now we can deduce that $\begin{matrix} A^{δ k, ε} = c_{ε} + α_{0} c_{ε}^{2} | D | + α_{1} δ k c_{ε}^{2} | D | + O (ε c_{ε}^{2} + c_{ε}^{2} δ^{2} k^{2}) . \end{matrix}$ Now we are able to solve for the zeros in $δ^{2} k^{2} - A^{δ k, ε}$ , and thus get the characteristic values of $A^{δ k, ε}$ . To this end, consider that $\sqrt{1 + x} = 1 + \frac{1}{2} x - O (x^{2})$ for $| x | < 1$ , thus $\begin{matrix} (3.25) & A_{⋆}^{δ k, ε} : = \sqrt{A^{δ k, ε}} = \sqrt{c_{ε}} (1 + c_{ε} \frac{α_{0}}{2} | D | + δ k c_{ε} \frac{α_{1}}{2} | D | + O (ε c_{ε} + c_{ε} δ^{2} k^{2})), \end{matrix}$ which leads us to the factorization, $\begin{matrix} δ^{2} k^{2} - A^{δ k, ε} = (δ k - A_{⋆}^{δ k, ε}) \cdot (δ k + A_{⋆}^{δ k, ε}) . \end{matrix}$ Thus, the roots for $δ^{2} k^{2} - A^{δ k, ε}$ are those for $δ k - A_{⋆}^{δ k, ε}$ and $k + A_{⋆}^{δ k, ε}$ .
Proposition 3.20.
There are exactly two characteristic values for the operator $A^{δ k, ε}$ in $K_{C}$ . Those are approximated by $\begin{array}{l} δ k_{+}^{δ, ε} = δ k_{0}^{δ, ε} + \frac{α_{0} | D |}{2} c_{ε}^{3 / 2} + \frac{α_{1} | D |}{2} c_{ε}^{2} + O (c_{ε}^{5 / 2}), \\ δ k_{-}^{δ, ε} = - δ k_{0}^{δ, ε} - \frac{α_{0} | D |}{2} c_{ε}^{3 / 2} + \frac{α_{1} | D |}{2} c_{ε}^{2} + O (c_{ε}^{5 / 2}), \\ δ k_{0}^{δ, ε} : = c_{ε}^{1 / 2} : = \sqrt{\frac{- π}{2 | D | log (ε / 2)}} . \end{array}$
Proof.
We define $\begin{matrix} A_{⋆, (0)}^{δ k, ε} : = \sqrt{c_{ε}} (1 + c_{ε} \frac{α_{0}}{2} | D | + δ k c_{ε} \frac{α_{1}}{2} | D |) . \end{matrix}$ For ε small enough, the zeros of $δ k - A_{⋆, (0)}^{δ k, ε}$ are exactly the zeros of $δ k - A_{⋆}^{δ k, ε}$ , up to a term in $O (c_{ε}^{5 / 2})$ . This follows readily from Rouché’s Theorem.

As for the zeros of $δ k - A_{⋆, (0)}^{δ k, ε}$ , we obtain them through the approach $δ k = γ_{1} c_{ε}^{1 / 2} + γ_{2} c_{ε}^{1} + γ_{3} c_{ε}^{3 / 2} + γ_{4} c_{ε}^{2}$ . Inserting the approach into $A_{⋆, (0)}^{δ k, ε}$ and solving $δ k - A_{⋆, (0)}^{δ k, ε} = 0$ , we obtain $γ_{1} = 1$ , $γ_{2} = 0$ , $γ_{3} = \frac{α_{0} | D |}{2}$ , $γ_{4} = \frac{α_{1} | D |}{2}$ . An analogous argumentation leads to the zeros of $δ k + A_{⋆, (0)}^{δ k, ε}$ . □

With Proposition 3.20 we immediately obtain Theorem 3.1.
3.3.4. Inversion of $A^{δ k, ε}$ – Solving the first order linear equation

We know now that $A^{δ k, ε}$ is invertible for $δ k \in K_{C}$ , except at the characteristic values $k = k_{+}^{δ, ε}$ and $k = k_{-}^{δ, ε}$ and at the pole, $k = 0$ . Let us examine, how we can express ${(A^{δ k, ε})}^{- 1} [f^{δ k}]$ , where $f^{δ k}$ is given by (3.11).

First consider that we already know that the equation $A^{δ k, ε} [μ] = f^{δ k}$ is equivalent to $\begin{matrix} (3.26) & \frac{2}{π} μ + \frac{{(μ, 1)}_{ε} {({\tilde{L}}^{δ k, ε})}^{- 1} [1]}{| D | δ^{2} k^{2}} = {({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}] . \end{matrix}$ Applying ${(\cdot, 1)}_{ε}$ on both sides, we obtain $\begin{matrix} {(μ, 1)}_{ε} (1 - \frac{A^{δ k, ε}}{δ^{2} k^{2}}) = \frac{π}{2} {({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], 1)}_{ε} . \end{matrix}$ And thus $\begin{matrix} (3.27) & {(μ, 1)}_{ε} = \frac{δ^{2} k^{2}}{δ^{2} k^{2} - A^{δ k, ε}} \frac{π}{2} {({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], 1)}_{ε} . \end{matrix}$

Lemma 3.21.
For $δ k \in K_{C}$ and $ε \to 0$ we have that $\begin{matrix} \frac{1}{δ^{2} k^{2} - A^{δ k, ε}} = \frac{1}{(δ k - δ k_{+}^{δ, ε}) (δ k - δ k_{-}^{δ, ε})} (1 + O (c_{ε})) . \end{matrix}$

We refer to [4, Lemma 3.21] for the detailed proof. Lemma 3.22.
We have $\begin{array}{l} {({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}] (t) = f^{δ k} (0) {(L^{ε})}^{- 1} [1] + O (δ) O_{X^{ε}} (c_{ε}^{2}) + O (δ) O_{X^{ε}} (ε), \\ {({\tilde{L}}^{δ k, ε})}^{- 1} [1] (t) = {(L^{ε})}^{- 1} [1] + O_{X^{ε}} (c_{ε}^{2}), \end{array}$ for $ε \to 0$ in the $X^{ε}$ norm and $δ \to 0$ .
Proof.
We write $f^{δ k} (τ) = f^{δ k} (0) + τ g^{δ k} (τ)$ with a smooth function $g^{δ k} \in C^{1, 1 / 2} ([- ε, ε])$ . We readily see that $f^{δ k} = O (δ)$ and $g^{δ k} = O (δ)$ . Recall that $\begin{matrix} {({\tilde{L}}^{δ k, ε})}^{- 1} = \sum_{l = 0}^{\infty} {(- S^{δ k, ε})}^{l} {(L^{ε})}^{- 1} . \end{matrix}$ Using that $‖ S^{δ k, ε} ‖_{L (X^{ε}, X^{ε})} = O (c_{ε})$ and according to Lemma 3.13, $‖ {(L^{ε})}^{- 1} [1] ‖_{X^{ε}} = O (c_{ε})$ , we have for $n \in N$ that $\begin{array}{l} {‖ {(S^{δ k, ε})}^{n} {(L^{ε})}^{- 1} [1] ‖}_{X^{ε}} = O (c_{ε}^{n + 1}), \\ {‖ {(S^{δ k, ε})}^{n} {(L^{ε})}^{- 1} [τ g^{δ k} (τ)] ‖}_{X^{ε}} = O (c_{ε}^{n}) {‖ {(L^{ε})}^{- 1} [τ g^{δ k} (τ)] ‖}_{X^{ε}} . \end{array}$ Consider that $ε = O (c_{ε}^{n})$ , thus it is enough to show $\begin{matrix} {‖ {(L^{ε})}^{- 1} [τ g^{δ k} (τ)] ‖}_{X^{ε}} = O (δ) O (ε) . \end{matrix}$ To this end we have to mimic [6, Proof of Lemma 5.4]. For more details, we refer to [4, Lemma 3.22]. □
Proposition 3.23.
Let $δ k \in K_{C} ∖ {0, δ k_{+}^{δ, ε}, δ k_{-}^{δ, ε}}$ , there exists a unique solution to the equation $A^{δ k, ε} [μ] = f^{δ k}$ . Moreover, the solution can be written as $μ = μ_{⋆} + μ_{\sim}$ , where $\begin{array}{l} μ_{⋆} = \frac{π c_{ε}}{2} \frac{f^{δ k} (0) {(L^{ε})}^{- 1} [1]}{δ k_{+}^{δ, ε} - δ k_{-}^{δ, ε}} (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}}) + \frac{π}{2} f^{δ k} (0) {(L^{ε})}^{- 1} [1], \\ μ_{\sim} = O (δ) (O_{X^{ε}} (c_{ε}^{3}) (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}}) + O_{X^{ε}} (c_{ε}^{2}) + O_{X^{ε}} (ε)) . \end{array}$
Proof.
From $A^{δ k, ε} [μ] = f^{δ k}$ we get that $\begin{matrix} (3.28) & \frac{2}{π} μ + \frac{{(μ, 1)}_{ε} {({\tilde{L}}^{δ k, ε})}^{- 1} [1]}{| D | δ^{2} k^{2}} = {({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}] . \end{matrix}$ After rearranging, see (3.26), we derive $\begin{matrix} {(μ, 1)}_{ε} = \frac{δ^{2} k^{2}}{δ^{2} k^{2} - A^{δ k, ε}} \frac{π}{2} {({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], 1)}_{ε} . \end{matrix}$ Then, we obtain by applying Lemma 3.21 that $\begin{matrix} {(μ, 1)}_{ε} = \frac{π}{2} {({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], 1)}_{ε} \frac{δ^{2} k^{2}}{(δ k - δ k_{+}^{δ, ε}) (δ k - δ k_{-}^{δ, ε})} (1 + O (c_{ε})) . \end{matrix}$ Using that $\frac{1}{(x - a) (x - b)} = \frac{1}{a - b} (\frac{1}{x - a} - \frac{1}{x - b})$ , we have $\begin{matrix} (3.29) & {(μ, 1)}_{ε} = \frac{π}{2} {({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], 1)}_{ε} \frac{δ^{2} k^{2}}{δ k_{+}^{δ, ε} - δ k_{-}^{δ, ε}} (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}}) (1 + O (c_{ε})) . \end{matrix}$ To get the solution we use (3.28), but insert (3.29) into it to arrive at $\begin{array}{l} \frac{2}{π} μ & = - \frac{π}{2} \frac{{({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], 1)}_{ε} {({\tilde{L}}^{δ k, ε})}^{- 1} [1]}{| D | (δ k_{+}^{δ, ε} - δ k_{-}^{δ, ε})} (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}}) (1 + O (c_{ε})) \\ + {({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}] . \end{array}$ Then from Lemma 3.22 we get after rearranging that $\begin{array}{l} \frac{2}{π} μ & = \frac{- π}{2 | D |} \frac{f^{δ k} (0) {({(L^{ε})}^{- 1} [1], 1)}_{ε} {(L^{ε})}^{- 1} [1]}{δ k_{+}^{δ, ε} - δ k_{-}^{δ, ε}} (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}}) + f^{δ k} (0) {(L^{ε})}^{- 1} [1] \\ + O (δ) O_{X^{ε}} (c_{ε}^{3}) (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}}) + O (δ) O_{X^{ε}} (c_{ε}^{2}) + O (δ) O_{X^{ε}} (ε) . \end{array}$ With that, and using ${({(L^{ε})}^{- 1} [1], 1)}_{ε} = - \frac{2 | D |}{π} c_{ε}$ , the proof for Proposition 3.23 readily follows. □

3.3.5. Asymptotic expansion of our solution to the physical problem

In Proposition 3.10 we established $\begin{matrix} (3.30) & A^{δ k, ε} [\partial_{ν} u^{δ k} |_{Λ}] (τ) + \sum_{n = 1}^{\infty} δ^{n} 2 {\hat{Γ}}_{+, n}^{k, ε} [\partial_{ν} u^{δ k} |_{Λ}] (τ) = f^{δ k} (τ), \end{matrix}$ and we know that for $δ k \in K_{C} ∖ {0, δ k_{+}^{δ, ε}, δ k_{-}^{δ, ε}}$ that $A^{δ k, ε}$ is invertible, see Proposition 3.23. Then for δ small enough we can use the Neumann series and obtain $\begin{array}{l} {(A^{δ k, ε} + \sum_{n = 1}^{\infty} 2 δ^{n} {\hat{Γ}}_{+, n}^{k, ε})}^{- 1} & = \sum_{m = 0}^{\infty} {(- {(A^{δ k, ε})}^{- 1} \sum_{n = 1}^{\infty} 2 δ^{n} {\hat{Γ}}_{+, n}^{k, ε})}^{m} {(A^{δ k, ε})}^{- 1} \\ = {(A^{δ k, ε})}^{- 1} - 2 δ {(A^{δ k, ε})}^{- 1} {\hat{Γ}}_{+, 1}^{k, ε} + O (δ^{2}) O_{L (X^{ε}, X^{ε})} (1) . \end{array}$

Thus we have from solving (3.30) $\begin{matrix} \partial_{ν} u^{δ k} |_{Λ} = {(A^{δ k, ε})}^{- 1} [f^{δ k}] + O (δ^{2}), \end{matrix}$ where we use that $f^{δ k} = O (δ)$ , that ${\hat{Γ}}_{+, 1}^{k, ε}$ is linear, and the formula for ${(A^{δ k, ε})}^{- 1}$ . With Proposition 3.23 we split ${(A^{δ k, ε})}^{- 1} [f^{δ k}]$ into $μ_{⋆}^{δ k}$ and $μ_{\sim}^{δ k}$ and thus we have for $t \in (- ε, ε)$ $\begin{matrix} \partial_{ν} u^{δ k} ((\begin{matrix} t \\ h \end{matrix})) = μ_{⋆}^{δ k} (t) + μ_{\sim}^{δ k} (t) + O (δ^{2}), \end{matrix}$ and we see from Proposition 3.23 also that $\partial_{ν} u^{δ k} |_{Λ} = O (δ)$ .

Now we want to calculate the first order expansion term in δ for the solution in the far-field. Let $z \in R^{2}$ , where $z_{2} ≫ 1$ . The following asymptotic expansion holds.

Lemma 3.24.
We have for $z \in R_{+}^{2}$ , $z_{2} > 1$ , $\begin{array}{l} u_{s}^{δ k} (z) & = u_{RHS, p}^{δ k} (z) + u_{RHS, e}^{δ k} (z) + S_{+, p}^{δ k} [\partial_{ν} u^{δ k} |_{Λ}] (z) + S_{+, e}^{δ k} [\partial_{ν} u^{δ k} |_{Λ}] (z) \\ (3.31) & + T_{+, p}^{δ k} [\partial_{ν} u^{δ k} |_{Λ}] (z) + T_{+, e}^{δ k} [\partial_{ν} u^{δ k} |_{Λ}] (z) + O (δ^{2}), \end{array}$ where for $μ \in X^{ε}$ $\begin{array}{l} S_{+, p}^{δ k} [μ] (z) : = \int_{Λ} μ (y) Γ_{+, p}^{δ k} (z, y) d σ_{y}, S_{+, e}^{δ k} [μ] (z) : = \int_{Λ} μ (y) Γ_{+, e}^{δ k} (z, y) d σ_{y}, \\ T_{+, p}^{δ k} [μ] (z) : = - \int_{Λ} \int_{\partial D} μ (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+, p}^{δ k} (z, w) d σ_{w} d σ_{y}, \\ T_{+, e}^{δ k} [μ] (z) : = - \int_{Λ} \int_{\partial D} μ (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+, e}^{δ k} (z, w) d σ_{w} d σ_{y}, \end{array}$ and $\begin{array}{l} \begin{matrix} u_{RHS, p}^{δ k} (z) & = e^{i δ (k_{2} z_{2} - k_{1} z_{1})} [\int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) \frac{sin (δ k_{2} y_{2}) e^{i δ k_{1} y_{1}}}{δ k_{2} p} d σ_{y} \\ - \int_{\partial D} \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} \frac{i k_{1}}{p k_{2}} sin (δ k_{2} w_{2}) \\ \frac{1}{p} cos (δ k_{2} w_{2}) \end{matrix}) e^{i δ k_{1} w_{1}} d σ_{w} d σ_{y}], \end{matrix} \\ \begin{matrix} u_{RHS, e}^{δ k} (z) & = [- \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) Γ_{+, e}^{δ k} (z, y) d σ_{y} \\ + \int_{\partial D} \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+, e}^{δ k} (z, w) d σ_{w} d σ_{y}] . \end{matrix} \end{array}$
Proof.
We define $u_{f}^{δ k} (z) : = u^{δ k} (z) + f^{δ k} (z) = : u_{s}^{δ k} (z) - u_{RHS}^{δ k} (z)$ . Then we have from Proposition 3.5 that $\begin{array}{l} u_{f}^{δ k} (z) & = \int_{Λ} \partial_{ν} u^{δ k} (y) N_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} \\ = \int_{Λ} \partial_{ν} u^{δ k} (y) Γ_{+}^{δ k} (z, y) d σ_{y} + \int_{Λ} \partial_{ν} u^{δ k} (y) R_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} \\ = : S_{+}^{δ k} [\partial_{ν} u^{δ k} |_{Λ}] + T_{+}^{δ k} [\partial_{ν} u^{δ k} |_{Λ}] . \end{array}$ Using the splitting $Γ_{+}^{k} = Γ_{+, p}^{k} + Γ_{+, e}^{k}$ , see Proposition 2.4, we already obtain $S_{+, p}^{δ k}$ and $S_{+, e}^{δ k}$ . To study the terms $u_{RHS}^{δ k}$ and $T_{+}^{δ k}$ , consider that $\begin{matrix} R_{Ω, +}^{k} (z, y) = - \int_{\partial E} N_{\partial Ω, \times}^{k} (y, w) \partial_{ν_{y}} Γ_{+}^{k} (z, w) d σ_{w}, \end{matrix}$ from Proposition 2.9. Thus $\begin{matrix} \int_{Λ} \partial_{ν} u^{δ k} (y) R_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} = - \int_{Λ} \int_{\partial D} \partial_{ν} u^{δ k} (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+}^{δ k} (z, w) d σ_{w} d σ_{y}, \end{matrix}$ and $\begin{array}{l} \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) R_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} \\ (3.32) & = - \int_{Λ} \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+}^{δ k} (z, w) d σ_{w} d σ_{y} . \end{array}$ Using again the splitting $\nabla Γ_{+}^{k} = \nabla Γ_{+, p}^{k} + \nabla Γ_{+, e}^{k}$ and the explicit formula for $Γ_{+, p}^{k}$ and $\nabla Γ_{+, p}^{k}$ we obtain the formulas in Lemma 3.24. □

Let us approximate $S_{+}^{δ k}$ and $T_{+}^{δ k}$ . Let $z_{2} > 1$ . We define $\begin{array}{l} S_{+, p, 0}^{δ k} [μ] (z) : = - e^{i (δ k_{2} z_{2} - δ k_{1} z_{1})} \frac{1}{p} \int_{Λ} y_{2} μ (y) d σ_{y}, \\ (3.33) & \begin{matrix} S_{+, e, 0}^{δ k} [μ] (z) & : = e^{- i δ k_{1} z_{1}} (\int_{Λ} μ (y) Γ_{+}^{0} (z, y) d σ_{y} + \frac{1}{p} \int_{Λ} y_{2} μ (y) d σ_{y}) \\ = - e^{- i δ k_{1} z_{1}} \int_{Λ} μ (y) \sum_{\begin{array}{c} n \in Z ∖ {0} \\ l : = 2 π n / p \end{array}} \frac{1}{p | l |} e^{i l (z_{1} - y_{1})} e^{- | l | z_{2}} sinh (| l | y_{2}) d σ_{y}, \end{matrix} \\ T_{+, p, 0}^{δ k} [μ] (z) : = e^{i (δ k_{2} z_{2} - δ k_{1} z_{1})} \frac{1}{p} \int_{Λ} \int_{\partial D} μ (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) d σ_{w} d σ_{y}, \\ \begin{matrix} T_{+, e, 0}^{δ k} [μ] (z) & : = - e^{- i δ k_{1} z_{1}} (\int_{Λ} \int_{\partial D} μ (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+}^{0} (z, w) d σ_{w} d σ_{y} \\ + e^{- i δ k_{2} z_{2}} T_{+, p, 0}^{δ k} [μ] (z)) . \end{matrix} \end{array}$
Lemma 3.25.
There exist $C_{(3.34)} > 0$ , $C_{(3.35)} > 0$ , $C_{(3.36)} > 0$ and $C_{(3.37)} > 0$ such that, for all $z \in R_{+}^{2}$ with $z_{2} > 1$ and all $μ \in X^{ε}$ , it holds that $\begin{array}{l} (3.34) & | (S_{+, p}^{δ k} - S_{+, p, 0}^{δ k}) [μ] (z) | + \frac{1}{δ} | \nabla (S_{+, p}^{δ k} - S_{+, p, 0}^{δ k}) [μ] (z) | ⩽ C_{(3.34)} ‖ μ ‖_{X^{ε}} δ^{2}, \\ (3.35) & | (S_{+, e}^{δ k} - S_{+, e, 0}^{δ k}) [μ] (z) | + \frac{1}{δ} | \nabla (S_{+, e}^{δ k} - S_{+, e, 0}^{δ k}) [μ] (z) | ⩽ C_{(3.35)} ‖ μ ‖_{X^{ε}} δ, \\ (3.36) & | (T_{+, p}^{δ k} - T_{+, p, 0}^{δ k}) [μ] (z) | + \frac{1}{δ} | \nabla (T_{+, p}^{δ k} - T_{+, p, 0}^{δ k}) [μ] (z) | ⩽ C_{(3.36)} ‖ μ ‖_{X^{ε}} δ, \\ (3.37) & | (T_{+, e}^{δ k} - T_{+, e, 0}^{δ k}) [μ] (z) | + \frac{1}{δ} | \nabla (T_{+, e}^{δ k} - T_{+, e, 0}^{δ k}) [μ] (z) | ⩽ C_{(3.37)} ‖ μ ‖_{X^{ε}} δ . \end{array}$
Proof.
Let us consider $S_{+}^{δ k}$ first. Using the following splitting for $Γ_{+}^{δ k} (z, y)$ , which is given in Proposition 2.4, we have $\begin{matrix} Γ_{+}^{δ k} (z, y) = Γ_{+, p}^{δ k} (z, x) + Γ_{+, e}^{δ k} (z, x), \end{matrix}$ where $Γ_{+, p}^{δ k} (z, x) = - \frac{sin (δ k_{2} x_{2})}{δ k_{2} p} e^{i δ (k_{2} z_{2} - k_{1} z_{1})} e^{i δ k_{1} x_{1}}$ . $S_{+, p, 0}^{δ k}$ is the zeroth order term of the Taylor expansion with respect to δ of $Γ_{+, p}^{δ k} (z, x)$ , but without the $e^{i δ (k_{2} z_{2} - k_{1} z_{1})}$ term, and $S_{+, e, 0}^{δ k}$ is the zeroth order term of the Taylor expansion with respect to δ of $Γ_{+, e}^{δ k} (z, x)$ , but without the $e^{- i δ k_{1} z_{1}}$ term. To see that, write $S_{+, e, 0}^{δ k}$ using (2.1) for $z_{2} > 1$ , and rewrite $\int_{Λ} \partial_{ν} u^{δ k} (y) Γ_{+}^{0} (z, y) d σ_{y}$ with it. With that, we obtain the formula in (3.33).

Inserting these exact formulas into the expressions in Lemma 3.25 and using $\int_{- ε}^{ε} μ (t) φ (t) d t ⩽ ‖ φ ‖_{C^{0}} ‖ μ ‖_{X^{ε}} \sqrt{π}$ , where $φ \in C^{0} ([- ε, ε])$ and where we used the $L^{2}$ -Cauchy–Schwarz inequality, yields the desired estimations for $S_{+}^{δ k}$ . For $T_{+}^{δ k}$ it works analogously. □

We define $\begin{array}{l} u_{S_{p}^{⋆}}^{δ k} (z) : = S_{+, p, 0}^{δ k} [μ_{⋆}^{δ k}] (z), u_{T_{p}^{⋆}}^{δ k} (z) : = T_{+, p, 0}^{δ k} [μ_{⋆}^{δ k}] (z), \\ u_{S_{e}^{⋆}}^{δ k} (z) : = S_{+, e, 0}^{δ k} [μ_{⋆}^{δ k}] (z), u_{T_{e}^{⋆}}^{δ k} (z) : = T_{+, e, 0}^{δ k} [μ_{⋆}^{δ k}] (z) . \end{array}$ We then have the following proposition: Proposition 3.26.
Let $V_{r} : = {z \in R_{+}^{2} ∣ z_{2} > r}$ . There exists a constant $C_{(3.38)} > 0$ such that $\begin{array}{l} {‖ u_{s}^{δ k} - (u_{S_{p}^{⋆}}^{δ k} + u_{S_{e}^{⋆}}^{δ k} + u_{T_{p}^{⋆}}^{δ k} + u_{T_{e}^{⋆}}^{δ k} + u_{RHS, p}^{δ k} + u_{RHS, e}^{δ k}) ‖}_{L^{\infty} (V_{r})} \\ + \frac{1}{δ} {‖ \nabla [u_{s}^{δ k} - (u_{S_{p}^{⋆}}^{δ k} + u_{S_{e}^{⋆}}^{δ k} + u_{T_{p}^{⋆}}^{δ k} + u_{T_{e}^{⋆}}^{δ k} + u_{RHS, p}^{δ k} + u_{RHS, e}^{δ k})] ‖}_{L^{\infty} (V_{r})} \\ (3.38) & ⩽ C_{(3.38)} (δ c_{ε}^{3} (\frac{1}{δ k - δ k_{+}^{δ, ε}} - \frac{1}{δ k - δ k_{-}^{δ, ε}}) + δ c_{ε}^{2} + δ ε^{1} + δ^{2}), \end{array}$ for δ, ε small enough and r large enough.
Proof.
According to (3.31), we have $\begin{matrix} u_{s}^{δ k} - u_{RHS, p}^{δ k} - u_{RHS, e}^{δ k} = (S_{+, e}^{δ k} + S_{+, p}^{δ k} + T_{+, e}^{δ k} + T_{+, p}^{δ k}) [u^{δ k} |_{Λ}] + O (δ^{2}) . \end{matrix}$ With Lemma 3.25 and the fact that $u^{δ k} |_{Λ} = μ_{⋆}^{δ k} + μ_{\sim}^{δ k} + O (δ^{2})$ , $‖ μ_{⋆}^{δ k} ‖_{X^{ε}} = O (δ)$ and $‖ μ_{\sim}^{δ k} ‖_{X^{ε}} = O (δ)$ , we readily proved Proposition 3.26. □
Proof of Theorem 3.2.
We see that $u_{S_{e}^{⋆}}^{δ k}$ , $u_{T_{e}^{⋆}}^{δ k}$ and $u_{RHS, e}^{δ k}$ are exponentially decaying in $z_{2}$ and are of order $O (δ)$ , thus their $L^{\infty} (V_{r})$ -norm is of order $δ e^{- C r}$ for some constant $C > 0$ . Then with Proposition 3.26 and the change into the macroscopic view with $U^{k} (x) = u^{δ k} (x / δ)$ , we obtain Theorem 3.2. □

3.3.6. Evaluating the impedance boundary condition

We switch back to the macroscopic variable $U^{k} (x) = u^{δ k} (x / δ)$ . We approximate our solution in the far-field with the function $\begin{array}{l} U_{app}^{k} (z) & : = (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (z / δ) + u_{S_{p}^{⋆}}^{δ k} (z / δ) + u_{T_{p}^{⋆}}^{δ k} (z / δ) + u_{RHS, p}^{δ k} (z / δ) \\ (3.39) & = - 2 i a_{0} e^{- i k_{1} z_{1}} sin (k_{2} z_{2}) + e^{i (k_{2} z_{2} - k_{1} z_{1})} C_{(3.40)}^{δ k}, \end{array}$ where $\begin{array}{rcl} C_{(3.40)}^{δ k} & : = & - \frac{h}{p} \int_{Λ} μ_{⋆}^{δ k} (y) d σ_{y} + \frac{1}{p} \int_{Λ} \int_{\partial D} μ_{⋆}^{δ k} (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) d σ_{w} d σ_{y} \\ + \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) \frac{sin (δ k_{2} y_{2}) e^{i δ k_{1} y_{1}}}{δ k_{2} p} d σ_{y} \\ - \int_{\partial D} \int_{\partial D} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \\ (3.40) & \cdot (\begin{matrix} \frac{i k_{1}}{p k_{2}} sin (δ k_{2} w_{2}) \\ \frac{1}{p} cos (δ k_{2} w_{2}) \end{matrix}) e^{i δ k_{1} w_{1}} d σ_{w} d σ_{y} . \end{array}$ Consider that $C_{(3.40)}^{δ k} = O (δ)$ since $μ_{⋆}^{δ k} = O (δ)$ and $(u_{0}^{δ k} - u_{0}^{δ k} \circ P) = O (δ)$ and the other factors are of size $O (1)$ .

We want that $U_{app}^{k}$ satisfies the impedance boundary condition, that is $U_{app}^{k} (z) + δ c_{IBC} \partial_{z_{2}} U_{app}^{k} (z) = 0$ at $\partial R_{+}^{2}$ . From (3.39), we obtain the condition $\begin{matrix} e^{- i k_{1} z_{1}} C_{(3.40)}^{δ k} + δ c_{IBC} (- 2 i a_{0} k_{2} e^{- i k_{1} z_{1}} + i k_{2} e^{- i k_{1} z_{1}} C_{(3.40)}^{δ k}) = 0, for all z_{2} \in R . \end{matrix}$ After rearranging the terms, we obtain $\begin{matrix} c_{IBC} = \frac{- C_{(3.40)}^{δ k}}{i δ k_{2} (C_{(3.40)}^{δ k} - 2 a_{0})} . \end{matrix}$ Using that $\frac{1}{1 + O (δ)} = 1 - O (δ)$ , we have $c_{IBC} = \frac{C_{(3.40)}^{δ k}}{2 i a_{0} δ k_{2}} + O (δ)$ . This proves Theorem 3.3.

3.4. Numerical illustrations

In this subsection we compute the impedance boundary condition constant $c_{IBC}$ with numerical means using Theorem 3.3. We use two geometries, both rectangles, but with different sizes and for each geometry we have different ranges for the wave vector k and the gap length ε. This is because the resonance value $k_{res}^{δ, ε} : = k_{+}^{δ, ε}$ is proportional to the square root of the geometry area, and the same holds for the width of the resonance peak of $c_{IBC}$ . However, we do not have to consider different values of δ, since according to Theorem 3.3 all computations are done with the input $δ k$ , thus a different δ would only scale the first coordinate axis, but would not change the shape of the curve itself. We fix $δ = 0.01$ .

We implement $Γ_{♯}^{k}$ using Edwald’s Method see [15] or [3, Chapter 2.13.3]. We implement the remainders using Lemmas 2.10, 2.11 and 2.12.

The first geometry has the following set-up. It is a rectangle with length 0.9 and height 0.9. The period is $p = 1$ and $h = 1$ . The amplitude of the incident wave is $a_{0} = 1$ . The number of points, with which we approximate the boundary of the rectangle, is 200. For the wave vector $k = k_{2}$ , we take 341 equidistant points on the interval $[30, 200]$ . For ε we pick 5 values, those are ${0.1, 0.05, 0.03, 0.01, 0.001}$ . The result can be seen in Fig. 2.

Fig. 2.

The plot of the absolute value of the variable $c_{IBC}$ depending on the wave vector k for the first geometry. For every value of ε, the rounded value of the resonance value $k_{res}^{δ ε}$ is displayed.

The second geometry has the following set-up. It is a rectangle with length 0.2 and height 0.3. The period is $p = 1$ and $h = 0.5$ . The amplitude of the incident wave is $a_{0} = 1$ . The number of points, with which we approximate the boundary of the rectangle, is 200. For the wave vector $k = k_{2}$ , we take 301 equidistant points on the interval $[100, 400]$ . For ε we pick 5 values, those are ${0.1, 0.05, 0.03, 0.01, 0.001}$ . Consider that the case $ε = 0.1$ means that the whole upper boundary of our rectangle is the gap. The result can be seen in Fig. 3.

Fig. 3.

The plot of the absolute value of the variable $c_{IBC}$ depending on the wave vector k for the second geometry. For every value of ε, the rounded value of the resonance value $k_{res}^{δ ε}$ is displayed.

Consider that in Fig. 2, for $ε ⩽ 0.01$ , the resonance splits, so we get two peaks, due to the strong interaction between the main resonator and the neighbouring ones, while, in the second geometry, it seems that the resonators are at a large enough distance from each other so that the neighbouring resonators do not affect the main one. See Fig. 3.

4. Two periodically arranged Helmholtz resonators

In this section, we look at two domains $D_{1}$ and $D_{2}$ both bounded and simply connected domains, which have the same height h. We repeat $D_{1} \cup D_{2}$ periodically along the $x_{1}$ -axis, with period p and scale the whole geometry by a factor of δ. Additionally, $D_{1}$ and $D_{2}$ have each a gap called $Λ_{1}$ and $Λ_{2}$ on their boundary, which allows the incident wave $U_{0}^{k}$ to pass through. The incident wave rebounds inside $D_{1}$ and $D_{2}$ and leaves at the gaps, which then leads to the scattered wave $U_{s}^{k}$ .

We will have a good approximation of the scattered wave in the far-field. Moreover, this approximation satisfies the Helmholtz equation with a Robin boundary condition at the $x_{1}$ -axis which approximates a Dirichlet or a Neumann boundary condition depending on the magnitude of the incoming wave vector k and δ.

4.1. Mathematical description of the physical problem

4.1.1. Geometry

Before we consider the periodic and macroscopic problem, we first define the geometry of our Helmholtz resonators in the unit cell, shown in Fig. 4. Let $D_{1}, D_{2} ⋐ (- p / 2, p / 2) \times (0, 1)$ be two open, bounded, and simply connected domains, such that $\overline{D_{1}}$ and $\overline{D_{2}}$ do not intersect and where $p \in R$ is close enough to 1. We assume that $D_{1} \cup D_{2}$ is a $C^{2}$ -domain. We define $Λ_{1} \subset \partial D$ and $Λ_{2} \subset \partial D$ to be the gap of $D_{1}$ and $D_{2}$ respectively, where $Λ_{1}$ and $Λ_{2}$ are both line segment parallel to the $x_{1}$ -axis. $Λ_{1}$ is centered at ${(- ξ, h)}^{T}$ and $Λ_{2}$ is centered at ${(ξ, h)}^{T}$ , where $h \in (0, 1)$ and $ξ \in (0, p)$ , and $Λ_{1}$ and $Λ_{2}$ have both length $2 ε$ , where $ε \in (0, 1)$ small enough. To facilitate future computations we assume that $h / p ⩾ 1 / 2$ .

Fig. 4.

The physical setup. In (a) we have the microscopic, non-periodic view. In (b) we have the macroscopic, periodic view.

Now we define the macroscopic view, that is, we shrink our domain by the factor $δ \in (0, \infty)$ . We define the collection of periodically arranged Helmholtz resonators $Ω^{δ}$ , with period $δ p$ , and the collection of gaps of those Helmholtz resonators $Ξ^{δ}$ , where a single gap has length $2 δ ε$ , as $\begin{matrix} Ω^{δ} : = ⋃_{n \in Z} δ (D_{1} \cup D_{2} + n (\begin{matrix} p \\ 0 \end{matrix})), Ξ^{δ} : = ⋃_{n \in Z} δ (Λ_{1} \cup Λ_{2} + n (\begin{matrix} p \\ 0 \end{matrix})) . \end{matrix}$ The incident wave $U_{0}^{k} (x) : = a_{0} e^{- i k_{1} x_{1}} e^{- i k_{2} x_{2}}$ is exactly as in the one Helmholtz resonator case, see Section 3.1.2.

4.1.2. The resulting wave

With the geometry and the incident wave, we model the electromagnetic scattering problem and the resulting wave $U^{k} : R_{+}^{2} ∖ \partial Ω^{δ} \to C$ by the following system of equations: $\begin{matrix} (4.1) & \{\begin{matrix} (△ + k^{2}) U^{k} = 0 & in R_{+}^{2} ∖ \partial Ω^{δ}, \\ U^{k} |_{+} - U^{k} |_{-} = 0 & on Ξ^{δ}, \\ \partial_{ν} U^{k} |_{+} - \partial_{ν} U^{k} |_{-} = 0 & on Ξ^{δ}, \\ \partial_{ν} U^{k} |_{+} = 0 & on \partial Ω^{δ} ∖ Ξ^{δ}, \\ \partial_{ν} U^{k} |_{-} = 0 & on \partial Ω^{δ} ∖ Ξ^{δ}, \\ U^{k} = 0 & on \partial R_{+}^{2}, \end{matrix} \end{matrix}$ where $\cdot |_{+}$ denotes the limit from outside of $Ω^{δ}$ and $\cdot |_{-}$ denotes the limit from inside of $Ω^{δ}$ , and $\partial_{ν}$ denotes the normal derivative on $\partial Ω^{δ}$ . Similar to diffraction problems for gratings, the above system of equations is complemented by a certain outgoing radiation condition imposed on the scattered field $U_{s}^{k} : = U^{k} - (U_{0}^{k} - U_{0}^{k} \circ P)$ and quasi-periodicity on $U^{k}$ . More precisely, we are interested in the quasi-periodic solutions, that is $\begin{matrix} U^{k} (x + (\begin{matrix} p \\ 0 \end{matrix})) = e^{- i k_{1} p} U^{k} (x) for x \in R_{+}^{2}, \end{matrix}$ and solutions satisfying the outgoing radiation condition, thus $| \partial_{x_{2}} U_{s}^{k} - i k_{2} U_{s}^{k} | \to 0$ , for $x_{2} \to \infty$ .

Then the outgoing radiation condition can be imposed by assuming that all the modes in the Rayleigh-Bloch expansion are either decaying exponentially or propagating along the $x_{2}$ -direction. Since in our case we assume that the period of the resonator structure $δ p$ is much smaller than k, the outgoing radiation condition takes the following specific form: $\begin{matrix} (U^{k} - U_{0}^{k}) (x) \sim a e^{- i k_{1} x_{1}} e^{i k_{2} x_{2}} as x_{2} \to \infty, \end{matrix}$ thus $U_{s}^{k} (x) \sim (a + 1) e^{- i k_{1} x_{1}} e^{i k_{2} x_{2}}$ as $x_{2} \to \infty$ , for some constant amplitude $a \in R$ . Consider also that in absence of Helmholtz resonators the solution to (4.1) is given by $U_{0}^{k} - U_{0}^{k} \circ P$ . Analogous to Section 3.1.4, we can consider the resulting wave $U^{k}$ in its microscopic view $u^{δ k} : R_{+}^{2} ∖ \partial Ω^{1} \to C$ .

4.2. Main results

We assume that $δ k \in K : = {\hat{k} \in R ∣ 0 \neq | \hat{k} | < k_{D_{1} \cup D_{2}, min, △} / 2 and | \hat{k} |^{2} < inf {| l - \hat{k} e_{1} |^{2} ∣ l : = 2 π n / p, n \in Z ∖ {0}}}$ , where $k_{D_{1} \cup D_{2}, min, △}$ is defined as the first non-zero eigenvalue of the operator $- △$ with Neumann conditions on the boundary $\partial D_{1} \cup \partial D_{2}$ . If we would extend the domain $K$ to $K_{C} : = {k_{*} \in C ∣ \sqrt{k_{*} {\bar{k}}_{*}} < k_{D_{1} \cup D_{2}, min, △} / 2}$ , we would obtain the following resonance values for our physical problem:

Theorem 4.1.
There exists exactly four resonance values in $K_{C}$ for $U^{k}$ . These are for $j \in {1, 2}$ $\begin{array}{l} δ k_{j, +}^{δ, ε} = δ k_{⋆, j}^{δ, ε} (1 + \frac{{(π c_{ε})}^{\frac{3}{2}}}{4} e_{j}^{T} {(Y_{⋆, ⋆}^{ε})}^{- 1} A_{⋆, (1)} {(Y_{⋆, ⋆}^{ε})}_{j}) + O (c_{ε}^{5 / 2}), \\ δ k_{j, -}^{δ, ε} = δ k_{⋆, j}^{δ, ε} (- 1 + \frac{{(π c_{ε})}^{\frac{3}{2}}}{4} e_{j}^{T} {(Y_{⋆, ⋆}^{ε})}^{- 1} A_{⋆, (1)} {(Y_{⋆, ⋆}^{ε})}_{j}) + O (c_{ε}^{5 / 2}), \end{array}$ where $\begin{array}{l} δ k_{⋆, 1}^{δ, ε} = \sqrt{\frac{π}{2} c_{ε}} (\frac{1}{2} (k_{⋆, tr 1}^{ε} + k_{⋆, tr 2}^{ε}) + {[\frac{1}{4} {(k_{⋆, tr 1}^{ε} - k_{⋆, tr 2}^{ε})}^{2} - k_{⋆, \det}^{ε}]}^{1 / 2}), \\ δ k_{⋆, 2}^{δ, ε} = \sqrt{\frac{π}{2} c_{ε}} (\frac{1}{2} (k_{⋆, tr 1}^{ε} + k_{⋆, tr 2}^{ε}) - {[\frac{1}{4} {(k_{⋆, tr 1}^{ε} - k_{⋆, tr 2}^{ε})}^{2} - k_{⋆, \det}^{ε}]}^{1 / 2}), \\ k_{⋆, tr 1}^{ε} = \frac{1}{\sqrt{| D_{1} |}} (1 + \frac{{(α_{(0)})}_{1, 1} π c_{ε}}{4}), k_{⋆, tr 2}^{ε} = \frac{1}{\sqrt{| D_{2} |}} (1 + \frac{{(α_{(0)})}_{2, 2} π c_{ε}}{4}), \\ k_{⋆, \det}^{ε} = \frac{π^{2} c_{ε}^{2}}{4} \frac{{(T_{(0)})}_{1, 2} {(T_{(0)})}_{2, 1}}{(\sqrt{| D_{1} |} + \sqrt{| D_{2} |})}, c_{ε} = \frac{- 1}{log (ε / 2)} . \end{array}$ Here, $Y_{⋆, ⋆}^{ε} = [{(Y_{⋆, ⋆}^{ε})}_{1}, {(Y_{⋆, ⋆}^{ε})}_{2}]$ , with ${(Y_{⋆, ⋆}^{ε})}_{j}$ being the normalized eigenvector to the eigenvalue $δ k_{⋆, j}^{δ, ε}$ of $A_{⋆, ⋆}^{ε}$ , where $A_{⋆, ⋆}^{ε}$ is given in ( 4.31 ) and $A_{⋆, D}$ , $A_{⋆, (0)}$ and $A_{⋆, (1)}$ are given in Lemma 4.15 , where ${(T_{(1)})}_{2, 1}$ , ${(T_{(1)})}_{1, 2}$ , ${(T_{(0)})}_{2, 1}$ , ${(T_{(0)})}_{1, 2}$ , ${(α_{(1)})}_{2, 2}$ , ${(α_{(1)})}_{1, 1}$ , ${(α_{(0)})}_{2, 2}$ and ${(α_{(0)})}_{1, 1}$ are given in ( 4.24 )–( 4.29 ).

We have the following approximation for the resulting wave $U^{k}$ :
Theorem 4.2.
Let $V_{r} : = {z \in R_{+}^{2} ∣ z_{2} > r}$ . There exist constants $C_{(4.2)}, {\tilde{C}}_{(4.2)} > 0$ such that $\begin{array}{l} {‖ U_{s}^{k} - (U_{S_{p}^{⋆}}^{k} + U_{T_{p}^{⋆}}^{k} + U_{RHS, p}^{k}) ‖}_{L^{\infty} (V_{r})} + {‖ \nabla [U_{s}^{k} - (U_{S_{p}^{⋆}}^{k} + U_{T_{p}^{⋆}}^{k} + U_{RHS, p}^{k})] ‖}_{L^{\infty} (V_{r})} \\ (4.2) & ⩽ C_{(4.2)} (\frac{δ}{| log (ε) |^{3}} (‖ {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} ‖) + \frac{δ}{| log (ε) |^{2}} + δ ε + δ e^{- {\tilde{C}}_{(4.2)} r} + δ^{2}), \end{array}$ for ε small enough, where $\begin{array}{l} U_{S_{p}^{⋆}}^{k} (z) = - e^{i (k_{2} z_{2} - k_{1} z_{1})} \frac{h}{p} \sum_{j \in {1, 2}} \int_{Λ_{j}} {(μ_{⋆}^{δ k})}_{j} (y) d σ_{y}, \\ U_{T_{p}^{⋆}}^{k} (z) = e^{i (k_{2} z_{2} - k_{1} z_{1})} \frac{1}{p} \sum_{j, \hat{j} \in {1, 2}} \int_{Λ_{j}} \int_{\partial D_{\hat{j}}} {(μ_{⋆}^{δ k})}_{j} (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) d σ_{w} d σ_{y}, \\ \begin{matrix} U_{RHS, p}^{k} (z) & = e^{i (k_{2} z_{2} - k_{1} z_{1})} [\sum_{j \in {1, 2}} \int_{\partial D_{j}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) \frac{sin (δ k_{2} y_{2}) e^{i δ k_{1} y_{1}}}{δ k_{2} p} d σ_{y} \\ - \sum_{j, \hat{j} \in {1, 2}} \int_{\partial D_{j}} \int_{\partial D_{\hat{j}}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \\ \cdot (\begin{matrix} \frac{i k_{1}}{p k_{2}} sin (δ k_{2} w_{2}) \\ \frac{1}{p} cos (δ k_{2} w_{2}) \end{matrix}) e^{i δ k_{1} w_{1}} d σ_{w} d σ_{y}] . \end{matrix} \end{array}$ Here, for $y = ({(t_{1} - ξ, h)}^{T}, {(t_{2} + ξ, h)}^{T}) \in Λ_{1} \times Λ_{2}$ , we have $\begin{matrix} μ_{⋆} (y) = [\begin{matrix} \frac{- 1}{\sqrt{ε^{2} - t_{1}^{2}}} (c_{ε}^{2} \frac{π}{4} {(A_{D} (Y_{⋆, ⋆}^{ε}) {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} {(Y_{⋆, ⋆}^{ε})}^{- 1} {[f_{D_{1}}^{δ k}, f_{D_{2}}^{δ k}]}^{T})}_{1} + \frac{1}{2} c_{ε} f_{D_{1}}^{δ k}) \\ \frac{- 1}{\sqrt{ε^{2} - t_{2}^{2}}} (c_{ε}^{2} \frac{π}{4} {(A_{D} (Y_{⋆, ⋆}^{ε}) {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} {(Y_{⋆, ⋆}^{ε})}^{- 1} {[f_{D_{1}}^{δ k}, f_{D_{2}}^{δ k}]}^{T})}_{2} + \frac{1}{2} c_{ε} f_{D_{2}}^{δ k}) \end{matrix}], \end{matrix}$ where $f_{D_{1}}^{δ k}, f_{D_{2}}^{δ k} \in C$ are given by $\begin{array}{l} f_{D_{1}}^{δ k} = 2 i a_{0} sin (δ k_{2} h) e^{- i δ k_{1} ξ} \\ - 2 a_{0} δ \int_{\partial D_{1} \cup \partial D_{2}} ν_{y} \cdot (\begin{matrix} k_{1} e^{- i δ k_{1} y_{1}} sin (δ k_{2} y_{2}) \\ i k_{2} e^{- i δ k_{1} y_{1}} cos (δ k_{2} y_{2}) \end{matrix}) N_{\partial Ω^{1}, +}^{δ k} ((\begin{matrix} - ξ \\ h \end{matrix}), y) d σ_{y}, \\ f_{D_{2}}^{δ k} = 2 i a_{0} sin (δ k_{2} h) e^{i δ k_{1} ξ} \\ - 2 a_{0} δ \int_{\partial D_{1} \cup \partial D_{2}} ν_{y} \cdot (\begin{matrix} k_{1} e^{- i δ k_{1} y_{1}} sin (δ k_{2} y_{2}) \\ i k_{2} e^{- i δ k_{1} y_{1}} cos (δ k_{2} y_{2}) \end{matrix}) N_{\partial Ω^{1}, +}^{δ k} ((\begin{matrix} ξ \\ h \end{matrix}), y) d σ_{y} . \end{array}$ Here, $Y_{⋆, ⋆}^{ε}$ is given as in Theorem 4.1 and $\begin{array}{l} M_{+}^{δ k} = [\begin{matrix} δ k - δ k_{1, +}^{δ, ε} & 0 \\ 0 & δ k - δ k_{2, +}^{δ, ε} \end{matrix}], M_{-}^{δ k} = [\begin{matrix} δ k - δ k_{1, -}^{δ, ε} & 0 \\ 0 & δ k - δ k_{2, -}^{δ, ε} \end{matrix}], \\ A_{D} = [\begin{matrix} 1 / | D_{1} | & 0 \\ 0 & 1 / | D_{2} | \end{matrix}] . \end{array}$

We see from Theorem 4.2 that the function $\begin{matrix} U_{app}^{k} (z) : = (U_{0}^{k} - U_{0}^{k} \circ P) (z) + U_{S_{p}^{⋆}}^{k} (z) + U_{T_{p}^{⋆}}^{k} (z) + U_{RHS, p}^{k} (z), \end{matrix}$ gives an accurate approximation of $U^{k}$ in the far-field. Moreover, it satisfies the Helmholtz equation in $R_{+}^{2}$ with the boundary condition $\begin{matrix} U_{app}^{k} (z) + δ c_{IBC} \partial_{z_{2}} U_{app}^{k} (z) = 0, for z \in \partial R_{+}^{2}, \end{matrix}$ and $U_{app}^{k} - U_{0}^{k}$ satisfies the outgoing radiation condition.

Using Theorem 4.2 we can express $c_{IBC}$ as follows. Theorem 4.3.
The constant in the impedance boundary condition is given as $\begin{matrix} c_{IBC} = \frac{1}{2 i a_{0} δ k_{2}} C_{(4.4)}^{δ k} + O (δ), \end{matrix}$ where $C_{(4.4)}^{δ k} \in R$ is defined as $\begin{array}{l} C_{(4.4)}^{δ k} & = - \frac{h}{p} \sum_{j \in {1, 2}} \int_{Λ_{j}} {(μ_{⋆}^{δ k})}_{j} (y) d σ_{y} + \sum_{j \in {1, 2}} \int_{\partial D_{j}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) \frac{sin (δ k_{2} y_{2}) e^{i δ k_{1} y_{1}}}{δ k_{2} p} d σ_{y} \\ + \frac{1}{p} \sum_{j, \hat{j} \in {1, 2}} \int_{Λ_{j}} \int_{\partial D_{\hat{j}}} {(μ_{⋆}^{δ k})}_{j} (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) d σ_{w} d σ_{y} \\ - \sum_{j, \hat{j} \in {1, 2}} \int_{\partial D_{j}} \int_{\partial D_{\hat{j}}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \\ (4.3) & \cdot (\begin{matrix} \frac{i k_{1}}{p k_{2}} sin (δ k_{2} w_{2}) \\ \frac{1}{p} cos (δ k_{2} w_{2}) \end{matrix}) e^{i δ k_{1} w_{1}} d σ_{w} d σ_{y}, \end{array}$ from which we see that $\begin{matrix} (4.4) & C_{(4.4)}^{δ k} = O (δ) (\frac{1}{log (ε / 2)} + \frac{1}{{(log (ε / 2))}^{2}} O (‖ {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} ‖)) . \end{matrix}$

4.3. Proof of the main results

We want to proof Theorem 3.1–3.3. First, we express the resulting wave outside the Helmholtz resonators and the resulting wave inside of the Helmholtz resonators through operators acting on the resulting wave, but restricted on the gap. This leads us to a condition with the linear operator $A^{δ k, ε}$ , whose solution is the resulting wave on the gap up to a term of order $δ^{2}$ . We solve this linear system based on the procedure given in [9]. We will see that it is solvable for a complex wave vector near 0 except in five points, four of which are the resonances of our system. Then we recover the resulting wave outside the resonators up to a term of order $δ^{2}$ . We will see, that we can split the resulting wave into a propagating wave and a evanescent one. The propagating one leads us to the impedance boundary condition constant $c_{IBC}$ .

4.3.1. Collapsing the wave-informations on the two gaps

Let us consider the resulting wave $u^{δ k}$ in the microscopic view. We will keep the microscopic view until Section 4.3.6. We only look on the main strip $Y : = {y \in R_{+}^{2} ∣ | y_{1} | < p / 2}$ . D is the Helmholtz resonator on that strip and Λ the gap on $\partial D$ . Furthermore, we fix $k_{1} / k = : e_{1}$ and $k_{2} / k = : e_{2}$ and assume that $δ k \in K : = {\hat{k} \in R ∣ 0 \neq | \hat{k} | < k_{D_{1} \cup D_{2}, min, △} / 2 and | \hat{k} |^{2} < inf {| l - \hat{k} e_{1} |^{2} ∣ l : = 2 π n / p, n \in Z ∖ {0}}}$ . Consider that $u^{δ k}$ is continuous on the gap $Λ_{1} \cup Λ_{2}$ , thus $u^{δ k} (z)$ is well-defined for $z \in Λ_{1} \cup Λ_{2}$ .

Proposition 4.4.
Let $j \in {1, 2}$ and let $N_{D_{j}}^{k}$ be as in Definition 2.7 . Let $z \in D_{j}$ . Then, $\begin{matrix} u^{δ k} (z) = - \int_{Λ_{j}} \partial_{ν} u^{δ k} (y) N_{D_{j}}^{δ k} (z, y) d σ_{y} . \end{matrix}$ Let $z \in Λ_{j}$ . Then, $\begin{matrix} (4.5) & u^{δ k} (z) = - \int_{Λ_{j}} \partial_{ν} u^{δ k} (y) N_{\partial D_{j}}^{δ k} (z, y) d σ_{y} . \end{matrix}$
Proposition 4.5.
Let $N_{Ω, +}^{k}$ , $N_{Ω, \times}^{k}$ be as in Definition 2.8 . Let $z \in Y ∖ \overline{D}$ . Then, $\begin{matrix} u_{s}^{δ k} (z) = \int_{Λ_{1} \cup Λ_{2}} \partial_{ν} u^{δ k} (y) N_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} - \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} . \end{matrix}$ Let $z \in Λ$ . Then, $\begin{matrix} (4.6) & u_{s}^{δ k} (z) = \int_{Λ_{1} \cup Λ_{2}} \partial_{ν} u^{δ k} (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} - \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} . \end{matrix}$

Using that $u^{δ k}$ is continuous on the gap we can deduce from the following proposition a necessary condition for $\partial_{ν} u^{δ k} |_{Λ}$ . Assume we can obtain a solution $\partial_{ν} u^{δ k}$ from that condition, then from Propositions 4.4 and 4.5 we can recover the resulting wave on Y.
Proposition 4.6 (Gap-Formula).

Let $j, \hat{j} \in {1, 2}$ , $j \neq \hat{j}$ and let $z \in Λ_{j}$ then $\begin{array}{l} \int_{Λ_{j}} \partial_{ν} u^{δ k} (y) (N_{\partial Ω^{1}, +}^{δ k} (z, y) + N_{\partial D_{j}}^{δ k} (z, y)) d σ_{y} + \int_{Λ_{\hat{j}}} \partial_{ν} u^{δ k} (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} \\ (4.7) & = \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} - (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (z) . \end{array}$

Consider that the right-hand-side in (4.7) does not depend on $u^{δ k}$ and it is computable.

The proofs for Proposition (4.4)–(4.6) are analogous to the one Helmholtz resonator case, that is Proposition (3.4)–(3.6).

4.3.2. Expanding the gap-formula in terms of delta

We define $f_{\partial Ω^{1}}^{δ k} : Λ_{1} \cup Λ_{2} \to C$ as the right-hand-side of the Gap-Formula 3.6, that is $\begin{matrix} (4.8) & f_{\partial Ω^{1}}^{δ k} (z) : = \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} - (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (z) . \end{matrix}$ We define $f_{D_{1}}^{δ k} (τ)$ as $f_{\partial Ω^{1}}^{δ k} ({(τ - ξ, h)}^{T})$ and $f_{D_{2}}^{δ k} (τ)$ as $f_{\partial Ω^{1}}^{δ k} ({(τ + ξ, h)}^{T})$ , for $τ \in (- ε, ε)$ .

Let us define the following operator-spaces and their respective norms:

Definition 4.7.
Recall Definition 3.7, we define $X^{ε} : = X^{ε} \times X^{ε}, Y^{ε} : = Y^{ε} \times Y^{ε}$ and $\begin{array}{l} ‖ μ ‖_{X^{ε}} : = ‖ μ_{1} ‖_{X^{ε}} + ‖ μ_{2} ‖_{X^{ε}}, for μ = (μ_{1}, μ_{2}) \in X^{ε}, \\ ‖ μ ‖_{Y^{ε}} : = ‖ μ_{1} ‖_{Y^{ε}} + ‖ μ_{2} ‖_{Y^{ε}}, for μ = (μ_{1}, μ_{2}) \in Y^{ε} . \end{array}$
Definition 4.8.
Let $μ \in X^{ε}$ and g be a real function and $α > 0$ . We say $μ = O_{X^{ε}} (g (α))$ for $α \to 0$ if $\frac{‖ μ ‖_{X^{ε}}}{g (α)}$ is bounded as $α \to 0$ .

With those spaces we can define the following operators:
Definition 4.9.
The following operators are defined as functions from $X^{ε}$ to $Y^{ε}$ or from $X^{ε}$ to $Y^{ε}$ . Let $n \in N ∖ {0}$ , and $μ = (μ_{1}, μ_{2}) \in X^{ε}$ then $\begin{array}{l} L^{ε} [μ] (τ) : = [\begin{matrix} L^{ε} & 0 \\ 0 & L^{ε} \end{matrix}] [\begin{matrix} μ_{1} \\ μ_{2} \end{matrix}] (τ) = [\begin{matrix} L^{ε} [μ_{1}] (τ) \\ L^{ε} [μ_{2}] (τ) \end{matrix}], \\ K^{ε} [μ] (τ) : = [\begin{matrix} K_{1}^{ε} [μ_{1}] (τ) \\ K_{2}^{ε} [μ_{2}] (τ) \end{matrix}] : = [\begin{matrix} \int_{- ε}^{ε} \frac{μ_{1} (t)}{| D_{1} |} d t \\ \int_{- ε}^{ε} \frac{μ_{2} (t)}{| D_{2} |} d t \end{matrix}], \\ R^{δ k, ε} [μ] (τ) : = [\begin{matrix} R_{1, 1}^{δ k, ε} & R_{1, 2}^{δ k, ε} \\ R_{2, 1}^{δ k, ε} & R_{2, 2}^{δ k, ε} \end{matrix}] [\begin{matrix} μ_{1} \\ μ_{2} \end{matrix}] (τ), \\ \begin{matrix} R_{0, 0}^{δ k, ε} [μ_{1}] (τ) & : = \int_{- ε}^{ε} μ_{1} (t) [\frac{1}{π} log (\frac{π}{p}) + \frac{1}{π} log (sinc | \frac{π}{p} (τ - t) |) \\ - \frac{1}{2 π} log (sinh {(\frac{π}{p} 2 h)}^{2} + sin {(\frac{π}{p} (τ - t))}^{2})] d t, \end{matrix} \\ \begin{matrix} R_{1, 1}^{δ k, ε} [μ_{1}] (τ) & : = R_{0, 0}^{δ k, ε} [μ_{1}] (τ) \\ + \int_{- ε}^{ε} μ_{1} (t) [R_{\partial D_{1}}^{δ k} (\begin{matrix} (\begin{matrix} τ - ξ \\ h \end{matrix}), (\begin{matrix} t - ξ \\ h \end{matrix}) \end{matrix}) + R_{\partial Ω^{1}, +}^{δ k} (\begin{matrix} (\begin{matrix} τ - ξ \\ h \end{matrix}), (\begin{matrix} t - ξ \\ h \end{matrix}) \end{matrix})] d t, \end{matrix} \\ R_{1, 2}^{δ k, ε} [μ_{2}] (τ) : = \int_{- ε}^{ε} μ_{2} (t) N_{\partial Ω^{1}, +}^{δ k} (\begin{matrix} (\begin{matrix} τ - ξ \\ h \end{matrix}), (\begin{matrix} t + ξ \\ h \end{matrix}) \end{matrix}) d t, \\ R_{2, 1}^{δ k, ε} [μ_{1}] (τ) : = \int_{- ε}^{ε} μ_{1} (t) N_{\partial Ω^{1}, +}^{δ k} (\begin{matrix} (\begin{matrix} τ + ξ \\ h \end{matrix}), (\begin{matrix} t - ξ \\ h \end{matrix}) \end{matrix}) d t, \\ \begin{matrix} R_{2, 2}^{δ k, ε} [μ_{2}] (τ) & : = R_{0, 0}^{δ k, ε} [μ_{2}] (τ) \\ + \int_{- ε}^{ε} μ_{2} (t) [R_{\partial D_{2}}^{δ k} (\begin{matrix} (\begin{matrix} τ + ξ \\ h \end{matrix}), (\begin{matrix} t + ξ \\ h \end{matrix}) \end{matrix}) + R_{\partial Ω^{1}, +}^{δ k} (\begin{matrix} (\begin{matrix} τ + ξ \\ h \end{matrix}), (\begin{matrix} t + ξ \\ h \end{matrix}) \end{matrix})] d t, \end{matrix} \\ \begin{matrix} {\hat{Γ}}_{+, n}^{k, ε} [μ] (τ) & : = [\int_{- ε}^{ε} μ_{1} (t) Γ_{+, n}^{k} (\begin{matrix} (\begin{matrix} τ - ξ \\ h \end{matrix}), (\begin{matrix} t - ξ \\ h \end{matrix}) \end{matrix}) d t, \\ \int_{- ε}^{ε} μ_{2} (t) Γ_{+, n}^{k} (\begin{matrix} (\begin{matrix} τ + ξ \\ h \end{matrix}), (\begin{matrix} t + ξ \\ h \end{matrix}) \end{matrix}) d t]^{T}, \end{matrix} \\ A^{δ k, ε} [μ] (τ) : = \frac{2}{π} L^{ε} [μ] (τ) + \frac{K^{ε} [μ] (τ)}{δ^{2} k^{2}} + R^{δ k, ε} [μ] (τ), \end{array}$ where $Γ_{+, n}^{k}$ is given in Lemma 2.2.

For $τ \in {(ε, ε)}^{2}$ , we also define $f^{δ k} (τ) : = {(f_{D_{1}}^{δ k} (τ_{1}), f_{D_{2}}^{δ k} (τ_{2}))}^{T}$ . Later, we will show that $\begin{matrix} [\begin{matrix} \partial_{ν} u^{δ k} |_{Λ_{1}} \\ \partial_{ν} u^{δ k} |_{Λ_{2}} \end{matrix}] = {(A^{δ k, ε})}^{- 1} [f^{δ k}] + O (δ^{2}) . \end{matrix}$
Proposition 4.10.
Let $2 ε < p$ , let $τ \in {(- ε, ε)}^{2}$ , then $\begin{matrix} A^{δ k, ε} [\begin{matrix} \partial_{ν} u^{δ k} |_{Λ_{1}} \\ \partial_{ν} u^{δ k} |_{Λ_{2}} \end{matrix}] (τ) + \sum_{n = 1}^{\infty} δ^{n} 2 {\hat{Γ}}_{+, n}^{k, ε} [\begin{matrix} \partial_{ν} u^{δ k} |_{Λ_{1}} \\ \partial_{ν} u^{δ k} |_{Λ_{2}} \end{matrix}] (τ) = f^{δ k} (τ) . \end{matrix}$
Proof.
Let $z : = {(τ \pm ξ, h)}^{T} \in Λ_{j}$ , where $j, \hat{j} \in {1, 2}$ , $j \neq \hat{j}$ . From Proposition 4.6 we have $\begin{matrix} \int_{Λ_{j}} \partial_{ν} u^{δ k} (y) (N_{\partial Ω^{1}, +}^{δ k} (z, y) + N_{\partial D_{j}}^{δ k} (z, y)) d σ_{y} + \int_{Λ_{\hat{j}}} \partial_{ν} u^{δ k} (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} = f_{\partial Ω^{1}}^{δ k} (z) . \end{matrix}$ Using Lemma 2.2 we can rearrange the last equation and obtain $\begin{array}{l} f_{\partial Ω^{1}}^{δ k} (z) & = \int_{Λ_{j}} \partial_{ν} u^{δ k} (y) (2 Γ_{+}^{0} (z, y) + \frac{1}{π} log | z - y | + \frac{1 / | D_{j} |}{δ^{2} k^{2}} + R_{\partial Ω^{1}, +}^{δ k} (z, y) + R_{\partial D_{j}}^{δ k} (z, y) \\ (4.9) & + \sum_{n = 1}^{\infty} δ^{n} 2 Γ_{+, n}^{k} (z, y)) d σ_{y} + \int_{Λ_{\hat{j}}} \partial_{ν} u^{δ k} (y) N_{\partial Ω^{1}, +}^{δ k} (z, y) d σ_{y} . \end{array}$ Using that Λ is a line segment parallel to the $x_{1}$ -axis and writing $y = y_{1} = {(t - ξ, h)}^{T}$ , on the gap $Λ_{1}$ , and writing $y = y_{2} = {(t + ξ, h)}^{T}$ , on the gap $Λ_{2}$ , we have that $d σ_{y} = d t$ . Using (2.7) for $Γ_{+}^{0} (z, y)$ and using that the expansion in δ (Lemma 2.2) is uniform, we can interchange the infinite sum and the integration. Let $μ_{j} (t) : = \partial_{ν} u^{δ k} (y) |_{Λ_{j}}$ , we have that $\begin{array}{l} f_{\partial Ω^{1}}^{δ k} (z) & = \int_{- ε}^{ε} μ_{j} (t) [\frac{1}{π} log | τ - t | + \frac{1}{2 π} log (sin {(\frac{π}{p} (τ - t))}^{2}) \\ - \frac{1}{2 π} log (sinh {(\frac{π}{p} 2 h)}^{2} + sin {(\frac{π}{p} (τ - t))}^{2}) + R_{\partial Ω^{1}, +}^{δ k} (z, y_{j}) + R_{\partial D_{j}}^{δ k} (z, y_{j})] d t \\ (4.10) & + R_{j, \hat{j}}^{δ k, ε} [μ_{\hat{j}}] (τ) + \frac{K^{ε} [μ_{j}] (τ)}{δ^{2} k^{2}} + \sum_{n = 1}^{\infty} δ^{n} 2 {\hat{Γ}}_{+, n}^{k, ε} [μ_{j}] (τ) . \end{array}$ Now consider that for $2 ε < p$ , we obtain that $\begin{matrix} \frac{2}{π} L^{ε} [μ_{j}] (τ) + R_{j, j}^{δ k, ε} [μ_{j}] (τ) + R_{j, \hat{j}}^{δ k, ε} [μ_{\hat{j}}] (τ) + \frac{K_{j}^{ε} [μ_{j}] (τ)}{δ^{2} k^{2}} + \sum_{n = 1}^{\infty} δ^{n} {\hat{Γ}}_{+, n}^{k, ε} [μ_{j}] (τ) = f_{\partial Ω^{1}}^{δ k} (z) . \end{matrix}$ With Definition 4.9 we have proven Proposition 4.10. □

From Lemma 3.11 we readily get that for $0 < ε < 2$ , the operator $L^{ε} : X^{ε} \to Y^{ε}$ is linear, bounded and invertible and has the inverse $\begin{matrix} (4.11) & {(L^{ε})}^{- 1} [μ] (t) = [\begin{matrix} {(L^{ε})}^{- 1} [μ_{1}] (t_{1}) \\ {(L^{ε})}^{- 1} [μ_{2}] (t_{2}) \end{matrix}] . \end{matrix}$

Since $R_{\partial D_{j}}^{δ k}$ and $R_{\partial Ω^{1}, +}^{δ k}$ are continuous, $R^{δ k, ε}$ is a compact operator. Thus we have that $\frac{2}{π} L^{ε} + R^{δ k, ε}$ is a Fredholm operator of index zero. Thus for the operator $A^{δ k, ε}$ , extending the domain $K$ to the complex numbers in a disk-shaped form, we will see that $A^{δ k, ε}$ is invertible except for a finite amount of values of $δ k$ . Some of those values are the resonances of our physical system. To that end, we will need the following result.
4.3.3. Characteristic values of $A^{δ k, ε}$ and the four resonance values

Let us first look at the characteristic values of $\begin{matrix} Q^{δ k, ε} [μ] : = \frac{2}{π} L^{ε} [μ] + \frac{K^{ε} [μ]}{δ^{2} k^{2}}, \end{matrix}$ where $μ \in X^{ε}$ and $δ k \in K_{C} : = {k_{*} \in C ∣ \sqrt{k_{*} {\bar{k}}_{*}} < k_{D_{1} \cup D_{2}, min, △} / 2}$ . For $μ, λ \in X^{ε}$ we define ${(μ, λ)}_{ε \oplus ε} : = {(μ_{1}, λ_{1})}_{ε} + {(μ_{2}, λ_{2})}_{ε}$ , where ${(\cdot, \cdot)}_{ε}$ is the $L^{2} ((- ε, ε))$ inner-product. We also define $1 : = {[1, 1]}^{T} \in X^{ε}$ , $e_{1} : = {[1, 0]}^{T} \in X^{ε}$ and $e_{2} : = {[0, 1]}^{T} \in X^{ε}$ .

Lemma 4.11.
$Q^{δ k, ε}$ has exactly the four characteristic values $\pm k_{j, 0}^{δ, ε}$ for $j \in {1, 2}$ with the characteristic functions $μ_{j, 0}^{δ, ε}$ , where $\begin{array}{l} k_{j, 0}^{δ, ε} = \frac{1}{δ} {(- \frac{π}{2 | D_{j} |} {({(L^{ε})}^{- 1} [1], 1)}_{ε})}^{1 / 2} = \frac{1}{δ} {(- \frac{π}{2 | D_{j} | log (ε / 2)})}^{1 / 2}, \\ μ_{j, 0}^{ε} = - \frac{π}{2 | D_{j} |} \frac{{(L^{ε})}^{- 1} [1]}{{(δ k_{j, 0}^{δ, ε})}^{2}}, \end{array}$ after imposing ${(μ_{1}, 1)}_{ε} = 1$ , ${(μ_{2}, 1)}_{ε} = 1$ .

Consider that $k_{j, 0}^{δ, ε}$ is real and positive.
Proof.
We are looking for $μ = {[μ_{1}, μ_{2}]}^{T} \in X^{ε}$ such that $\begin{matrix} Q^{δ k, ε} [μ] = 0 . \end{matrix}$ Since ${(L^{ε})}^{- 1} [0] = 0$ , the last equation is equivalent to $\begin{matrix} (4.12) & \frac{2}{π} μ + \frac{{(L^{ε})}^{- 1} K^{ε} [μ]}{{(δ k)}^{2}} = 0 . \end{matrix}$ Applying once ${(\cdot, e_{1})}_{ε \oplus ε}$ and once ${(\cdot, e_{2})}_{ε \oplus ε}$ on both sides, we obtain $\begin{matrix} (4.13) & {(μ_{j}, 1)}_{ε} (\frac{2}{π} + \frac{1}{| D_{j} |} \frac{{({(L^{ε})}^{- 1} [1], 1)}_{ε}}{{(δ k)}^{2}}) = 0, for j \in {1, 2} . \end{matrix}$ If ${(μ_{j}, 1)}_{ε} = 0$ , then $L^{ε} [μ_{j}] = 0$ , because of the condition $Q^{δ k, ε} [μ] = 0$ , and then $μ_{j} = 0$ , since $L^{ε}$ is invertible and linear. But 0 cannot be a characteristic function, by definition. Hence $(μ_{j}, 1) \neq 0$ . Thus the second factor in (4.13) has to be zero. This leads us to $\begin{matrix} δ^{2} k^{2} = - \frac{π}{2 | D_{j} |} {({(L^{ε})}^{- 1} [1], 1)}_{ε}, for j \in {1, 2} . \end{matrix}$ Using Lemma 3.12, we can calculate that ${({(L^{ε})}^{- 1} [1], 1)}_{ε} = \frac{1}{log (ε / 2)}$ , and obtain the characteristic values.

As for the characteristic functions, we rewrite (4.12) as $\begin{matrix} \frac{μ_{j}}{{(μ_{j}, 1)}_{ε}} = - \frac{π}{2 | D_{j} |} \frac{{(L^{ε})}^{- 1} [1]}{{(δ k)}^{2}}, for j \in {1, 2} . \end{matrix}$ Imposing the normalization on μ we have proven our statement. □

Let us look at the characteristic values of $A^{δ k, ε}$ . Denote ${\tilde{L}}^{δ k, ε} : = L^{ε} + \frac{π}{2} R^{δ k, ε}$ and $S^{δ k, ε} : = \frac{π}{2} {(L^{ε})}^{- 1} R^{δ k, ε}$ , where we fixed the angles of the incoming wave vector, but let the magnitude be complex. Using that $R^{δ k, ε}$ is in $C^{1, η}$ , for $η \in (0, 1)$ , and $L^{ε}$ invertible and using Lemma 3.14, we can apply the Neumann series, whenever ε is small enough, and thus we have $\begin{matrix} (4.14) & {({\tilde{L}}^{δ k, ε})}^{- 1} = {(I + S^{δ k, ε})}^{- 1} {(L^{ε})}^{- 1} = \sum_{l = 0}^{\infty} {(- S^{δ k, ε})}^{l} {(L^{ε})}^{- 1}, \end{matrix}$ where $I$ denotes the identity function in $X^{ε}$ .

We then define the $R^{2 \times 2}$ -matrix $A^{δ k, ε}$ as $\begin{matrix} (4.15) & {(A^{δ k, ε})}_{j, \hat{j}} : = - \frac{1}{| D_{\hat{j}} |} {({(S^{δ k, ε})}^{- 1} [e_{j}], e_{\hat{j}})}_{ε \oplus ε} = - \frac{π}{2 | D_{\hat{j}} |} {({(L^{ε})}^{- 1} R_{j, \hat{j}}^{δ k, ε} [1], 1)}_{ε}, \end{matrix}$ for $j, \hat{j} \in {1, 2}$ .
Lemma 4.12.
Any characteristic value of $A^{δ k, ε}$ is a characteristic value of the $R^{2 \times 2}$ -matrix $δ^{2} k^{2} I_{2} - A^{δ k, ε}$ .
Proof.
Suppose ${(δ k)}^{2}$ is a characteristic value of $A^{δ k, ε}$ . Substituting $L^{ε}$ with ${\tilde{L}}^{δ k, ε}$ in Lemma 4.11, we readily see that $\begin{matrix} (4.16) & \frac{2}{π} μ + \frac{{({\tilde{L}}^{δ k, ε})}^{- 1} K^{ε} [μ]}{{(δ k)}^{2}} = 0 . \end{matrix}$ Applying ${(\cdot, e_{1})}_{ε \oplus ε}$ on both sides, we obtain $\begin{matrix} (4.17) & (\frac{2}{π} {(μ_{1}, 1)}_{ε} + \frac{{(μ_{1}, 1)}_{ε}}{| D_{1} |} \frac{{({(L^{ε})}^{- 1} R_{1, 1}^{δ k, ε} [1], 1)}_{ε}}{{(δ k)}^{2}} + \frac{{(μ_{2}, 1)}_{ε}}{| D_{2} |} \frac{{({(L^{ε})}^{- 1} R_{1, 2}^{δ k, ε} [1], 1)}_{ε}}{{(δ k)}^{2}}) = 0 . \end{matrix}$ We readily see, $\begin{array}{l} {(δ k)}^{2} {(μ_{1}, 1)}_{ε} = - \frac{π {(μ_{1}, 1)}_{ε}}{2 | D_{1} |} {({(L^{ε})}^{- 1} R_{1, 1}^{δ k, ε} [1], 1)}_{ε} - \frac{π {(μ_{2}, 1)}_{ε}}{2 | D_{2} |} {({(L^{ε})}^{- 1} R_{1, 2}^{δ k, ε} [1], 1)}_{ε}, \\ {(δ k)}^{2} {(μ_{2}, 1)}_{ε} = - \frac{π {(μ_{1}, 1)}_{ε}}{2 | D_{1} |} {({(L^{ε})}^{- 1} R_{2, 1}^{δ k, ε} [1], 1)}_{ε} - \frac{π {(μ_{2}, 1)}_{ε}}{2 | D_{2} |} {({(L^{ε})}^{- 1} R_{2, 2}^{δ k, ε} [1], 1)}_{ε} . \end{array}$ Thus, $\begin{matrix} {(δ k)}^{2} [\begin{matrix} {(μ_{1}, 1)}_{ε} \\ {(μ_{2}, 1)}_{ε} \end{matrix}] = A^{δ k, ε} [\begin{matrix} {(μ_{1}, 1)}_{ε} \\ {(μ_{2}, 1)}_{ε} \end{matrix}] . \end{matrix}$ Hence, if ${(δ k)}^{2}$ is a characteristic value of $A^{δ k, ε}$ then it also is a characteristic value of ${(δ k)}^{2} - A^{δ k, ε}$ . □

We define $\begin{matrix} c_{ε} : = \frac{- 1}{log (ε / 2)} . \end{matrix}$
Proposition 4.13.
There exist four characteristic values, counting multiplicity, for the operator $A^{δ k, ε}$ function in $K_{C}$ . Moreover, they have the asymptotic $\begin{matrix} δ k = \pm δ k_{0}^{δ, ε} + δ O (c_{ε}), for ε \to 0 . \end{matrix}$
Proof.
This follows readily using the generalized Rouché’s theorem [6, Theorem 1.15]. We refer to [4, Proposition 4.13] for the detailed proof. □

Let us give an asymptotic expression for those characteristic values. Let $l ⩾ 1$ be an integer, we define the $R^{2 \times 2}$ -matrix $S_{(l)}^{δ k, ε}$ as $\begin{matrix} {(S_{(l)}^{δ k, ε})}_{j, \hat{j}} : = - \frac{π}{2 | D_{\hat{j}} |} {({(- S^{δ k, ε})}^{l} {(L^{ε})}^{- 1} [e_{j}], e_{\hat{j}})}_{ε \oplus ε}, \end{matrix}$ for $j, \hat{j} \in {1, 2}$ . Because of (4.14), we can write $A^{δ k, ε} = \frac{π}{2} c_{ε} [\begin{matrix} \frac{1}{| D_{1} |} & 0 \\ 0 & \frac{1}{| D_{2} |} \end{matrix}] + \sum_{l = 1}^{\infty} S_{(l)}^{δ k, ε}$ . We want to give a second order analytic expression for ${(S_{(1)}^{δ k, ε})}_{j, \hat{j}} = \frac{π}{2 | D_{\hat{j}} |} {(\frac{π}{2} {(L^{ε})}^{- 1} R^{δ k, ε} {(L^{ε})}^{- 1} [e_{j}], e_{\hat{j}})}_{ε \oplus ε}$ . To this end, we define $\begin{array}{l} (4.18) & {(S_{(1, 1)}^{δ k, ε})}_{j, \hat{j}} = \frac{π}{2 | D_{\hat{j}} |} {(\frac{π}{2} {(L^{ε})}^{- 1} R_{1}^{δ k, ε} {(L^{ε})}^{- 1} [e_{j}], e_{\hat{j}})}_{ε \oplus ε}, \\ (4.19) & {(S_{(1, 2)}^{δ k, ε})}_{j, \hat{j}} = \frac{π}{2 | D_{\hat{j}} |} {(\frac{π}{2} {(L^{ε})}^{- 1} R_{2}^{δ k, ε} {(L^{ε})}^{- 1} [e_{j}], e_{\hat{j}})}_{ε \oplus ε}, \\ (4.20) & {(S_{(1, 3)}^{δ k, ε})}_{j, \hat{j}} = \frac{π}{2 | D_{\hat{j}} |} {(\frac{π}{2} {(L^{ε})}^{- 1} R_{3}^{δ k, ε} {(L^{ε})}^{- 1} [e_{j}], e_{\hat{j}})}_{ε \oplus ε}, \end{array}$ where $\begin{array}{l} (4.21) & R_{1}^{δ k, ε} [μ] (τ) : = [\begin{matrix} {(α_{(0)})}_{1, 1} {(μ_{1}, 1)}_{ε} & {(T_{(0)})}_{1, 2} {(μ_{2}, 1)}_{ε} \\ {(T_{(0)})}_{2, 1} {(μ_{1}, 1)}_{ε} & {(α_{(0)})}_{2, 2} {(μ_{2}, 1)}_{ε} \end{matrix}], \\ (4.22) & R_{2}^{δ k, ε} [μ] (τ) : = δ k [\begin{matrix} {(α_{(1)})}_{1, 1} {(μ_{1}, 1)}_{ε} & {(T_{(1)})}_{1, 2} {(μ_{2}, 1)}_{ε} \\ {(T_{(1)})}_{2, 1} {(μ_{1}, 1)}_{ε} & {(α_{(1)})}_{2, 2} {(μ_{2}, 1)}_{ε} \end{matrix}], \\ (4.23) & {(R_{3}^{δ k, ε})}_{j, \hat{j}} [μ_{\hat{j}}] (τ) : = \int_{- ε}^{ε} μ_{\hat{j}} (t) (t {\tilde{\partial}}_{t} R_{j, \hat{j}}^{δ k} (τ, t) + τ {\tilde{\partial}}_{τ} R_{j, \hat{j}}^{δ k} (τ, t) + δ^{2} k^{2} {\tilde{\partial}}_{δ k}^{2} R_{j, \hat{j}}^{δ k} (τ, t)) d t . \end{array}$ Here, $R_{j, \hat{j}}^{δ k} (τ, t)$ denotes the kernel of ${(R^{δ k, ε})}_{j, \hat{j}}$ , see Definition 4.9, and $\tilde{\partial}$ denotes the derivative part of the remainder in Taylor’s theorem in the Peano form and where $\begin{array}{l} (4.24) & \begin{matrix} {(α_{(0)})}_{1, 1} & : = R_{\partial D_{1}}^{0} (\begin{matrix} (\begin{matrix} - ξ \\ h \end{matrix}), (\begin{matrix} - ξ \\ h \end{matrix}) \end{matrix}) + R_{\partial Ω^{1}, +}^{0} (\begin{matrix} (\begin{matrix} - ξ \\ h \end{matrix}), (\begin{matrix} - ξ \\ h \end{matrix}) \end{matrix}) \\ + \frac{1}{π} log (\frac{π}{p}) - \frac{1}{2 π} log (sinh {(\frac{π}{p} 2 h)}^{2}), \end{matrix} \\ (4.25) & \begin{matrix} {(α_{(0)})}_{2, 2} & : = R_{\partial D_{2}}^{0} (\begin{matrix} (\begin{matrix} ξ \\ h \end{matrix}), (\begin{matrix} ξ \\ h \end{matrix}) \end{matrix}) + R_{\partial Ω^{1}, +}^{0} (\begin{matrix} (\begin{matrix} ξ \\ h \end{matrix}), (\begin{matrix} ξ \\ h \end{matrix}) \end{matrix}) \\ + \frac{1}{π} log (\frac{π}{p}) - \frac{1}{2 π} log (sinh {(\frac{π}{p} 2 h)}^{2}), \end{matrix} \\ (4.26) & {(α_{(1)})}_{1, 1} : = \partial_{δ k} R_{\partial D_{1}}^{0} (\begin{matrix} (\begin{matrix} - ξ \\ h \end{matrix}), (\begin{matrix} - ξ \\ h \end{matrix}) \end{matrix}) + \partial_{δ k} R_{\partial Ω^{1}, +}^{0} (\begin{matrix} (\begin{matrix} - ξ \\ h \end{matrix}), (\begin{matrix} - ξ \\ h \end{matrix}) \end{matrix}), \\ (4.27) & {(α_{(1)})}_{2, 2} : = \partial_{δ k} R_{\partial D_{2}}^{0} (\begin{matrix} (\begin{matrix} ξ \\ h \end{matrix}), (\begin{matrix} ξ \\ h \end{matrix}) \end{matrix}) + \partial_{δ k} R_{\partial Ω^{1}, +}^{0} (\begin{matrix} (\begin{matrix} ξ \\ h \end{matrix}), (\begin{matrix} ξ \\ h \end{matrix}) \end{matrix}), \\ (4.28) & {(T_{(0)})}_{1, 2} : = N_{\partial Ω^{1}, +}^{0} (\begin{matrix} (\begin{matrix} - ξ \\ h \end{matrix}), (\begin{matrix} + ξ \\ h \end{matrix}) \end{matrix}), {(T_{(0)})}_{2, 1} : = N_{\partial Ω^{1}, +}^{0} (\begin{matrix} (\begin{matrix} + ξ \\ h \end{matrix}), (\begin{matrix} - ξ \\ h \end{matrix}) \end{matrix}), \\ (4.29) & {(T_{(1)})}_{1, 2} : = \partial_{δ k} N_{\partial Ω^{1}, +}^{0} (\begin{matrix} (\begin{matrix} - ξ \\ h \end{matrix}), (\begin{matrix} + ξ \\ h \end{matrix}) \end{matrix}), {(T_{(1)})}_{2, 1} : = \partial_{δ k} N_{\partial Ω^{1}, +}^{0} (\begin{matrix} (\begin{matrix} + ξ \\ h \end{matrix}), (\begin{matrix} - ξ \\ h \end{matrix}) \end{matrix}) . \end{array}$ We define $\begin{array}{l} α_{(0)} : = [\begin{matrix} {(α_{(0)})}_{1, 1} / | D_{1} | & 0 \\ 0 & {(α_{(0)})}_{2, 2} / | D_{2} | \end{matrix}], T_{(0)} : = [\begin{matrix} 0 & {(T_{(0)})}_{1, 2} / | D_{2} | \\ {(T_{(0)})}_{2, 1} / | D_{1} | & 0 \end{matrix}], \\ α_{(1)} : = [\begin{matrix} {(α_{(1)})}_{1, 1} / | D_{1} | & 0 \\ 0 & {(α_{(1)})}_{2, 2} / | D_{2} | \end{matrix}], T_{(1)} : = [\begin{matrix} 0 & {(T_{(1)})}_{1, 2} / | D_{2} | \\ {(T_{(1)})}_{2, 1} / | D_{1} | & 0 \end{matrix}] . \end{array}$ We obtain the second order analytic expression with the following lemma:
Lemma 4.14.
For all $l \in N$ , we have that $S_{(l)}^{δ k, ε} = O (c_{ε}^{l + 1})$ , as well as $S_{(1, 1)}^{δ k, ε} = \frac{π^{2}}{2} c_{ε}^{2} (α_{(0)} + T_{(0)})$ , $S_{(1, 2)}^{δ k, ε} = δ k \frac{π^{2}}{2} c_{ε}^{2} (α_{(1)} + T_{(1)})$ , $S_{(1, 3)}^{δ k, ε} = O (ε c_{ε}^{2} + c_{ε}^{2} δ^{2} k^{2})$ .
Proof.
The proof follows from straightforward calculation using Lemma 3.14 and the expressions in (4.21)–(4.23) and (4.18)–(4.20). For $S_{(1, 3)}^{δ k, ε}$ , we can use the argument in (3.23). □

Now we can deduce that $\begin{matrix} (4.30) & A^{δ k, ε} = \frac{π}{2} c_{ε} [\begin{matrix} \frac{1}{| D_{1} |} & 0 \\ 0 & \frac{1}{| D_{2} |} \end{matrix}] + \frac{π^{2}}{4} c_{ε}^{2} (α_{(0)} + T_{(0)}) + δ k \frac{π^{2}}{4} c_{ε}^{2} (α_{(1)} + T_{(1)}) + O (ε c_{ε}^{2} + c_{ε}^{2} δ^{2} k^{2}) . \end{matrix}$
Lemma 4.15.
There exists a $R^{2 \times 2}$ -matrix $A_{⋆}^{δ k, ε}$ such that $A^{δ k, ε} = {(A_{⋆}^{δ k, ε})}^{2}$ and $\begin{matrix} A_{⋆}^{δ k, ε} = \sqrt{\frac{π}{2} c_{ε}} (A_{⋆, D} + \frac{π c_{ε}}{2} A_{⋆, (0)} + \frac{δ k π c_{ε}}{2} A_{⋆, (1)}) + O (c_{ε}^{3 / 2} δ^{2} k^{2}) + O (c_{ε}^{5 / 2}), \end{matrix}$ where $\begin{array}{l} A_{⋆, D} = [\begin{matrix} \frac{1}{\sqrt{| D_{1} |}} & 0 \\ 0 & \frac{1}{\sqrt{| D_{2} |}} \end{matrix}], A_{⋆, (0)} = [\begin{matrix} \frac{{(α_{(0)})}_{1, 1}}{2 \sqrt{| D_{1} |}} & \frac{{(T_{(0)})}_{1, 2} \sqrt{| D_{1} | / | D_{2} |}}{\sqrt{| D_{1} |} + \sqrt{| D_{2} |}} \\ \frac{{(T_{(0)})}_{2, 1} \sqrt{| D_{2} | / | D_{1} |}}{\sqrt{| D_{1} |} + \sqrt{| D_{2} |}} & \frac{{(α_{(0)})}_{2, 2}}{2 \sqrt{| D_{2} |}} \end{matrix}], \\ A_{⋆, (1)} = [\begin{matrix} \frac{{(α_{(1)})}_{1, 1}}{2 \sqrt{| D_{1} |}} & \frac{{(T_{(1)})}_{1, 2} \sqrt{| D_{1} | / | D_{2} |}}{\sqrt{| D_{1} |} + \sqrt{| D_{2} |}} \\ \frac{{(T_{(1)})}_{2, 1} \sqrt{| D_{2} | / | D_{1} |}}{\sqrt{| D_{1} |} + \sqrt{| D_{2} |}} & \frac{{(α_{(1)})}_{2, 2}}{2 \sqrt{| D_{2} |}} \end{matrix}] . \end{array}$
Proof.
We use the following approach: $\begin{matrix} A_{⋆}^{δ k, ε} = c_{ε}^{1 / 2} A_{⋆, 1}^{δ k} + c_{ε} A_{⋆, 2}^{δ k} + c_{ε}^{3 / 2} A_{⋆, 3}^{δ k} + c_{ε}^{2} A_{⋆, 4}^{δ k} + O (c_{ε}^{5 / 2}) . \end{matrix}$ Thus, $\begin{array}{l} A^{δ k, ε} & = {(A_{⋆}^{δ k, ε})}^{2} = {(c_{ε}^{1 / 2} A_{⋆, 1}^{δ k} + c_{ε} A_{⋆, 2}^{δ k} + c_{ε}^{3 / 2} A_{⋆, 3}^{δ k} + c_{ε}^{2} A_{⋆, 4}^{δ k} + O (c_{ε}^{2}))}^{2} \\ = {(A_{⋆, 1}^{δ k})}^{2} c_{ε} + (A_{⋆, 1}^{δ k} A_{⋆, 2}^{δ k} + A_{⋆, 2}^{δ k} A_{⋆, 1}^{δ k}) c_{ε}^{3 / 2} + ({(A_{⋆, 2}^{δ k})}^{2} + A_{⋆, 1}^{δ k} A_{⋆, 3}^{δ k} + A_{⋆, 3}^{δ k} A_{⋆, 1}^{δ k}) c_{ε}^{2} \\ + (A_{⋆, 4}^{δ k} A_{⋆, 1}^{δ k} + A_{⋆, 1}^{δ k} A_{⋆, 4}^{δ k} + A_{⋆, 2}^{δ k} A_{⋆, 3}^{δ k} + A_{⋆, 3}^{δ k} A_{⋆, 2}^{δ k}) + O (c_{ε}^{3}) . \end{array}$ Comparing this equation to (4.30), and comparing the matrix-coefficients to the same order, we readily obtain the desired matrices. □

With $A_{⋆}^{δ k, ε}$ we can write $δ^{2} k^{2} I_{2} - A^{δ k, ε} = (δ k I_{2} - A_{⋆}^{δ k, ε}) (δ k I_{2} + A_{⋆}^{δ k, ε})$ , thus it is enough to find the characteristic values of $(δ k I_{2} - A_{⋆}^{δ k, ε})$ and $(δ k I_{2} + A_{⋆}^{δ k, ε})$ to get the characteristic values of $δ^{2} k^{2} I_{2} - A^{δ k, ε}$ .

We define $\begin{matrix} (4.31) & A_{⋆, ⋆}^{ε} : = \sqrt{\frac{π}{2} c_{ε}} (A_{⋆, D} + \frac{π c_{ε}}{2} A_{⋆, (0)}) . \end{matrix}$ Consider that $A_{⋆}^{δ k, ε} = A_{⋆, ⋆}^{ε} + \frac{δ k {(π c_{ε})}^{3 / 2}}{2} A_{⋆, (1)} + O (c_{ε}^{5 / 2} + c_{ε}^{3 / 2} δ^{2} k^{2})$ .

Now for ε small enough, $A_{⋆, ⋆}^{ε}$ is diagonalizable with $A_{⋆, ⋆}^{ε} = Y_{⋆, ⋆}^{ε} M_{⋆, ⋆}^{ε} {(Y_{⋆, ⋆}^{ε})}^{- 1}$ where $Y_{⋆, ⋆}^{ε} = [{(Y_{⋆, ⋆}^{ε})}_{1}, {(Y_{⋆, ⋆}^{ε})}_{2}]$ , with ${(Y_{⋆, ⋆}^{ε})}_{1}$ being the normalized eigenvector to the eigenvalue $δ k_{⋆, 1}^{δ, ε}$ of $A_{⋆, ⋆}^{ε}$ and ${(Y_{⋆, ⋆}^{ε})}_{2}$ being the normalized eigenvector to the eigenvalue $δ k_{⋆, 2}^{δ, ε}$ of $A_{⋆, ⋆}^{ε}$ , and for ε small enough, but not zero, $δ k_{⋆, 1}^{δ, ε}$ and $δ k_{⋆, 2}^{δ, ε}$ are distinct, since $A_{⋆, (0)}$ is not zero and thus $A_{⋆, ⋆}^{ε}$ is not similar to $I_{2}$ . Then we sort $δ k_{⋆, 1}^{δ, ε}$ and $δ k_{⋆, 2}^{δ, ε}$ such that the real part of $δ k_{⋆, 1}^{δ, ε}$ is greater or equal to $δ k_{⋆, 2}^{δ, ε}$ and $M_{⋆, ⋆}^{ε} = [\begin{matrix} δ k_{⋆, 1}^{δ, ε} & 0 \\ 0 & δ k_{⋆, 2}^{δ, ε} \end{matrix}]$ . We can explicitly compute those values:
Lemma 4.16.
We have that $\begin{array}{l} δ k_{⋆, 1}^{δ, ε} = \sqrt{π c_{ε}} (\frac{1}{2} (k_{⋆, tr 1}^{ε} + k_{⋆, tr 2}^{ε}) + {[\frac{1}{4} {(k_{⋆, tr 1}^{ε} - k_{⋆, tr 2}^{ε})}^{2} - k_{⋆, \det}^{ε}]}^{1 / 2}), \\ δ k_{⋆, 2}^{δ, ε} = \sqrt{π c_{ε}} (\frac{1}{2} (k_{⋆, tr 1}^{ε} + k_{⋆, tr 2}^{ε}) - {[\frac{1}{4} {(k_{⋆, tr 1}^{ε} - k_{⋆, tr 2}^{ε})}^{2} - k_{⋆, \det}^{ε}]}^{1 / 2}), \end{array}$ where $\begin{array}{l} k_{⋆, tr 1}^{ε} = \frac{1}{\sqrt{| D_{1} |}} (1 + \frac{{(α_{(0)})}_{1, 1} π c_{ε}}{4}), k_{⋆, tr 2}^{ε} = \frac{1}{\sqrt{| D_{2} |}} (1 + \frac{{(α_{(0)})}_{2, 2} π c_{ε}}{4}), \\ k_{⋆, \det}^{ε} = \frac{π^{2} c_{ε}^{2}}{4} \frac{{(T_{(0)})}_{1, 2} {(T_{(0)})}_{2, 1}}{(\sqrt{| D_{1} |} + \sqrt{| D_{2} |})} . \end{array}$

With this lemma we especially see that $δ k_{⋆, 1}^{δ, ε} = O (c_{ε}^{1 / 2})$ and $δ k_{⋆, 2}^{δ, ε} = O (c_{ε}^{1 / 2})$ . Proposition 4.17.
There exists exactly 2 characteristic values for each of the matrix-valued functions $δ k I_{2} - A_{⋆}^{δ k, ε}$ and $δ k I_{2} + A_{⋆}^{δ k, ε}$ . For $j \in {1, 2}$ , these are $\begin{array}{l} (4.32) & δ k_{j, +}^{δ, ε} = δ k_{⋆, j}^{δ, ε} (1 + \frac{{(π c_{ε})}^{\frac{3}{2}}}{4} e_{j}^{T} {(Y_{⋆, ⋆}^{ε})}^{- 1} A_{⋆, (1)} {(Y_{⋆, ⋆}^{ε})}_{j}) + O (c_{ε}^{5 / 2}), \\ (4.33) & δ k_{j, -}^{δ, ε} = δ k_{⋆, j}^{δ, ε} (- 1 + \frac{{(π c_{ε})}^{\frac{3}{2}}}{4} e_{j}^{T} {(Y_{⋆, ⋆}^{ε})}^{- 1} A_{⋆, (1)} {(Y_{⋆, ⋆}^{ε})}_{j}) + O (c_{ε}^{5 / 2}) . \end{array}$
Proof.
The existence of the unperturbed can be established readily by definition and the existence of the perturbed characteristic values are given through the generalized Rouché’s theorem [6, Theorem 1.15]. Using the an asymptotic expansion for the eigenvectors and inserting these afterwards into $k_{j, +}^{δ, ε} - A_{⋆}^{δ k, ε} |_{k = k_{j, +}^{δ, ε}}$ we obtain the proposition. For details consider [4, Proposition 4.17]. □

4.3.4. Inversion of $A^{δ k, ε}$ – Solving the first order linear equation

We know now that $A^{δ k, ε}$ is invertible for $δ k \in K_{C}$ , except at the characteristic values $k = k_{j, +}^{δ, ε}$ and $k = k_{j, -}^{δ, ε}$ and at the pole, $k = 0$ . Let us examine, how we can express ${(A^{δ k, ε})}^{- 1} [f^{δ k}]$ , where $f^{δ k} (τ) : = {(f_{D_{1}}^{δ k} (τ_{1}), f_{D_{2}}^{δ k} (τ_{2}))}^{T}$ , which uses (4.8).

First consider that we already know that the equation $A^{δ k, ε} [μ] = f^{δ k}$ is equivalent to $\begin{matrix} (4.34) & \frac{2}{π} μ + \frac{{({\tilde{L}}^{δ k, ε})}^{- 1} K^{ε} [μ]}{{(δ k)}^{2}} = {({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}] . \end{matrix}$ Similar to the proof of Lemma 4.12, we get that $\begin{matrix} (4.35) & (δ^{2} k^{2} I_{2} - A^{δ k, ε}) [\begin{matrix} {(μ_{1}, 1)}_{ε} \\ {(μ_{2}, 1)}_{ε} \end{matrix}] = \frac{π}{2} (δ^{2} k^{2}) [\begin{matrix} {({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], e_{1})}_{ε \oplus ε} \\ {({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], e_{2})}_{ε \oplus ε} \end{matrix}] . \end{matrix}$ We define $\begin{matrix} f_{L}^{δ k} : = [\begin{matrix} {({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], e_{1})}_{ε \oplus ε} \\ {({({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}], e_{2})}_{ε \oplus ε} \end{matrix}], f_{0}^{δ k} : = [\begin{matrix} f_{D_{1}}^{δ k} (0) \\ f_{D_{2}}^{δ k} (0) \end{matrix}] . \end{matrix}$

Lemma 4.18.
The inverse of $δ^{2} k^{2} I_{2} - A^{δ k, ε}$ has the following representation: $\begin{matrix} {(δ^{2} k^{2} I_{2} - A^{δ k, ε})}^{- 1} = (Y_{⋆, ⋆}^{ε}) {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} {(Y_{⋆, ⋆}^{ε})}^{- 1} + M_{rest}^{δ k, ε}, \end{matrix}$ where $\begin{array}{l} M_{+}^{δ k} = [\begin{matrix} δ k - δ k_{1, +}^{δ, ε} & 0 \\ 0 & δ k - δ k_{2, +}^{δ, ε} \end{matrix}], M_{-}^{δ k} = [\begin{matrix} δ k - δ k_{1, -}^{δ, ε} & 0 \\ 0 & δ k - δ k_{2, -}^{δ, ε} \end{matrix}], \\ M_{rest}^{δ k, ε} = [\begin{matrix} \frac{O (c_{ε}^{1 / 2})}{δ k - δ k_{1, +}^{δ, ε}} + \frac{O (c_{ε}^{1 / 2})}{δ k - δ k_{1, -}^{δ, ε}} & \frac{O (c_{ε}^{1 / 2})}{δ k - δ k_{1, +}^{δ, ε}} + \frac{O (c_{ε}^{1 / 2})}{δ k - δ k_{2, -}^{δ, ε}} \\ \frac{O (c_{ε}^{1 / 2})}{δ k - δ k_{2, +}^{δ, ε}} + \frac{O (c_{ε}^{1 / 2})}{δ k - δ k_{1, -}^{δ, ε}} & \frac{O (c_{ε}^{1 / 2})}{δ k - δ k_{2, +}^{δ, ε}} + \frac{O (c_{ε}^{1 / 2})}{δ k - δ k_{2, -}^{δ, ε}} \end{matrix}] . \end{array}$
Proof.
Using $δ^{2} k^{2} I_{2} - A^{δ k, ε} = (δ k - A_{⋆}^{δ k, ε}) (δ k + A_{⋆}^{δ k, ε})$ together with $δ k_{j, +}^{δ, ε} Y_{j}^{ε} = A_{⋆}^{δ k, ε} |_{k = k_{j, +}^{δ, ε}} Y_{j}^{ε}$ and the asymptotic expansion $Y_{j}^{ε} = {(Y_{⋆, ⋆}^{ε})}_{j} + c_{ε}^{1 / 2} Y_{j, (1)}^{ε} + O (c_{ε})$ we readily get the result. For details consider [4, Lemma 4.18]. □
Proposition 4.19.
Let $δ k \in K_{C} ∖ {0, δ k_{1, +}^{δ, ε}, δ k_{2, +}^{δ, ε}, δ k_{1, -}^{δ, ε}, δ k_{2, -}^{δ, ε}}$ , there exists a unique solution to the equation $A^{δ k, ε} [μ] = f^{δ k}$ . Moreover, the solution can be written as $μ = μ_{⋆} + μ_{\sim}$ , where $\begin{array}{l} μ_{⋆} = c_{ε} \frac{π^{2}}{4} {[{(L^{ε})}^{- 1} [1] {(A_{D} ((Y_{⋆, ⋆}^{ε}) {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} {(Y_{⋆, ⋆}^{ε})}^{- 1}) f_{0}^{δ k})}_{j}]}_{j = 1, 2} + \frac{π}{2} {(L^{ε})}^{- 1} [f_{0}^{δ k}], \\ μ_{\sim} = O (δ) ((O_{X^{ε}} (c_{ε}^{5 / 2}) + O_{X^{ε}} (ε)) (‖ {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} ‖ + ‖ M_{rest}^{δ k, ε} ‖) + (O_{X^{ε}} (c_{ε}^{2}) + O_{X^{ε}} (ε))) . \end{array}$ where $A_{D} : = [\begin{matrix} 1 / | D_{1} | & 0 \\ 0 & 1 / | D_{2} | \end{matrix}]$ .
Proof.
From (4.35), we get that $(δ^{2} k^{2} I_{2} - A^{δ k, ε}) [\begin{matrix} {(μ_{1}, 1)}_{ε} \\ {(μ_{2}, 1)}_{ε} \end{matrix}] = \frac{π}{2} (δ^{2} k^{2}) f_{L}^{δ k}$ . Thus, Lemma 4.18 yields $\begin{matrix} [\begin{matrix} {(μ_{1}, 1)}_{ε} \\ {(μ_{2}, 1)}_{ε} \end{matrix}] = ((Y_{⋆, ⋆}^{ε}) {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} {(Y_{⋆, ⋆}^{ε})}^{- 1} + M_{rest}^{δ k, ε}) \frac{π}{2} (δ^{2} k^{2}) f_{L}^{δ k} . \end{matrix}$ Note that similarly to Lemma 3.22, we have $\begin{array}{l} {({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}] & = [\begin{matrix} f_{D_{1}}^{δ k} (0) {(L^{ε})}^{- 1} [1] + O (δ) (O_{X^{ε}} (c_{ε}^{2}) + O_{X^{ε}} (ε)) \\ f_{D_{2}}^{δ k} (0) {(L^{ε})}^{- 1} [1] + O (δ) (O_{X^{ε}} (c_{ε}^{2}) + O_{X^{ε}} (ε)) \end{matrix}] \\ = {(L^{ε})}^{- 1} [f_{0}^{δ k}] + O (δ) (O_{X^{ε}} (c_{ε}^{2}) + O_{X^{ε}} (ε)) . \end{array}$ From (4.34), it follows that $\begin{array}{l} μ & = - \frac{π}{2} \frac{{({\tilde{L}}^{δ k, ε})}^{- 1} K^{ε} [μ]}{{(δ k)}^{2}} + \frac{π}{2} {({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}] \\ = - \frac{π}{2 {(δ k)}^{2}} {({\tilde{L}}^{δ k, ε})}^{- 1} [A_{D} ((Y_{⋆, ⋆}^{ε}) {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} {(Y_{⋆, ⋆}^{ε})}^{- 1} + M_{rest}^{δ k, ε}) \frac{π}{2} (δ^{2} k^{2}) f_{L}^{δ k}] \\ + \frac{π}{2} {({\tilde{L}}^{δ k, ε})}^{- 1} [f^{δ k}] \\ = - \frac{π^{2}}{4} {[{(L^{ε})}^{- 1} [1] {(A_{D} ((Y_{⋆, ⋆}^{ε}) {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} {(Y_{⋆, ⋆}^{ε})}^{- 1} + M_{rest}^{δ k, ε}) f_{L}^{δ k})}_{j}]}_{j = 1, 2} \\ \cdot (1 + O_{X^{ε}} (c_{ε}^{1}) + O_{X^{ε}} (ε)) + \frac{π}{2} ({(L^{ε})}^{- 1} [f_{0}^{δ k}] + O (δ) (O_{X^{ε}} (c_{ε}^{2}) + O_{X^{ε}} (ε))) . \end{array}$ Consider that $\begin{matrix} f_{L}^{δ k} = - c_{ε} f_{0}^{δ k} + O (δ) (O (c_{ε}^{2}) + O (ε)) . \end{matrix}$ This leads us to $\begin{array}{l} μ & = c_{ε} \frac{π^{2}}{4} {[{(L^{ε})}^{- 1} [1] {(A_{D} ((Y_{⋆, ⋆}^{ε}) {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} {(Y_{⋆, ⋆}^{ε})}^{- 1}) f_{0}^{δ k})}_{j}]}_{j = 1, 2} + \frac{π}{2} {(L^{ε})}^{- 1} [f_{0}^{δ k}] \\ + O (δ) (O_{X^{ε}} (c_{ε}^{5 / 2}) + O_{X^{ε}} (ε)) (‖ {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} ‖ + ‖ M_{rest}^{δ k, ε} ‖) \\ + O (δ) (O_{X^{ε}} (c_{ε}^{2}) + O_{X^{ε}} (ε)) . \end{array}$ This proves Proposition 4.19. □

4.3.5. Asymptotic expansion of our solution to the physical problem

In Proposition 4.10 we established $\begin{matrix} (4.36) & A^{δ k, ε} [\begin{matrix} \partial_{ν} u^{δ k} |_{Λ_{1}} \\ \partial_{ν} u^{δ k} |_{Λ_{2}} \end{matrix}] (τ) + \sum_{n = 1}^{\infty} δ^{n} 2 {\hat{Γ}}_{+, n}^{k, ε} [\begin{matrix} \partial_{ν} u^{δ k} |_{Λ_{1}} \\ \partial_{ν} u^{δ k} |_{Λ_{2}} \end{matrix}] (τ) = f^{δ k} (τ) . \end{matrix}$ On the other hand, we know that for $δ k \in K_{C} ∖ {0, δ k_{1, +}^{δ, ε}, δ k_{2, +}^{δ, ε}, δ k_{1, -}^{δ, ε}, δ k_{2, -}^{δ, ε}}$ that $A^{δ k, ε}$ is invertible, Proposition 4.19. Then for δ small enough we can use the Neumann series and obtain $\begin{array}{l} {(A^{δ k, ε} + \sum_{n = 1}^{\infty} 2 δ^{n} {\hat{Γ}}_{+, n}^{k, ε})}^{- 1} & = {(I + {(A^{δ k, ε})}^{- 1} \sum_{n = 1}^{\infty} 2 δ^{n} {\hat{Γ}}_{+, n}^{k, ε})}^{- 1} {(A^{δ k, ε})}^{- 1} \\ = {(A^{δ k, ε})}^{- 1} - 2 δ {(A^{δ k, ε})}^{- 1} {\hat{Γ}}_{+, 1}^{k, ε} + O (δ^{2}) O_{L (X^{ε}, X^{ε})} (1) . \end{array}$

Thus we have from solving (4.36) $[\begin{matrix} \partial_{ν} u^{δ k} |_{Λ_{1}} \\ \partial_{ν} u^{δ k} |_{Λ_{2}} \end{matrix}] = {(A^{δ k, ε})}^{- 1} [f^{δ k}] + O (δ^{2})$ , where we use the formula for ${(A^{δ k, ε})}^{- 1}$ and the facts that $f^{δ k} = O (δ)$ and ${\hat{Γ}}_{+, 1}^{k, ε}$ is linear. With Proposition 4.19 we split ${(A^{δ k, ε})}^{- 1} [f^{δ k}]$ into $μ_{⋆}^{δ k}$ and $μ_{\sim}^{δ k}$ and thus we have for $t \in (- ε, ε)$ $\begin{matrix} \partial_{ν} u^{δ k} |_{Λ_{j}} ((\begin{matrix} t \pm ξ \\ h \end{matrix})) = {(μ_{⋆}^{δ k})}_{j} (t) + {(μ_{\sim}^{δ k})}_{j} (t) + O (δ^{2}) . \end{matrix}$ We also see from Proposition 4.19 that $\partial_{ν} u^{δ k} |_{Λ_{j}} = O (δ)$ .

Now we want to calculate the first order expansion term in δ for the solution in the far-field. Let $z \in R^{2}$ , where $z_{2} ≫ 1$ .

Lemma 4.20.
We have for $z \in R_{+}^{2}$ , $z_{2} > 1$ , $\begin{array}{l} u_{s}^{δ k} (z) & = u_{RHS, p}^{δ k} (z) + u_{RHS, e}^{δ k} (z) + (S_{+, p}^{δ k} + S_{+, e}^{δ k}) [\partial_{ν} u^{δ k} |_{Λ_{1} \cup Λ_{2}}] (z) \\ (4.37) & + (T_{+, p}^{δ k} + T_{+, e}^{δ k}) [\partial_{ν} u^{δ k} |_{Λ_{1} \cup Λ_{2}}] (z) + O (δ^{2}), \end{array}$ where for $μ \in X^{ε}$ $\begin{array}{l} S_{+, p}^{δ k} [μ] (z) : = \int_{Λ_{1} \cup Λ_{2}} μ (y) Γ_{+, p}^{δ k} (z, y) d σ_{y}, S_{+, e}^{δ k} [μ] (z) : = \int_{Λ_{1} \cup Λ_{2}} μ (y) Γ_{+, e}^{δ k} (z, y) d σ_{y}, \\ T_{+, p}^{δ k} [μ] (z) : = - \int_{Λ_{1} \cup Λ_{2}} \int_{\partial D_{1} \cup \partial D_{2}} μ (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+, p}^{δ k} (z, w) d σ_{w} d σ_{y}, \\ T_{+, e}^{δ k} [μ] (z) : = - \int_{Λ_{1} \cup Λ_{2}} \int_{\partial D_{1} \cup \partial D_{2}} μ (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+, e}^{δ k} (z, w) d σ_{w} d σ_{y}, \end{array}$ and $\begin{array}{l} \begin{matrix} u_{RHS, p}^{δ k} (z) & = e^{i δ (k_{2} z_{2} - k_{1} z_{1})} [\int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) \frac{sin (δ k_{2} y_{2}) e^{i δ k_{1} y_{1}}}{δ k_{2} p} d σ_{y} \\ - \int_{\partial D_{1} \cup \partial D_{2}} \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \\ \cdot (\begin{matrix} \frac{i k_{1}}{p k_{2}} sin (δ k_{2} w_{2}) \\ \frac{1}{p} cos (δ k_{2} w_{2}) \end{matrix}) e^{i δ k_{1} w_{1}} d σ_{w} d σ_{y}], \end{matrix} \\ \begin{matrix} u_{RHS, e}^{δ k} (z) & = [- \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) Γ_{+, e}^{δ k} (z, y) d σ_{y} \\ + \int_{\partial D_{1} \cup \partial D_{2}} \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+, e}^{δ k} (z, w) d σ_{w} d σ_{y}] . \end{matrix} \end{array}$
Proof.
We define $u_{f}^{δ k} (z) : = u^{δ k} (z) + f_{\partial Ω^{1}}^{δ k} (z) = : u_{s}^{δ k} (z) - u_{RHS}^{δ k} (z)$ . Then we have from Proposition 4.5 that $\begin{array}{l} u_{f}^{δ k} (z) & = \int_{Λ_{1} \cup Λ_{2}} \partial_{ν} u^{δ k} (y) N_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} \\ = \int_{Λ_{1} \cup Λ_{2}} \partial_{ν} u^{δ k} (y) Γ_{+}^{δ k} (z, y) d σ_{y} + \int_{Λ_{1} \cup Λ_{2}} \partial_{ν} u^{δ k} (y) R_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} \\ = : S_{+}^{δ k} [\partial_{ν} u^{δ k} |_{Λ_{1} \cup Λ_{2}}] + T_{+}^{δ k} [\partial_{ν} u^{δ k} |_{Λ_{1} \cup Λ_{2}}] . \end{array}$ Using the splitting $Γ_{+}^{k} = Γ_{+, p}^{k} + Γ_{+, e}^{k}$ , see Proposition 2.4, we already obtain $S_{+, p}^{δ k}$ and $S_{+, e}^{δ k}$ . To study the terms $u_{RHS}^{δ k}$ and $T_{+}^{δ k}$ , we consider in view of Proposition 2.9 that $R_{Ω, +}^{k} (z, y) = - \int_{\partial E} N_{\partial Ω, \times}^{k} (y, w) \partial_{ν_{y}} Γ_{+}^{k} (z, w) d σ_{w}$ . Hence $\begin{matrix} \int_{Λ_{1} \cup Λ_{2}} \partial_{ν} u^{δ k} (y) R_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} = - \int_{Λ_{1} \cup Λ_{2}} \int^{\partial D_{1} \cup \partial D_{2}} \partial_{ν} u^{δ k} (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+}^{δ k} (z, w) d σ_{w} d σ_{y}, \end{matrix}$ and $\begin{array}{l} \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) R_{Ω^{1}, +}^{δ k} (z, y) d σ_{y} \\ (4.38) & = - \int_{Λ_{1} \cup Λ_{2}} \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+}^{δ k} (z, w) d σ_{w} d σ_{y} . \end{array}$ Using again the splitting $\nabla Γ_{+}^{k} = \nabla Γ_{+, p}^{k} + \nabla Γ_{+, e}^{k}$ and the explicit formula for $Γ_{+, p}^{k}$ and $\nabla Γ_{+, p}^{k}$ we obtain the formulas stated in Lemma 4.20. □

Let us approximate $S_{+}^{δ k}$ and $T_{+}^{δ k}$ . Let $z_{2} > 1$ . We define $\begin{array}{l} S_{+, p, 0}^{δ k} [μ] (z) : = - e^{i (δ k_{2} z_{2} - δ k_{1} z_{1})} \frac{1}{p} \int_{Λ_{1} \cup Λ_{2}} y_{2} μ (y) d σ_{y}, \\ S_{+, e, 0}^{δ k} [μ] (z) : = e^{- i δ k_{1} z_{1}} (\int_{Λ_{1} \cup Λ_{2}} μ (y) Γ_{+}^{0} (z, y) d σ_{y} + \frac{1}{p} \int_{Λ_{1} \cup Λ_{2}} y_{2} μ (y) d σ_{y}), \\ T_{+, p, 0}^{δ k} [μ] (z) : = e^{i (δ k_{2} z_{2} - δ k_{1} z_{1})} \frac{1}{p} \int_{Λ_{1} \cup Λ_{2}} \int_{\partial D_{1} \cup \partial D_{2}} μ (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) d σ_{w} d σ_{y}, \\ \begin{matrix} T_{+, e, 0}^{δ k} [μ] (z) & : = - e^{- i δ k_{1} z_{1}} (\int_{Λ_{1} \cup Λ_{2}} \int_{\partial D_{1} \cup \partial D_{2}} μ (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) \partial_{ν_{w}} Γ_{+}^{0} (z, w) d σ_{w} d σ_{y} \\ + e^{- i δ k_{2} z_{2}} T_{+, p, 0}^{δ k} [μ] (z)) . \end{matrix} \end{array}$
Lemma 4.21.
There exist $C_{(4.39)} > 0$ , $C_{(4.40)} > 0$ , $C_{(4.41)} > 0$ and $C_{(4.42)} > 0$ such that, for all $z \in R_{+}^{2}$ with $z_{2} > 1$ and all $μ \in X^{ε}$ , it holds that $\begin{array}{l} (4.39) & | (S_{+, p}^{δ k} - S_{+, p, 0}^{δ k}) [μ] (z) | + \frac{1}{δ} | \nabla (S_{+, p}^{δ k} - S_{+, p, 0}^{δ k}) [μ] (z) | ⩽ C_{(4.39)} ‖ μ ‖_{X^{ε}} δ^{2}, \\ (4.40) & | (S_{+, e}^{δ k} - S_{+, e, 0}^{δ k}) [μ] (z) | + \frac{1}{δ} | \nabla (S_{+, e}^{δ k} - S_{+, e, 0}^{δ k}) [μ] (z) | ⩽ C_{(4.40)} ‖ μ ‖_{X^{ε}} δ, \\ (4.41) & | (T_{+, p}^{δ k} - T_{+, p, 0}^{δ k}) [μ] (z) | + \frac{1}{δ} | \nabla (T_{+, p}^{δ k} - T_{+, p, 0}^{δ k}) [μ] (z) | ⩽ C_{(4.41)} ‖ μ ‖_{X^{ε}} δ, \\ (4.42) & | (T_{+, e}^{δ k} - T_{+, e, 0}^{δ k}) [μ] (z) | + \frac{1}{δ} | \nabla (T_{+, e}^{δ k} - T_{+, e, 0}^{δ k}) [μ] (z) | ⩽ C_{(4.42)} ‖ μ ‖_{X^{ε}} δ . \end{array}$

This proof is analogous to Lemma 3.25. For details consider also [4, Lemma 4.21]. We define $u_{S_{p}^{⋆}}^{δ k} (z) : = S_{+, p, 0}^{δ k} [μ_{⋆}^{δ k}] (z)$ , $u_{T_{p}^{⋆}}^{δ k} (z) : = T_{+, p, 0}^{δ k} [μ_{⋆}^{δ k}] (z)$ , $u_{S_{e}^{⋆}}^{δ k} (z) : = S_{+, e, 0}^{δ k} [μ_{⋆}^{δ k}] (z)$ , $u_{T_{e}^{⋆}}^{δ k} (z) : = T_{+, e, 0}^{δ k} [μ_{⋆}^{δ k}] (z)$ . We then have the following proposition: Proposition 4.22.
Let $V_{r} : = {z \in R_{+}^{2} ∣ z_{2} > r}$ . There exists a constant $C_{(4.43)} > 0$ such that $\begin{array}{l} {‖ u_{s}^{δ k} - (u_{S_{p}^{⋆}}^{δ k} + u_{S_{e}^{⋆}}^{δ k} + u_{T_{p}^{⋆}}^{δ k} + u_{T_{e}^{⋆}}^{δ k} + u_{RHS, p}^{δ k} + u_{RHS, e}^{δ k}) ‖}_{L^{\infty} (V_{r})} \\ + \frac{1}{δ} {‖ \nabla [u_{s}^{δ k} - (u_{S_{p}^{⋆}}^{δ k} + u_{S_{e}^{⋆}}^{δ k} + u_{T_{p}^{⋆}}^{δ k} + u_{T_{e}^{⋆}}^{δ k} + u_{RHS, p}^{δ k} + u_{RHS, e}^{δ k})] ‖}_{L^{\infty} (V_{r})} \\ (4.43) & ⩽ C_{(4.43)} (δ c_{ε}^{3} (‖ {(M_{+}^{δ k} M_{-}^{δ k})}^{- 1} ‖) + δ c_{ε}^{2} + δ ε^{1} + δ^{2}), \end{array}$ for ε small enough.
Proof.
According to (4.37), we have $\begin{matrix} u_{s}^{δ k} - u_{RHS, p}^{δ k} - u_{RHS, e}^{δ k} = (S_{+, e}^{δ k} + S_{+, p}^{δ k} + T_{+, e}^{δ k} + T_{+, p}^{δ k}) [u^{δ k} |_{Λ_{1} \cup Λ_{2}}] + O (δ^{2}) . \end{matrix}$ With Lemma 4.21 and the fact that $u^{δ k} |_{Λ_{1} \cup Λ_{2}} = μ_{⋆}^{δ k} + μ_{\sim}^{δ k} + O (δ^{2})$ , $‖ μ_{⋆}^{δ k} ‖_{X^{ε}} = O (δ)$ and $‖ μ_{\sim}^{δ k} ‖_{X^{ε}} = O (δ)$ , we readily proof Proposition 4.22. □
Proof of Theorem 4.2.
We see that $u_{S_{e}^{⋆}}^{δ k}$ , $u_{T_{e}^{⋆}}^{δ k}$ and $u_{RHS, e}^{δ k}$ are exponentially decaying in $z_{2}$ and are of order $O (δ)$ , thus their $L^{\infty} (V_{r})$ -norm is of order $δ e^{- C r}$ for some constant $C > 0$ . Then with Proposition 4.22 and the change into the macroscopic view with $U^{k} (x) = u^{δ k} (x / δ)$ , we obtain Theorem 4.2. □

4.3.6. Evaluating the impedance boundary condition

We switch back to the macroscopic variable $U^{k} (x) = u^{δ k} (x / δ)$ . We approximate our solution in the far-field with the function $\begin{array}{l} U_{app}^{k} (z) & : = (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (z / δ) + u_{S_{p}^{⋆}}^{δ k} (z / δ) + u_{T_{p}^{⋆}}^{δ k} (z / δ) + u_{RHS, p}^{δ k} (z / δ) \\ (4.44) & = - 2 i a_{0} e^{- i k_{1} z_{1}} sin (k_{2} z_{2}) + e^{i (k_{2} z_{2} - k_{1} z_{1})} C_{(4.45)}^{δ k}, \end{array}$ where $\begin{array}{l} C_{(4.45)}^{δ k} & : = - \frac{h}{p} \int_{Λ_{1} \cup Λ_{2}} μ_{⋆}^{δ k} (y) d σ_{y} + \frac{1}{p} \int_{Λ_{1} \cup Λ_{2}} \int_{\partial D_{1} \cup \partial D_{2}} μ_{⋆}^{δ k} (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \cdot (\begin{matrix} 0 \\ 1 \end{matrix}) d σ_{w} d σ_{y} \\ + \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) \frac{sin (δ k_{2} y_{2}) e^{i δ k_{1} y_{1}}}{δ k_{2} p} d σ_{y} \\ - \int_{\partial D_{1} \cup \partial D_{2}} \int_{\partial D_{1} \cup \partial D_{2}} \partial_{ν} (u_{0}^{δ k} - u_{0}^{δ k} \circ P) (y) N_{\partial Ω^{1}, \times}^{δ k} (y, w) ν_{w} \\ (4.45) & \cdot (\begin{matrix} \frac{i k_{1}}{p k_{2}} sin (δ k_{2} w_{2}) \\ \frac{1}{p} cos (δ k_{2} w_{2}) \end{matrix}) e^{i δ k_{1} w_{1}} d σ_{w} d σ_{y} . \end{array}$ Consider that $C_{(4.45)}^{δ k} = O (δ)$ since $μ_{⋆}^{δ k} = O (δ)$ and $(u_{0}^{δ k} - u_{0}^{δ k} \circ P) = O (δ)$ and the other factors are of size $O (1)$ .

We want that $U_{app}^{k}$ satisfies the impedance boundary condition, that is $U_{app}^{k} (z) + δ c_{IBC} \partial_{z_{2}} U_{app}^{k} (z) = 0$ at $\partial R_{+}^{2}$ . By (4.44), we obtain the condition $\begin{matrix} e^{- i k_{1} z_{1}} C_{(4.45)}^{δ k} + δ c_{IBC} (- 2 i a_{0} k_{2} e^{- i k_{1} z_{1}} + i k_{2} e^{- i k_{1} z_{1}} C_{(4.45)}^{δ k}) = 0, for all z_{2} \in R . \end{matrix}$ After rearranging the terms, we obtain $\begin{matrix} c_{IBC} = \frac{- C_{(4.45)}^{δ k}}{i δ k_{2} (C_{(4.45)}^{δ k} - 2 a_{0})} . \end{matrix}$ Using that $\frac{1}{1 + O (δ)} = 1 - O (δ)$ we have $c_{IBC} = \frac{C_{(4.45)}^{δ k}}{2 i a_{0} δ k_{2}} + O (δ)$ . This proves Theorem 4.3.

4.4. Numerical illustrations

In this subsection we compute the impedance boundary condition constant $c_{IBC}$ with numerical means using Theorem 4.3. We use two geometries, both build up upon rectangles, but with different sizes and for each geometry we have different ranges for the wave number and the gap length.

We fix $δ = 0.01$ .

Again, we implement $Γ_{♯}^{k}$ using Edwald’s Method see [15] or [6, Chapter 7.3.2]. We implement the remainders using Lemmas 2.10, 2.11 and 2.12.

The first geometry has the following set-up. There are two rectangles both with length 0.9 and height 0.9, whose gaps are centered at ${(- 0.5, 1)}^{T}$ and ${(0.5, 1)}^{T}$ , respectively. The period is $p = 2$ and $h = 1$ . The amplitude of the incident wave is $a_{0} = 1$ . The number of points, with which we approximate the boundary of each rectangle, is 200. For the wave vector k, we take 341 equidistant points in the interval $[30, 200]$ . For ε we pick 5 values, those are ${0.1, 0.05, 0.03, 0.01, 0.001}$ . The result can be seen in Fig. 5. Note that because of our choice of the direction of incidence, which is orthogonal to the array of resonators, only one resonance is excited.

Fig. 5.

The plot of the absolute value of the variable $c_{IBC}$ depending on the wave vector k for the first geometry. For every value of ε, the rounded values of the resonance values $k_{1, res}^{δ, ε} : = k_{+, 1}^{δ, ε}$ and $k_{2, res}^{δ, ε} : = k_{+, 2}^{δ, ε}$ are displayed.

Fig. 6.

The plot of the absolute value of the variable $c_{IBC}$ depending on the wave vector k for the second geometry. For every value of ε, the rounded values of the resonance values $k_{1, res}^{δ, ε} : = k_{+, 1}^{δ, ε}$ and $k_{2, res}^{δ, ε} : = k_{+, 2}^{δ, ε}$ are displayed.

The second geometry has the following set-up. There are two rectangles both with length 0.2 and height 0.3, whose gaps are centered at ${(- 0.5, 1)}^{T}$ and ${(0.5, 1)}^{T}$ , respectively. The period is $p = 2$ and $h = 0.5$ . The amplitude of the incident wave is $a_{0} = 1$ . The number of points, with which we approximate the boundary of each rectangle, is 200. For the wave vector k, we take 301 equidistant points in the interval $[100, 400]$ . For ε we pick 5 values, those are ${0.1, 0.05, 0.03, 0.01, 0.001}$ . Consider that the case $ε = 0.1$ implies that the whole upper boundary of both rectangles are gaps. The result can be seen in Fig. 6.

Consider that the first geometry is the same geometry as in the one resonator case up to a translated origin and thus Fig. 5, has the same appearance as Fig. 2.

5. Conclusion

In this paper, we have established a mathematical theory of micro-scaled periodically arranged Helmholtz resonators and derived expansions of the scattered fields at the subwavelength resonances in terms of the size of the gap opening. We have highlighted the mechanism of the Neumann functions to exploit the intrinsic properties of the wave behaviour near and away from the gaps.

With this knowledge, we were able to answer both question; how can we model an array of Helmholtz resonators and how does the resonance behave in the two arrangements, that is one periodically arranged Helmholtz resonator and pairs of periodically arranged Helmholtz resonators.

Our approach opens many new avenues for mathematical imaging and focusing of waves in complex media. Whereas the results in Sections 3 and 4 can also be used for different industrial objectives, the main application is the construction of a tunable metasurface, which leads, as discussed in [5], to the enhancement of the transmission between a source and a receiver placed in a cavity through specific cavity eigenmodes.

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