A stable, unified model for resonant Faraday cages

Abstract

We study some effective transmission conditions able to reproduce the effect of a periodic array of Dirichlet wires on wave propagation, in particular when the array delimits an acoustic Faraday cage able to resonate. In the study of Hewett & Hewitt (2016 Proc. R. Soc. A 472, 20160062 (doi:10.1098/rspa.2016.0062)) different transmission conditions emerge from the asymptotic analysis whose validity depends on the frequency, specifically the distance to a resonance frequency of the cage. In practice, dealing with such conditions is difficult, especially if the problem is set in the time domain. In the present study, we demonstrate the validity of a simpler unified model derived in Marigo & Maurel (2016 Proc. R. Soc. A 472, 20160068 (doi:10.1098/rspa.2016.0068)), where unified means valid whatever the distance to the resonance frequencies. The effectiveness of the model is discussed in the harmonic regime owing to explicit solutions. It is also exemplified in the time domain, where a formulation guaranteeing the stability of the numerical scheme has been implemented.

1. Introduction

A Faraday’s cage is a cavity whose metallic boundary acts as a shield against external electrical discharges. When this boundary is punctured the shielding is not perfect anymore. Originally set in the static case, Faraday cage structures have been studied in electromagnetism for applications in filtering and shielding by gratings and grids. In this context, the question of the radiation leakage, for instance from microwave oven doors, has been addressed theoretically [1–3] and experimentally [4]. More recently, the problem has been revisited and extended to the acoustic case, i.e. governed by the Helmholtz equation [5–7], and to the electromagnetic case, i.e. governed by Maxwell’s equations [8]. The aim is to evaluate the effectiveness of the shielding when waves propagate outside a cage made of a periodic arrangement of wires; in two dimensions, the distance between the wires is h and their typical extent is e. The low-frequency regime is considered, i.e. kh = O(ε) and ε ≪ 1, where k is the typical wavenumber imposed by the source. In addition, in most of these studies, the thin wire approximation is used, which means that e/h ≪ 1; hence, the shielding is controlled by the number N of wires per unit length. Martin [5] derived explicit solutions using multiple scattering theory for N point-like wires evenly distributed on a line (not a cage) or on a large circle (a cage) [5]. For a cage, Hewett and co-authors showed that the asymptotic limit of large N results in a continuum model involving homogenized transmission conditions across an effective interface [6,7]. For thin wires, these conditions are derived using the specific scaling (e/h) = O(e^−(c/ε)) for every c > 0, introduced in [9] for thin Dirichlet fibres in a large volume. This scaling allows the coupling between the two sides of the array to be captured in the limit problem, that is, the zero-order problem provided by the asymptotic analysis. This is a key point in the present context. Indeed, for thick wires with (e/h) = O(1), the coupling cannot be captured by the limit problem; at zero order, the array is crudely replaced by a Dirichlet wall. (In [8], this is used to discriminate between different geometries of arrays producing an efficient shielding.) The coupling is captured at the first order. For the problem of scattering by a linear array in free space considered in [5], this is incidental: the zero-order model predicts zero transmission and the first-order model a small transmission. However, when the array delimits a cavity, the limit problem is ill-posed at the resonances of the close cavity. To overcome this difficulty Hewett & Hewitt [7] derived a model with two different conditions, one valid far from the resonances and the other valid near the resonances.

The problem has been approached differently by Marigo and Maurel [10,11], who derived effective conditions without considering a specific wave problem. These unified conditions, which gather the zero- and first-order conditions, have been further applied to the scattering by a linear array in the free space (not a cage) [10] and to the scattering by the same array at a distance D of a Dirichlet wall with kD = O(1) (a one-dimensional cage); see appendix B in [11]. The underlying idea is that the effective conditions remain valid in any wave problem as long as this problem does not involve a length comparable to that of the array spacing (the microstructure). We can notice that the same idea applies to classical transmission conditions across a flat interface: while no interface is truly flat, at least at the atomic scale, it is commonly admitted that what matters is that it appears to be flat at the scale of the wavelength. Accordingly, their validity is not interrogated whenever a new problem is considered. Intuitively, this should be true after any homogenization process if the resulting conditions have the same, good, properties as the actual ones. In [11] the construction of unified conditions, avoiding an iterative resolution, provides a well-posed effective problem. This being said, well-posedness does not imply effectiveness of the model to approximate the actual solution. We address this question in the case of a cage by constructing asymptotic expansions of the solutions of the actual problem and of the unified, approximate, problem. By construction, the expansions coincide far from the resonances; more interestingly they also coincide near and at the resonances. It is good news for homogenization that no complication occurs in a simple configuration where the unique additional length scale, that of the resonant cavity, is at the largest, wavelength, scale. More generally, this interrogates the pertinence of an iterative resolution which reconstitutes faithfully the expansion of the solution. Indeed, such an expansion can fail in satisfying basic conservation laws. By contrast, a model using unified conditions is shown to enjoy the same good properties of the actual problem (conservation of the fluxes in the harmonic regime and conservation of the energy in the time domain).

The rest of the paper is organized as follows. In §2, we recall the results of the homogenization procedure for an array in free space. The result is the zero- and first-order transmission conditions as used in an iterative resolution and the construction of unified, up-to-first order conditions. We move on to the Faraday cage in §3, where we prove that the solution of the unified problem efficiently approximates the actual solution far, near and on the resonances. Numerical results are collected in §§4 and 5. In the harmonic regime (§4), explicit solutions of the effective models are available. We illustrate the effectiveness of our (unified) model that we compare with the three solutions resulting from an iterative resolution being piecewise-valid far, near and on the resonance. (It is shown that the three solutions coincide with Taylor expansions of the unified one.) In §5, we consider the transient regime which poses the problem of the stability of effective models. The initial formulation of the unified model is shown to belong to a family of formulations with the same asymptotic limit. We show that only a part of this family ensures stability of the model when it is associated with a law of energy conservation involving a positive effective energy. The numerical implementation of a stable effective model allows us to show its effectiveness by comparison with direct numerics. We also illustrate the problem of numerical instability for an unstable formulation and the link with the behaviour of the effective, interfacial energy. Concluding remarks and extensions of the present work are given in §6. We also collect additional calculations and results in four appendices.

We denote (p, u) as the fields satisfying the wave equation in the time domain

$$\frac{\mathrm{\partial }\mathit{u}}{\mathrm{\partial }t}=-\mathbf{\nabla }p,\frac{\mathrm{\partial }p}{\mathrm{\partial }t}+\mathrm{div}\mathit{u}=0$$ 1.1

(the wave speed is c = 1), and in the harmonic regime with time dependance e^−iωt, p satisfies the Helmholtz equation

$$\mathrm{\Delta }p+k^{2}p=0,$$ 1.2

with k = ω/c. We denote x = (x₁, x₂) as the spatial coordinate and we consider an array of wires whose boundaries are associated with the Dirichlet boundary condition p = 0. The array with spacing h is included in x₁ ∈ ( − e₁, e₁) and it can be unbounded or bounded along x₂. Within the array, a wire has a typical extent 2e₁ = O(h) along x₁ and 2e₂ = O(h) along x₂ (figure 1). To conduct the asymptotic analysis, we introduce the non-dimensional coordinate x = x/L, x = (x₁, x₂), with L of the order of a typical wavelength imposed by the source, and we introduce the small parameter

$$\epsilon =\frac{h}{L}\ll 1.$$

Figure 1.

A linear array in free space (not a cage). Left, the actual problem on p^ε in rescaled coordinate x = x/L. Right, the resulting effective problem on p^ap. The asymptotic homogenization, symbolized by the arrow, involves elementary problems on (Q⁻, Q⁺) set in an elementary cell $\mathsf{Y}$ containing a single wire. (Online version in colour.)

2. Reminder: a linear array in the free space (not a cage)

Here we consider the free space and an array of wires evenly distributed on the interface $\mathrm{\Gamma }=\{(0,x_2)\in {\mathbb{R}}^2,x_2\in \mathbb{R}\}$, hence unbounded along x₂.

(a). Iterative or unified transmission conditions

The asymptotic analyses conducted in [7,10,11] are basically identical and they provide the same results. With p = p₀ + εp₁ + O(ε²), the zero-order term p₀ is found to satisfy the wave equation (1.1) in the transient regime or (1.2) in the harmonic regime with boundary conditions

$$p_0(0^{\pm },x_2)=0.$$ 2.1

The first-order term p₁ is found to satisfy (1.1) or (1.2) with boundary conditions fed by p₀, specifically

$$\begin{array}{rl}\hspace{0.17em}& p_1(0^{-},x_2)=\mathcal{A}\frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}(0^{+},x_2)-\mathcal{B}\frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}(0^{-},x_2)\\ \text{and}& p_1(0^{+},x_2)=\mathcal{B}\frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}(0^{+},x_2)-\mathcal{A}\frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}(0^{-},x_2).\end{array}\}$$ 2.2

The two real constants $(\mathcal{A},\mathcal{B})$ are given by the elementary solution Q⁺ set in the rescaled coordinate y = x/ε in the domain $\mathsf{Y}=\{(y_1,y_2)\in \mathbb{R}\times (0,1)\setminus \overline{\hat{\mathsf{Y}}}\}$, where $\hat{\mathsf{Y}}\subset (-(e_1/h),e_1/h)\times (0,1)$ is a canonical wire that we suppose to be symmetric with respect to the axis of equation y₁ = 0 (inset of figure 1). The elementary solution Q⁺ is the 1-periodic w.r.t the y₂ solution to

$$\mathrm{\Delta }{Q}^{+}=0\mathrm{in}\hspace{0.17em}\mathsf{Y},{\mathrm{Q}}^{+}=0\text{on}\hspace{0.17em}\mathrm{\partial }\hat{\mathsf{Y}},\text{and}\underset{{\mathrm{y}}_1\to -\mathrm{\infty }}{lim}\mathbf{\nabla }{\mathrm{Q}}^{+}=\mathbf{0},\underset{{\mathrm{y}}_1\to +\mathrm{\infty }}{lim}\mathbf{\nabla }{\mathrm{Q}}^{+}={\mathit{e}}_1,$$ 2.3

and the constants $\mathcal{B}$ and $\mathcal{A}$ are defined by

$$Q^{+}\underset{y_1\to -\mathrm{\infty }}{\sim }\mathcal{A},Q^{+}\underset{y_1\to +\mathrm{\infty }}{\sim }{y}_1+\mathcal{B}.$$ 2.4

A usual resolution consists in solving the problem on p₀ with (2.1), then that on p₁ using (2.2); we call the conditions (2.1) and (2.2) iterative transmission conditions. Following [12–14], unified transmission conditions are constructed by recombining (2.1) and (2.2) in

$$\begin{array}{rl}\hspace{0.17em}& p^{\text{ap},\epsilon }(0^{-},x_2)=\epsilon \mathcal{A}\frac{\mathrm{\partial }{p}^{\text{ap},\epsilon }}{\mathrm{\partial }{x}_1}(0^{+},x_2)-\epsilon \mathcal{B}\frac{\mathrm{\partial }{p}^{\text{ap},\epsilon }}{\mathrm{\partial }{x}_1}(0^{-},x_2)\\ \text{and}& p^{\text{ap},\epsilon }(0^{+},x_2)=\epsilon \mathcal{B}\frac{\mathrm{\partial }{p}^{\text{ap},\epsilon }}{\mathrm{\partial }{x}_1}(0^{+},x_2)-\epsilon \mathcal{A}\frac{\mathrm{\partial }{p}^{\text{ap},\epsilon }}{\mathrm{\partial }{x}_1}(0^{-},x_2).\end{array}\}$$ 2.5

Obviously, p^ap,ε = p₀ + εp₁ + O(ε²); hence both p^ap,ε and (p₀ + εp₁) approximate p up to O(ε²).

Remark 2.1. —

The above relations are written in the case of wires being symmetric with respect to y₁; otherwise, an additional elementary solution Q⁻ is involved, resulting in an additional constant; see (2.31) in [10]. The elementary solution Q⁻ is the 1-periodic w.r.t. the y₂ solution to

$$\mathrm{\Delta }{Q}^{-}=0\mathrm{in}\mathsf{Y},{\mathrm{Q}}^{-}=0\mathrm{on}\mathrm{\partial }\hat{\mathsf{Y}},\mathrm{and}\underset{{\mathrm{y}}_1\to -\mathrm{\infty }}{lim}\mathbf{\nabla }{\mathrm{Q}}^{-}={\mathit{e}}_1,\underset{{\mathrm{y}}_1\to +\mathrm{\infty }}{lim}\mathbf{\nabla }{\mathrm{Q}}^{-}=\mathbf{0}.$$ 2.6

For symmetric wires, Q⁻(y₁, y₂) = −Q⁺( − y₁, y₂); hence, $Q^{-}\underset{y_1\to -\mathrm{\infty }}{\sim }{y}_1-\mathcal{B}$ and $Q^{-}\underset{y_1\to +\mathrm{\infty }}{\sim }-\mathcal{A}$.

(b). Remark on a simple scattering problem

Let us stress a drawback of an iterative resolution when illuminating the array by an incident wave ${\text{e}}^{-\text{i}k{x}_1}$ (coming from +∞) at normal incidence on the array, hence a solution of the form

$$p^{\text{ap}}(\mathbf{\text{x}})={\text{e}}^{-\text{i}k{x}_1}+{\mathcal{R}}^{\text{ap}}\hspace{0.17em}{\text{e}}^{\text{i}k{x}_1}\text{for }{x}_1>0,p^{\text{ap}}(\mathbf{\text{x}})={\mathcal{T}}^{\text{ap}}\hspace{0.17em}{\text{e}}^{-\text{i}k{x}_1}\text{for }{x}_1<0.$$ 2.7

In an iterative resolution, ${\mathcal{R}}^{\text{ap}}$ is sought of the form ${\mathcal{R}}^{\text{ap}}={\mathcal{R}}_0+\epsilon {\mathcal{R}}_1$ up to first order (the same for ${\mathcal{T}}^{\text{ap}}$). The solution p₀ ruled by (2.1) provides ${\mathcal{R}}_0=-1$ and ${\mathcal{T}}_0=0$. Next p₁, of the form ${\mathcal{R}}_1\hspace{0.17em}{\text{e}}^{mk{x}_1}$ for x₁ > 0 and ${\mathcal{T}}_1\hspace{0.17em}{\text{e}}^{-\text{i}k{x}_1}$ for x₁ < 0, and ruled by (2.2), provides ${\mathcal{R}}_1=-2\text{i}k\mathcal{B}$ and ${\mathcal{T}}_1=-2\text{i}k\mathcal{A}$. Reconstructing ${\mathcal{R}}^{\text{ap}}$ and ${\mathcal{T}}^{\text{ap}}$ up to first order leads to

$${\mathcal{R}}^{\text{ap}}=({\mathcal{R}}_0+\epsilon {\mathcal{R}}_1)=-(1+2\text{i}kh\mathcal{B}),{\mathcal{T}}^{\text{ap}}=({\mathcal{T}}_0+\epsilon {\mathcal{T}}_1)=-2\text{i}kh\mathcal{A},$$ 2.8

which do not satisfy the conservation of the fluxes $|{\mathcal{R}}^{\text{ap}}{|}^2+|{\mathcal{T}}^{\text{ap}}{|}^2=1$. (They satisfy the conservation up to O(ε²).) Consider now the solution (2.7) ruled by the unified conditions (2.5). We obtain

$${\mathcal{R}}^{\text{ap}}=-\frac{1-(kh{)}^2({\mathcal{A}}^2-{\mathcal{B}}^2)}{1-2\text{i}kh\mathcal{B}+(kh{)}^2({\mathcal{A}}^2-{\mathcal{B}}^2)},{\mathcal{T}}^{\text{ap}}=\frac{-2\text{i}kh\mathcal{A}}{1-2\text{i}kh\mathcal{B}+(kh{)}^2({\mathcal{A}}^2-{\mathcal{B}}^2)},$$ 2.9

which satisfies exactly $|{\mathcal{R}}^{\text{ap}}{|}^2+|{\mathcal{T}}^{\text{ap}}{|}^2=1$. By construction, $({\mathcal{R}}_0+\epsilon {\mathcal{R}}_1)$ and $({\mathcal{T}}_0+\epsilon {\mathcal{T}}_1)$ in (2.8) are the expansions up to O(ε²) of ${\mathcal{R}}^{\text{ap}}$ and ${\mathcal{T}}^{\text{ap}}$ in (2.9). We may say that (2.9) contains non-valuable terms O(ε²); this has no consequence. In contrast, cancelling these terms a posteriori would affect the properties of the solution, here the conservation of the fluxes. We can say as a rule that the solutions of the unified problem do not have to be interrogated again.

3. A linear array on top of a Dirichlet wall (a cage)

We now envision a Faraday cage (figure 2). We consider two rectangular domains Ω⁺ = (0, d⁺) × (0, ℓ) and Ω⁻ = ( − d⁻, 0) × (0, ℓ), with d^±, ℓ > 0, that share the common interface $\mathrm{\Gamma }=\{(0,x_2)\in {\mathbb{R}}^2,x_2\in (0,\ell )\}.$ We denote by $\mathrm{\Omega }={\mathrm{\Omega }}^{+}\cup {\mathrm{\Omega }}^{-}\cup \mathrm{\Gamma }$ and by Γ^± the lateral boundaries of Ω, i.e.

$${\mathrm{\Gamma }}^{\pm }=\{(\pm d^{\pm },x_2)\in {\mathbb{R}}^2,x_2\in (0,\ell )\}.$$

A large positive integer N > 0 is the number of wires in (0, ℓ), whence $N\sim \frac{\ell }{\epsilon }$, and the analysis holds far from the lateral boundaries at x₂ = 0, 1. We puncture the domain Ω along the interface Γ by subtracting the set ${\mathcal{L}}^{\epsilon }={\cup }_{1\le i\le N}\epsilon \{\overline{\hat{\mathsf{Y}}}+\text{i}{\mathbf{\text{e}}}_2\}.$ Finally, our domain is the open set ${\mathrm{\Omega }}^{\epsilon }=\mathrm{\Omega }\setminus {\mathcal{L}}^{\epsilon }.$ On that domain, we consider the Helmholtz equation written in non-dimensional form with K = (kL)²: find p^ε as the solution to

$$\begin{array}{rl}\hspace{0.17em}& \mathrm{\Delta }{p}^{\epsilon }+K{p}^{\epsilon }=S\mathrm{in}{\mathrm{\Omega }}^{\epsilon },\\ \text{and}& {\mathrm{\partial }}_n{p}^{\epsilon }=i\sqrt{K}{p}^{\epsilon }\mathrm{on}{\mathrm{\Gamma }}^{+},{\mathrm{p}}^{\epsilon }=0\mathrm{on}\mathrm{\partial }{\mathrm{\Omega }}^{\epsilon }\setminus {\mathrm{\Gamma }}^{+},\end{array}\}$$ 3.1

where S is a source term compactly supported in Ω⁺ (figure 2). Providing S is regular enough, problem (3.1) is well-posed.

Figure 2.

A linear array on top of a Dirichlet wall Γ⁻. The actual problem (3.1) on p^ε and, as a result of asymptotic analysis, the limit problem (3.2) on p* and of the unified problem (3.3) on p^ap,ε. (Online version in colour.)

(a). The question put to the unified problem

As ε → 0, problem (3.1) ‘tends’ to the following limit problem: find p* solution to

$$\{\begin{array}{l}\mathrm{\Delta }{p}^{*}+K{p}^{*}=S\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ {\mathrm{\partial }}_n{p}^{*}=i\sqrt{K}{p}^{*}\mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},{\mathrm{p}}^{*}=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{+}\setminus {\mathrm{\Gamma }}^{+},\end{array}\{\begin{array}{ll}\mathrm{\Delta }{p}^{*}+K{p}^{*}=0& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p^{*}=0& \mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-},\end{array}$$ 3.2

and it turns out that the limit problem uncouples the behaviour of p* above and below Γ. (The limit problem is the zero-order problem, p* = p₀, involving the boundary condition (2.1).) As previously said, problem (3.2, right), which is posed in Ω⁻, may be ill-posed while the initial problem is not. Indeed, there exists a sequence of values $(K_n^{*}{)}_{n\in \mathbb{N}}$ where problem (3.2, right) admits a non-trivial kernel of finite dimension. These values are known as the eigenvalues of the Laplace operator in Ω⁻ with Dirichlet boundary conditions which are the perfect resonances of the closed cavity Ω⁻. This ill-posed limit problem reflects the presence of the quasi-resonances for the actual problem (3.1) having a small imaginary part due to radiative damping and a real part close to K*_n.

Away from these particular frequencies, we expect that the unified up-to-first-order problem (2.5) can be used. Specifically, p^ε can be approximated (with an error of order ε²) by solving the following problem: find p^ap,ε the solution to

$$\{\begin{array}{l}\begin{array}{l}\mathrm{\Delta }{p}^{\text{ap},\epsilon }+K{p}^{\text{ap},\epsilon }=S\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \frac{\mathrm{\partial }{p}^{\text{ap},\epsilon }}{\mathrm{\partial }{x}_1}=i\sqrt{K}{p}^{\text{ap},\epsilon }\mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},\\ p^{\text{ap},\epsilon }=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{+}\setminus (\mathrm{\Gamma }\cup {\mathrm{\Gamma }}^{+}),\end{array}\begin{array}{l}\mathrm{\Delta }{p}^{\text{ap},\epsilon }+K{p}^{\text{ap},\epsilon }=S\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p^{\text{ap},\epsilon }=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },\end{array}\\ p_{|0^{+}}^{\text{ap},\epsilon }=\epsilon \mathcal{B}{\frac{\mathrm{\partial }{p}^{\text{ap},\epsilon }}{\mathrm{\partial }{x}_1}}_{|0^{+}}-\epsilon \mathcal{A}{\frac{\mathrm{\partial }{p}^{\text{ap},\epsilon }}{\mathrm{\partial }{x}_1}}_{|0^{-}},p_{|0^{-}}^{\text{ap},\epsilon }=\epsilon \mathcal{A}{\frac{\mathrm{\partial }{p}^{\text{ap},\epsilon }}{\mathrm{\partial }{x}_1}}_{|0^{+}}-\epsilon \mathcal{B}{\frac{\mathrm{\partial }{p}^{\text{ap},\epsilon }}{\mathrm{\partial }{x}_1}}_{|0^{-}}\mathrm{on\hspace{0.17em}}\mathrm{\Gamma }.\end{array}$$ 3.3

We emphasize that, unlike the limit problem (3.2), problem (3.3) turns out to be well-posed for any dimensionless frequency K. However, this does not presume that it approximates p^ε when K is close to an eigenvalue K*_n of the limit problem (3.2, right). This is the question addressed in the rest of this section. To do so, we consider K*_n to be a simple eigenfrequency of the cavity problem (3.2, right) and the resonance frequency K_n of the actual cavity is close to that eigenfrequency. Being ‘far from’ or ‘close to’ the resonance can be measured by the relative amplitudes of the fields p^ε inside and outside the cavity. Outside the cavity, p^ε = O(1) imposed by the source, and we define three cases, as follows.

• —The off-resonance case (1). It corresponds to |K − K*_n| = O(1), hence far from the perfect resonance, and we shall see that p^ε = O(ε) is small within the cavity Ω⁻. This is the case where the validity of (3.3) is not really questioned.
• —
  The near-resonance case (2i) and on-resonance case (2ii). These cases correspond to K close to K*_n with

  $$K(\epsilon )=K_n^{*}+\epsilon {\kappa }_1+{\epsilon }^2{\kappa }_2$$ 3.4

  (figure 3). For an arbitrary κ₁, p^ε = O(1) becomes significant within the cavity Ω⁻; it is termed the near-resonance (2i). Arbitrary κ₁ means κ₁ ≠ κ, κ being a specific value which makes (K_n* + εκ) close to the actual resonance K_n; see (3.9). This on-resonance case (2ii) is characterized by large amplitudes p^ε = O(1/ε) inside the cavity. In this close vicinity of the actual resonance |K − K_n* − εκ| = O(ε²), the parameter κ₂ can be used to get the resonance curve as ${\alpha }_2({\kappa }_2)=\epsilon \sqrt{{\int }_{{\mathrm{\Omega }}^{-}}|p^{\epsilon }{|}^2}$.

Figure 3.

Resonance curve by means of max| p^ε| in the cavity Ω⁻ close to the resonance K_n (solid red line). The dashed grey line shows the off-resonance solution blowing up at resonance K_n* of the close cavity (its validity is limited to p^ε = O(ε)). The dashed green line shows the near-resonance solution blowing up at (K_n* + εκ) close to the actual resonance K_n (its validity is limited to p^ε = O(1)). The inset show the parametrization of the resonance curve in the on-resonance regime whose validity is limited to p^ε = O(1/ε). (Online version in colour.)

The performance of problem (3.3) to approximate p^ε in cases (1), (2i) and (2ii) is demonstrated below. The demonstration relies on the construction of asymptotic expansions of p^ε and p^ap,ε and we shall see that the leading orders of the two expansions coincide in the three cases. This was expected, by construction, in case (1) but it may appear surprising in the resonant cases (2).

Remark 3.1. —

We restrict ourselves to the case where K*_n is a simple eigenfrequency of the cavity problem (3.2, right). The result holds for multiple eigenfrequencies but requires more involved calculations.

(b). Asymptotic behaviour of the exact solution

The determination of the asymptotic behaviour of p^ε is based on the method of matched asymptotic expansions, which permits the boundary layer effect occurring in the vicinity of the wires to be captured. The main idea is to distinguish between two ‘far-field regions’ located far from Γ and a near-field area located in its vicinity, the two types of expansions matching in some intermediate area. In the present case, we can construct asymptotic expansions of the form

$$p^{\epsilon }=\{\begin{array}{ll}\sum _{m\ge m_0}{\epsilon }^m{p}_m(x_1,x_2)& \mathrm{far\; from}\hspace{0.17em}\mathrm{\Gamma }\mathit{\text{(far-field expansion)}},\\ \sum _{m\ge m_0}{\epsilon }^m{q}_m(\frac{x_1}{\epsilon },\frac{x_2}{\epsilon },x_2)& \mathrm{in\; the\; vicinity\; of\hspace{0.17em}}\mathrm{\Gamma }\mathit{\text{(near-field expansion), }}\end{array}$$ 3.5

where the integer m₀ will be equal to −1 or 0. Since we are interested in the macroscopic behaviour of p^ε, we present the results on the leading order in the far-field expansion; the step-by-step construction of the asymptotic expansion is technical and is given in appendix A.

(i). The off-resonance case (1)

The off-resonance case consists in taking a fixed K ≠ K*_n (independent of ε) that is not an eigenvalue of the cavity problem (3.2, left).

Lemma 3.2. —

Assume that K is not an eigenvalue of the cavity problem (3.2, left). Then, far from Γ, p^ε admits the following leading-order asymptotic:

$$p^{\epsilon }\sim \{\begin{array}{ll}{P}^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \epsilon \mathcal{A}{p}^{-}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\end{array}$$ 3.6

where P* is the unique solution to (3.2, left) and p⁻ is the unique solution to

$$\begin{array}{rl}\hspace{0.17em}& \mathrm{\Delta }{p}^{-}+K{p}^{-}=0\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-}\\ \text{and}& p^{-}=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },{\mathrm{p}}^{-}=\frac{\mathrm{\partial }{\mathrm{P}}^{*}}{\mathrm{\partial }{\mathrm{x}}_1}\mathrm{on\hspace{0.17em}}\mathrm{\Gamma }.\end{array}\}$$ 3.7

We point out that $p^{\epsilon }=\epsilon \mathcal{A}{p}^{-}$ is small in the cavity Ω⁻, which means that the array efficiently shields the domain Ω⁻.

(ii). The near-resonance case (2i): κ₁ ≠ κ

We consider K*_n to be a simple eigenfrequency of the cavity problem (3.2, right), which is for instance the case for the first eigenvalue in two dimensions. We denote by p_n* a corresponding real-valued eigenvector such that

$${\int }_{{\mathrm{\Omega }}^{-}}(p_n^{*}{)}^2\text{d}\mathit{x}=1.$$ 3.8

Remark 3.3. —

Obviously the orthogonality condition (3.8) does not entirely define p_n* among all the normalized eigenvectors associated with the eigenvalue K*_n. Indeed if p_n* fulfils (3.8), so does −p_n*. Naturally the arbitrary choice of p_n* does not change the results below.

Then, we take K(ε) of the form (3.4), which means that we consider frequencies located in an ε-neighbourhood of K*_n. At that stage, we introduce the real number κ

$$\kappa =-\mathcal{B}\hspace{0.17em}{\int }_{\mathrm{\Gamma }}{(\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2))}^2\mathrm{d}{x}_2\hspace{0.17em},$$ 3.9

with $\mathcal{B}$ the constant appearing in (3.3). We shall make the distinction between the two sub-cases (2i) κ₁ ≠ κ and (2ii) κ₁ = κ, and we start here with κ₁ ≠ κ.

Lemma 3.4. —

Assume that K is of the form (3.4) and that κ₁ ≠ κ. Then, far from Γ, p^ε admits the following behaviour:

$$p^{\epsilon }\sim \{\begin{array}{ll}{P}_n^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ {\alpha }_1{p}_n^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\end{array}\mathrm{with}{\alpha }_1=\frac{\mathcal{A}\mathrm{I}}{{\kappa }_1-\kappa },$$ 3.10

where P*_n is the unique solution to (3.2, left) for K = K_n*, $\mathcal{A}$ is the constant appearing in (3.3) and

$$I={\int }_{\mathrm{\Gamma }}\frac{\mathrm{\partial }{P}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)\mathrm{d}{x}_2\hspace{0.17em}.$$ 3.11

By contrast to the non-resonant case, p^ε = α₁p_n* is of order unity inside the cavity. There is no shielding effect as the fields are as intense inside and outside the cavity.

(iii). The resonant case (2ii): κ₁ = κ

Lemma 3.5. —

Assume that K is of the form (3.4) and that κ₁ = κ. Then p^ε has the following behaviour:

$$p^{\epsilon }\sim \{\begin{array}{ll}{P}_n^{*}+{\alpha }_2\mathcal{A}{P}^{+}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \frac{{\alpha }_2}{\epsilon }\hspace{0.17em}{p}_n^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\end{array}\mathrm{with}\hspace{0.17em}{\alpha }_2=\frac{\mathcal{A}\mathrm{I}}{{\kappa }_2-{\mathcal{A}}^2{\mathrm{I}}^{+}+{\mathcal{B}}^2{\mathrm{I}}^{-}},$$ 3.12

where I is defined in (3.11), I^± are defined by

$$I^{+}={\int }_{\mathrm{\Gamma }}\frac{\mathrm{\partial }{P}^{+}}{\mathrm{\partial }{x}_1}(0,x_2)\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)\mathrm{d}{x}_2\hspace{0.17em},I^{-}={\int }_{\mathrm{\Gamma }}\frac{\mathrm{\partial }{P}^{-}}{\mathrm{\partial }{x}_1}(0,x_2)\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)\hspace{0.17em}\mathrm{d}{x}_2\hspace{0.17em}$$ 3.13

and P⁺ and P⁻ are the (unique) solutions to

$$\{\begin{array}{ll}\mathrm{\Delta }{P}^{+}+K_n^{*}{P}^{+}=0& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \frac{\mathrm{\partial }{P}^{+}}{\mathrm{\partial }{x}_1}=i\sqrt{K_n^{*}}{P}^{+}& \mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},\\ P^{+}=-\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}& \mathrm{on\hspace{0.17em}}\mathrm{\Gamma },\\ P^{+}=0& \mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{+}\setminus ({\mathrm{\Gamma }}^{+}\cup \mathrm{\Gamma }),\end{array}\{\begin{array}{ll}\mathrm{\Delta }{P}^{-}+K_n^{*}{P}^{-}=-\frac{\kappa }{\mathcal{B}}\hspace{0.17em}{p}_n^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ P^{-}=0& \mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },\\ P^{-}=-\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}& \mathrm{on\hspace{0.17em}}\mathrm{\Gamma },\\ {\int }_{{\mathrm{\Omega }}^{-}}{P}^{-}{p}_n^{*}\hspace{0.17em}\mathrm{d}\mathit{x}\hspace{0.17em}=0.& \hspace{0.17em}\end{array}$$ 3.14

In the close vicinity of a resonance, the field p^ε = O(1/ε) in Ω⁻ becomes more intense within the cavity than outside and it would blow up as ε vanishes. Indeed, in the limit of zero ε, p^ε tends to p* and we recover a perfect resonance with unbounded amplitude inside the cavity.

(c). Asymptotic behaviours of the homogenized solution

The obtention of the asymptotic behaviour of the solution p^ap,ε to (3.3) also relies on the construction of its asymptotic expansion. However, this expansion is much simpler than that of p^ε. Indeed, the approximate problem does not see the wires and we have a classical expansion of the form

$$p^{\text{ap},\epsilon }=\sum _{m\ge m_0}{\epsilon }^m{p}_m^{\text{ap}}(x_1,x_2)\mathrm{in\hspace{0.17em}}\hspace{0.17em}{\mathrm{\Omega }}^{+}\cup {\mathrm{\Omega }}^{-},$$ 3.15

where m₀ is equal to −1 or 0. Below we show that we recover the off-resonance (3.6) and the two different resonant cases, (3.10) and (3.12), corresponding to (3.4).

(i). The off-resonance case

In that case, as for the exact problem, the formal expansion (3.15) is plugged into (3.3), and we find m₀ = 0 ($p_{-1}^{\text{ap}}=0$ in Ω⁺ and Ω⁻). Next, $p_0^{\text{ap}}$ is a solution to

$$\{\begin{array}{ll}\mathrm{\Delta }{p}_0^{\text{ap}}+K{p}_0^{\text{ap}}=S& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \frac{\mathrm{\partial }{p}_0^{\text{ap}}}{\mathrm{\partial }{x}_1}=i\sqrt{K}\hspace{0.17em}{p}_0^{\text{ap}}& \mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},{\mathrm{p}}_0^{\text{ap}}=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{+}\setminus {\mathrm{\Gamma }}^{+},\end{array}\{\begin{array}{ll}\mathrm{\Delta }{p}_0^{\text{ap}}+K{p}_0^{\text{ap}}=0& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_0^{\text{ap}}=0& \mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-},\end{array}$$ 3.16

and as K ≠ K_n* the solution reads

$$p_0^{\text{ap}}=\{\begin{array}{ll}{P}^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ 0& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-}.\end{array}$$

In Ω⁻, the first non-zero contribution appears at the order 1, with $p_1^{\text{ap}}$ a solution to

$$\{\begin{array}{l}\mathrm{\Delta }{p}_1^{\text{ap}}+K{p}_1^{\text{ap}}=0\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_1^{\text{ap}}=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },{\mathrm{p}}_1^{\text{ap}}=\mathcal{A}\hspace{0.17em}\frac{\mathrm{\partial }{\mathrm{P}}^{*}}{\mathrm{\partial }{\mathrm{x}}_1}\mathrm{on\hspace{0.17em}}\mathrm{\Gamma }.\end{array}$$

Remembering that we have denoted p⁻ as the unique solution to (3.7), we obtain that $p_1^{\text{ap}}=\mathcal{A}\hspace{0.17em}{p}^{-}$. Eventually, we get for $p^{\text{ap},\epsilon }=p_0^{\text{ap}}+\epsilon p_1^{\text{ap}}+O({\epsilon }^2)$ the leading-order asymptotic

$$p^{\text{ap},\epsilon }\sim \{\begin{array}{ll}{P}^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \epsilon \mathcal{A}\hspace{0.17em}{p}^{-}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\end{array}$$

which coincides with (3.6).

(ii). The near-resonance case κ₁ ≠ κ

In that case, inserting the formal expansion into (3.3) and separating the different powers of ε, we find again that m₀ = 0. But now, $p_0^{\text{ap}}$ is a solution to

$$\{\begin{array}{ll}\mathrm{\Delta }{p}_0^{\text{ap}}+K_n^{*}{p}_0^{\text{ap}}=S& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \frac{\mathrm{\partial }{p}_0^{\text{ap}}}{\mathrm{\partial }{x}_1}=i\sqrt{K_n^{*}}\hspace{0.17em}{p}_0^{\text{ap}}& \mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},{\mathrm{p}}_0^{\text{ap}}=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{+}\setminus {\mathrm{\Gamma }}^{+},\end{array}\{\begin{array}{ll}\mathrm{\Delta }{p}_0^{\text{ap}}+K_n^{*}{p}_0^{\text{ap}}=0& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_0^{\text{ap}}=0& \mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}.\end{array}$$ 3.17

The problem (3.17, left) is identical to (3.16, left) for K = K_n* and we have denoted P*_n as its unique solution. By contrast, the problem (3.17, right) is ill-posed at the eigenfrequency K_n* of the close cavity. However, as p_n* is the eigenvector associated with K_n*, we can write

$$p_0^{\text{ap}}=\{\begin{array}{ll}{P}_n^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ {\alpha }_1^{\text{ap}}\hspace{0.17em}{p}_n^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\end{array}$$

but ${\alpha }_1^{\text{ap}}$ is unknown. (This means that the problem at the dominant first order cannot be solved; it corresponds to the limit problem (3.2) at an eigenvalue of the Laplace operator in Ω⁻.) The value of ${\alpha }_1^{\text{ap}}$ is obtained from the first-order problem in Ω⁻; namely, find $p_1^{\text{ap}}$ as a solution to

$$\{\begin{array}{ll}\mathrm{\Delta }{p}_1^{\text{ap}}+K_n^{*}\hspace{0.17em}{p}_1^{\text{ap}}=-{\kappa }_1\hspace{0.17em}{\alpha }_1^{\text{ap}}{p}_n^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_1^{\text{ap}}=\mathcal{A}\frac{\mathrm{\partial }{P}_n^{*}}{\mathrm{\partial }{x}_1}-\mathcal{B}\hspace{0.17em}{\alpha }_1^{\text{ap}}\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}& \mathrm{on\hspace{0.17em}}\mathrm{\Gamma },{\mathrm{p}}_1^{\text{ap}}=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },\end{array}$$

which is solvable if and only if

$${\alpha }_1^{\text{ap}}=\frac{\mathcal{A}}{{\kappa }_1-\kappa }{\int }_{\mathrm{\Gamma }}\frac{\mathrm{\partial }{P}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)\mathrm{d}{x}_2\hspace{0.17em}$$

(making use of (3.8) and (3.9)) and we exactly recover (3.10) (with ${\alpha }_1^{\text{ap}}={\alpha }_1$).

(iii). The on-resonance case κ₁ = κ

Inserting the formal expansion (3.15) into (3.3), we find m₀ = −1 with $p_{-1}^{\text{ap}}$ the solution to

$$\{\begin{array}{ll}\mathrm{\Delta }{p}_{-1}^{\text{ap}}+K_n^{*}{p}_{-1}^{\text{ap}}=0& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \frac{\mathrm{\partial }{p}_{-1}^{\text{ap}}}{\mathrm{\partial }{x}_1}=i\sqrt{K_n^{*}}\hspace{0.17em}{p}_{-1}^{\text{ap}}& \mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},{\mathrm{p}}_{-1}^{\text{ap}}=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{+}\setminus {\mathrm{\Gamma }}^{+},\end{array}\{\begin{array}{ll}\mathrm{\Delta }{p}_{-1}^{\text{ap}}+K_n^{*}{p}_{-1}^{\text{ap}}=0& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_{-1}^{\text{ap}}=0& \mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}.\end{array}$$

The term $p_{-1}^{\text{ap}}$ plays basically the same role as $p_0^{\text{ap}}$ in the near-resonance case. It is the solution to (3.16, left) in Ω⁺ but without the source S; hence, $p_{-1}^{\text{ap}}=0$ in Ω⁺. It is a solution to the ill-posed problem (3.17, right) in Ω⁻; hence, it is equal to p_n* up to an unknown constant ${\alpha }_2^{\text{ap}}$ in Ω⁻, namely

$$p_{-1}^{\text{ap}}=\{\begin{array}{ll}0& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ {\alpha }_2^{\text{ap}}\hspace{0.17em}{p}_n^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-}.\end{array}$$

As in the near-resonant case, ${\alpha }_2^{\text{ap}}$ is obtained from the problem at the next order; in the present case, the zero order. At the zero order, $p_0^{\text{ap}}$ is a solution to

$$\{\begin{array}{l}\mathrm{\Delta }{p}_0^{\text{ap}}+K_n^{*}\hspace{0.17em}{p}_0^{\text{ap}}=S\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \frac{\mathrm{\partial }{p}_0^{\text{ap}}}{\mathrm{\partial }{x}_1}=i\sqrt{K_n^{*}}{p}_0^{\text{ap}}\mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},\\ p_0^{\text{ap}}=0\text{on }\mathrm{\partial }{\mathrm{\Omega }}^{+}\setminus ({\mathrm{\Gamma }}^{+}\cup \mathrm{\Gamma }),\\ p_0^{\text{ap}}=-\mathcal{A}\hspace{0.17em}{\alpha }_2^{\text{ap}}\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}\mathrm{on\hspace{0.17em}}\mathrm{\Gamma },\end{array}\{\begin{array}{l}\mathrm{\Delta }{p}_0^{\text{ap}}+K_n^{*}{p}_0^{\text{ap}}=-\kappa \hspace{0.17em}{\alpha }_2^{\text{ap}}{p}_n^{*}\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_0^{\text{ap}}=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },\\ p_0^{\text{ap}}=-\mathcal{B}\hspace{0.17em}{\alpha }_2^{\text{ap}}\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}\mathrm{on\hspace{0.17em}}\mathrm{\Gamma }.\end{array}$$

As P⁺ is the solution to (3.14, left) and P*_n is the solution to (3.2, left) for K = K_n*, we recover that

$$p_0^{\text{ap}}={\alpha }_2^{\text{ap}}\mathcal{A}{P}^{+}+P_n^{*}\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},$$

as in (3.12), but ${\alpha }_2^{\text{ap}}$ has to be determined. In addition, the problem in Ω⁻ is solvable by construction (with κ₁ = κ) and with P⁻ as the solution to (3.14, right) we have

$$p_0^{\text{ap}}={\alpha }_2^{\text{ap}}\mathcal{B}{P}^{-}+{\beta }^{\text{ap}}{p}_n^{*}\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},$$

where β^ap is unknown at this order (but we do not need to determine it). In the same way we determine ${\alpha }_1^{\text{ap}}$ in the near-resonance case, we determine the value of ${\alpha }_2^{\text{ap}}$ using the solvability condition for $p_1^{\text{ap}}$ in Ω⁻, the solution to

$$\{\begin{array}{l}\mathrm{\Delta }{p}_1^{\text{ap}}+K_n^{*}{p}_1^{\text{ap}}=-{\kappa }_1({\alpha }_2^{\text{ap}}\mathcal{B}{P}^{-}+{\beta }_{*}^{\text{ap}}{p}_n^{*})-{\kappa }_2{\alpha }_2^{\text{ap}}{p}_n^{*}\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_1^{\text{ap}}=0\mathrm{on\hspace{0.17em}}{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },{\mathrm{p}}_1^{\text{ap}}=\mathcal{A}({\alpha }_1^{\text{ap}}\mathcal{A}\frac{\mathrm{\partial }{\mathrm{P}}^{+}}{\mathrm{\partial }{\mathrm{x}}_1}+\frac{\mathrm{\partial }{\mathrm{P}}_n^{*}}{\mathrm{\partial }{\mathrm{x}}_1})-\mathcal{B}({\alpha }_2^{\text{ap}}\frac{\mathrm{\partial }{\mathrm{P}}^{-}}{\mathrm{\partial }{\mathrm{x}}_1}+{\beta }_{*}^{\text{ap}}\frac{\mathrm{\partial }{\mathrm{p}}_n^{*}}{\mathrm{\partial }{\mathrm{x}}_1})\mathrm{on\hspace{0.17em}}\mathrm{\Gamma },\end{array}$$

and we get that ${\displaystyle {\alpha }_2^{\text{ap}}=\mathcal{A}I/({\kappa }_2-{\mathcal{A}}^2{I}^{+}+{\mathcal{B}}^2{I}^{-})}$; hence, we entirely recover (3.12) (${\alpha }_2^{\text{ap}}={\alpha }_2$).

4. A one-dimensional cage in the harmonic regime

To begin with, we consider a one-dimensional cage, which means a cage laterally unbounded along x₂. In the harmonic regime, the time dependence is e^−iωt and the scattering problem is set for an incident wave with wavenumber $\mathbf{\text{k}}=k({\text{c}}_{\theta },{\text{s}}_{\theta })$, ${\text{c}}_{\theta }^2+{\text{s}}_{\theta }^2=1$ and k = ω/c (we use c = 1). The wave comes from x₁ = +∞ on the array, hence pseudo-periodic conditions along x₂ apply. The array is made of square wires e = e₁ = e₂ and spacing h = 1 (with arbitrary unit) at distance D = 7 of a Dirichlet wall Γ⁻. We call p the solution of the actual problem computed numerically using a modal method described in [10]; we call p^ap (omitting ε) the solution of the homogenized problem.

(a). Validity of the unified model, comparison with the iterative model

(i). Solution of the unified model

The solution of the scattering problem is explicit using (1.2) along with (2.5).¹ It reads $p^{\text{ap}}(\mathbf{\text{x}})=p^{\text{ap}}(x_1)\hspace{0.17em}{\text{e}}^{\text{i}k{\text{s}}_{\theta }{x}_2}$ with

$$p^{\text{ap}}(x_1)=\{\begin{array}{ll}{\text{e}}^{-\text{i}k{\text{c}}_{\theta }{x}_1}+{\mathcal{R}}^{\text{ap}}\hspace{0.17em}{\text{e}}^{\text{i}k{\text{c}}_{\theta }{x}_1}& \mathrm{for}{\mathrm{x}}_1\in (0,+\mathrm{\infty }),\\ {\mathcal{P}}^{\text{ap}}\mathrm{sin\hspace{0.17em}}(k{\text{c}}_{\theta }(x_1+D))& \mathrm{for}{\mathrm{x}}_1\in (-\mathrm{D},0),\end{array}$$ 4.1

and

$$\{\begin{array}{l}{\mathcal{R}}^{\text{ap}}=-\frac{\overline{z}}{z},{\mathcal{P}}^{\text{ap}}=-\frac{2\text{i}k{\text{c}}_{\theta }h\mathcal{A}}{z}\\ \text{and }z=(1-\text{i}k{\text{c}}_{\theta }h\mathcal{B})\mathrm{sin\hspace{0.17em}}(k{\text{c}}_{\theta }D)+k{\text{c}}_{\theta }h(\mathcal{B}+\text{i}k{\text{c}}_{\theta }h({\mathcal{A}}^2-{\mathcal{B}}^2))\cos (k{\text{c}}_{\theta }D).\end{array}$$ 4.2

Expectedly, $|{\mathcal{R}}^{\text{ap}}|=1$ and ${\mathcal{P}}^{\text{ap}}$ is bounded. In addition, defining the resonance as ${\mathcal{R}}^{\text{ap}}=1$ provides the resonance frequency k_n and the corresponding amplitude ${\mathcal{P}}_{\text{max}}^{\text{ap}}$ by means of the condition that z is purely imaginary

$$\tan (k_n{\text{c}}_{\theta }D)+k_n{\text{c}}_{\theta }h\mathcal{B}=0,{\mathcal{P}}_{\text{max}}^{\text{ap}}=\frac{2}{\mathcal{A}{k}_n{\text{c}}_{\theta }h\cos (k_n{\text{c}}_{\theta }D)},$$ 4.3

hence $k_n{\text{c}}_{\theta }D\sim n\pi (1-(h\mathcal{B}/D)+(h\mathcal{B}/D{)}^2)$. (We have checked that defining the resonances as the maximum amplitudes within the cavity provides almost the same results for $(k_n,{\mathcal{P}}_{\text{max}}^{\text{ap}})$.)

To begin with, we report in figure 4 the fields p(x) and p^ap(x) for three frequencies corresponding to the cases (1), (2i) and (2ii) studied in §3. Off-resonance, the incident wave is basically reflected as on a Dirichlet wall with ${\mathcal{R}}^{\text{ap}}\simeq -1$ resulting in a weak amplitude within the cavity. By contrast, on-resonance ${\mathcal{R}}^{\text{ap}}\simeq 1$ and the amplitude is large within the cavity. In between, near-resonance, the fields inside and outside the cavity are comparable in amplitude. It is noticeable that the agreement between p(x) and p^ap(x) from (4.1) with (4.2) is qualitatively very good.

Figure 4.

Wavefields in a one-dimensional cage—p(x) computed numerically (a) and p^ap(x) from (4.1) and (4.2) (b). The source is a plane wave of amplitude unity at an oblique incidence of 45^° on a periodic array of square wires with spacing h = 1 and with e = e₁ = e₂ = 0.1 ( x₂ ∈ (0, 10)); the Dirichlet wall Γ is at x₂ = −7. The amplitude within the cavity is O(ε) off-resonance (k = 0.700), O(1) near-resonance (k = 0.640) and O(1/ε) on-resonance (k = 0.633). (Online version in colour.)

The validity of (4.3) is further illustrated in figures 5 and 6. We have computed the maximum amplitude max|p| within the cavity against k ∈ (0, 2) for e = 0.01, 0.1 and 0.2 (solid blue lines in figure 5). Next, max| p^ap|, reported by the dashed black lines, is determined from (4.1) to (4.2) (the values of $(\mathcal{A},\mathcal{B})$ for e = 0.01, 0.1 and 0.2 are given in (B 1) in appendix B). Notably, for kc_θ D > π/2, we have max$|p^{\text{ap}}|={\mathcal{P}}^{\text{ap}}$. Classical trends of the resonance curves are observed. As e increases, the leakage of the cage decreases, resulting in resonances with higher quality factors, that is, thinner in frequency and higher in amplitude. These resonances take place close to k_n*c_θ D = nπ, in the reported case k₁* = 0.635, k₂* = 1.270 and k₃* = 1.904. The prediction of the unified problem appears to be excellent up to kh = 2, that is, beyond the expected range of validity of the homogenization (with kh = O(ε) assumed to be small), although a slight discrepancy is visible above kh ∼ 1 for e = 0.1 and 0.2.

Figure 5.

Maximum pressure in the cavity against the dimensionless frequency kh, max|p| from direct numerics (solid blue lines), max| p^ap,ε| from (4.1) and (4.2) (dashed black lines). The panels in (b) show a zoom of the panels in (a). (Online version in colour.)

Figure 6.

Variations of the resonance frequency k₁ and maximum amplitude ${\mathcal{P}}_{\text{max}}$ at the resonance against h. Solid lines, computed from direct numerics; dashed black lines, from (4.1). (Online version in colour.)

From figure 5, we have determined the resonance frequency k₁ and the corresponding amplitude at the resonance ${\mathcal{P}}_{\text{max}}$ against h ∈ (0, 3). The results shown in figure 6 demonstrate that the one-dimensional cage has resonances behaving as for its two-dimensional analogue [7] and conform to (4.3). As the effective problem tends to the limit problem for vanishing h, decreasing h makes k₁ closer to k₁*; next k₁ departs from k₁* essentially linearly with h. Amusingly, as in the two-dimensional cage, e ≃ 0.1 h is a critical value; below k₁ decreases with h, above it increases. From (4.3), k_n = k_n* is obtained for $\mathcal{B}=0$, and $\mathcal{B}=0$ for e ≃ 0.115 h. In any case, ${\mathcal{P}}_{\text{max}}\sim 1/h$ decreases with h as the leakage increases.

(ii). Comparison with the iterative model

The iterative model consists of three predictions: off-resonance, near-resonance and on-resonance. We have shown in §3 that the asymptotics of the unified model coincide with the iterative solution in the three regimes. However, this does not presume to know how the two solutions compare for a given h. As previously mentioned, the problem is one-dimensional along x₁, which makes explicit the various functions needed to define the solutions of the iterative model.² We give below the resulting expressions in dimensional coordinates with $p^{\epsilon }(x_1,x_2)=p^{\epsilon }(x_1)\hspace{0.17em}{\text{e}}^{\text{i}k{\text{s}}_{\theta }{x}_2}$.

• —
  Off-resonance case. In this case, $p^{\epsilon }\sim \epsilon \mathcal{A}{p}^{-}$ for x₁ < 0 (see (3.6)); hence, using², p^ε reads

  $$p^{\epsilon }(x_1)\sim \sum _{m=1}^{+\mathrm{\infty }}\frac{-4\text{i}(\mathcal{A}h/D)\hspace{0.17em}(k{\text{c}}_{\theta }D/m\pi {)}^3\mathrm{sin\hspace{0.17em}}(m\pi x_1/D)}{(k{\text{c}}_{\theta }D/m\pi {)}^2-1}-2\text{i}\frac{\mathcal{A}h}{D}(k{\text{c}}_{\theta }D)(1+\frac{x_1}{D}).$$ 4.4

  Expectedly, the amplitude within the cavity blows up at all the perfect resonances of the close cavity kc_θ D = mπ. (The solution is valid far from these resonances.)

Remark 4.1. —

The solution (4.4) is the Taylor expansion of p^ap(x₁) in (4.1) at first order in (k h); hence, z ≃ sin (kc_θ D) in (4.2).³

• —
  Near-resonance case. From (3.10) p^ε ∼ α₁p_n* for x₁ < 0 with ${\alpha }_1=\mathcal{A}I/({\kappa }_1-\kappa )$ with $I=-2\sqrt{2}\text{i}(n\pi {)}^2$, and from (3.9) $\kappa =-2\mathcal{B}(n\pi {)}^2$ and κ₁ = (K − K_n*)/ε = (D/h)((kc_θ D)² − (nπ)²); see². Hence, we have

  $$p^{\epsilon }(x_1)\sim \frac{{\displaystyle -4\text{i}\hspace{0.17em}(\mathcal{A}h/D)\mathrm{sin\hspace{0.17em}}(n\pi x_1/D)}}{{\displaystyle (k{\text{c}}_{\theta }D/n\pi {)}^2-(1-2(\mathcal{B}h/D))}}.$$ 4.5

  Again, this amplitude blows up, but now at a value closer to the actual resonance frequency k_n, namely for $k{\text{c}}_{\theta }D\simeq n\pi (1-\mathcal{B}(h/D))$ corresponding to the first-order expansion of the dispersion relation in (4.3). (The solution is valid far from this resonance.)

Remark 4.2. —

The solution (4.5) coincides with the Taylor expansion of p^ap(x₁) in (4.1) up to O(ε²), around the resonance of the close cavity, namely for $k{\text{c}}_{\theta }D=n\pi (1+\epsilon \hat{k})$.⁴

• —
  On-resonance case. Here, (3.12) applies; hence, p^ε ∼ (α₂/ε)p_n* for x₁ < 0 with ${\alpha }_2=\mathcal{A}I/({\kappa }_2-{\mathcal{A}}^2{I}^{+}+{\mathcal{B}}^2{I}^{-})$. Next, from (3.13) along with ², I⁺ = −2i(nπ)³ and I⁻ = −3(nπ)² (using⁵) and by construction ${\kappa }_2=(n\pi D/h{)}^2[(k{\text{c}}_{\theta }D/n\pi {)}^2-1+2(\mathcal{B}h/D)]$. It follows that

  $$p^{\epsilon }(x_1)\sim \frac{{\displaystyle -4\text{i}\hspace{0.17em}(\mathcal{A}h/D)\mathrm{sin\hspace{0.17em}}((n\pi x_1)/D)}}{{\displaystyle (k{\text{c}}_{\theta }D/n\pi {)}^2-(1-2(\mathcal{B}h/D)+3(\mathcal{B}h/D{)}^2)+2\text{i}n\pi (\mathcal{A}h/D{)}^2}}.$$ 4.6

  The leakage of the cavity is recovered, hence the amplitude within the cavity is finite. This solution is valid in the close vicinity of the resonance k_n.

Remark 4.3. —

Again, the solution (4.6) coincides with the Taylor expansion of p^ap(x₁) in (4.1) up to O(1), around the resonance nπ of the closed cavity $k{\text{c}}_{\theta }h=n\pi (1-\epsilon \mathcal{B}+{\epsilon }^2\tilde{k})$.⁶ We also obtain that the maximum amplitude is reached for

$$(k_n{\text{c}}_{\theta }{)}^2={\left(\frac{n\pi }{D}\right)}^2(1-2\frac{h\mathcal{B}}{D}+3{\left(\frac{h\mathcal{B}}{D}\right)}^2),$$

which is the Taylor expansion of the dispersion relation in (4.3).

We end this comparison by reporting in figure 7 the same numerical result as in figure 5. However, now we compare the actual solution with the solutions (4.4) (dashed grey lines) valid far from the resonances, (4.5) (dotted green lines) valid near the first resonance and (4.6) (dashed red lines in the insets) valid in its close vicinity. The results conform with those reported for the two-dimensional cage in [7] and conform with the idea of piecewise-valid solutions. Compared with the unified solution, which is valid in the whole range of frequency, the validity of each prediction is locally as good; the main difficulty in their use is to define the frequency ranges where each of them can be used. Eventually, it can be seen that the result for e = 0.01 is not satisfactory essentially because the off-resonance solution departs significantly from the actual one; this case, which can be treated within the ‘thin wire approximation’, is discussed in appendix C.

Figure 7.

Same representation as in figure 5. Direct numerics for the actual problem and from the iterative model: off-resonance curve from (4.4), near-resonance curve from (4.5) and on-resonance curve from (4.6). (Online version in colour.)

5. A two-dimensional cage in the transient regime

We now move on to the transient regime and consider a two-dimensional cage, that is, a cage laterally bounded along x₂. As the fields depend on time and on space, we consider p(x) → p(x, t).

(a). Enlarged formulations of the model

To begin with, we note that the transmission conditions cannot be used as written in (2.5), that is, written across a zero-thickness interface. Introduced in [12,13], enlarged formulations of the transmission conditions consist in modifying (2.5) by simple Taylor expansions of p^ap(0^±, x₂, t) around x₁ = ±a with a > 0 resulting in (in dimensional form)

$$\begin{array}{rl}\hspace{0.17em}& p^{\text{ap}}(-a,x_2,t)=h\mathcal{A}\frac{\mathrm{\partial }{p}^{\text{ap}}}{\mathrm{\partial }{x}_1}(a,x_2,t)-(h\mathcal{B}+a)\hspace{0.17em}\frac{\mathrm{\partial }{p}^{\text{ap}}}{\mathrm{\partial }{x}_1}(-a,x_2,t)\\ \text{and}& p^{\text{ap}}(a,x_2,t)=(h\mathcal{B}+a)\frac{\mathrm{\partial }{p}^{\text{ap}}}{\mathrm{\partial }{x}_1}(a,x_2,t)-h\mathcal{A}\frac{\mathrm{\partial }{p}^{\text{ap}}}{\mathrm{\partial }{x}_1}(-a,x_2,t),\end{array}\}$$ 5.1

and we see that the new p^ap approximates p up to O(ε²) if a = O(e₁) = O(h); see (B.9) in [11]. In other words, it exists as a family of effective models parametrized by a and having the same asymptotic limit. The advantage of the models in enlarged formulations (5.1) is that they enjoy ‘good’ energetic properties as soon as a ≥ a_c and we shall see that 0 ≤ a_c ≤ e₁. For the time being, we show that it is sufficient that a ≥ e₁: let us consider the balance of energy in Ω to be a bounded region of the space containing a segment Γ and ${\mathrm{\Omega }}_a=\mathrm{\Omega }\setminus {\mathrm{\Gamma }}_a$, where Γ_a = {(x₁, x₂) ∈ ( − a, a) × Γ} is the enlarged interface ruled by (5.1). Multiplying the first equation of (1.1) by u, the second by p and summing up, we get

$$\begin{array}{rl}\hspace{0.17em}& \frac{\text{d}}{\text{d}t}({\mathcal{E}}^{\text{ap}}+{\mathcal{E}}_{\mathrm{\Gamma }})=0,{\mathcal{E}}^{\text{ap}}(t;a)=\frac{1}{2}{\int }_{{\mathrm{\Omega }}_a}(({\mathit{u}}^{\text{ap}}{)}^2+(p^{\text{ap}}{)}^2)\mathrm{d}\mathbf{\text{x}}\hspace{0.17em}\\ \text{and}& {\mathcal{E}}_{\mathrm{\Gamma }}(t;a)=\frac{h}{2}{\int }_{\mathrm{\Gamma }}((\mathcal{B}+\frac{a}{h})({u_1^{\text{ap}}}_{|a}^2+{u_1^{\text{ap}}}_{|-a}^2)-2\mathcal{A}{u_1^{\text{ap}}}_{|-a}{u_1^{\text{ap}}}_{|a})\mathrm{d}{x}_2\hspace{0.17em}.\end{array}\}$$ 5.2

(${u_1}_{|\pm a}^{\text{ap}}$ stands for $u_1^{\text{ap}}(\pm a,x_2,t)$.) ‘Good’ energetic properties means that the effective interface provides a positive energy contribution ${\mathcal{E}}_{\mathrm{\Gamma }}$, a lack of which may lead to severe instability of any time-discretization scheme; this will be illustrated in §5c.

Lemma 5.1. —

The quadratic form $(\mathcal{B}+(a/h))({\alpha }^2+{\beta }^2)-2\mathcal{A}\alpha \beta$ is positive if a ≥ e₁.

Proof. —

We set Q = αQ⁺ + βQ⁻ with α, β two reals and Q⁺, Q⁻ defined in (2.3), (2.6). Q satisfies

$$\mathrm{\Delta }Q=0\mathrm{in\hspace{0.17em}}\mathsf{Y},\mathrm{Q}=0\text{on }\mathrm{\partial }\hat{\mathsf{Y}},\text{and }\underset{{\mathrm{y}}_1\to -\mathrm{\infty }}{lim}\mathbf{\nabla }\mathrm{Q}=\beta {\mathit{e}}_1,\underset{{\mathrm{y}}_1\to +\mathrm{\infty }}{lim}\mathbf{\nabla }\mathrm{Q}=\alpha {\mathit{e}}_1.$$ 5.3

We define H(y₁) with ${\displaystyle H(y_1<-e_1/h)=\beta (y_1+(e_1/h))}$, H(|y₁| ≤ e₁/h) = 0, H(y₁ > e₁/h) = α(y₁ − (e₁/h)) and q = Q − H being continuous across any interfaces (in particular at y₁ = ±e₁/h) and satisfying

$$q\underset{y_1\to -\mathrm{\infty }}{\sim }\alpha \mathcal{A}-\beta (\mathcal{B}+\frac{e_1}{h}),q\underset{y_1\to +\mathrm{\infty }}{\sim }\alpha (\mathcal{B}+\frac{e_1}{h})-\beta \mathcal{A}.$$ 5.4

We define the restriction ${\mathsf{Y}}^{*}\subset \mathsf{Y}$ with ${\mathsf{Y}}^{*}=\{(y_1,y_2)\in (-y^{*},y^{*})\times (0,1)\setminus \overline{\hat{\mathsf{Y}}}\}$ and y* > e₁. Integrating qΔQ over ${\mathsf{Y}}^{*}$ and using (5.3) and (5.4) gives

$$0={\int }_{{\mathsf{Y}}^{*}}q\mathrm{\Delta }Q\mathrm{d}\mathit{y}\hspace{0.17em}=-{\int }_{{\mathsf{Y}}^{*}}\mathbf{\nabla }q\cdot (\mathbf{\nabla }q+\mathbf{\nabla }H)\mathrm{d}\mathit{y}\hspace{0.17em}+({\alpha }^2+{\beta }^2)(\mathcal{B}+\frac{e_1}{h})-2\alpha \beta \mathcal{A}+o(y^{*}),$$

with ${\displaystyle \underset{y^{*}\to +\mathrm{\infty }}{lim}o(y^{*})=0}$. Next, integrating HΔQ over ${\mathsf{Y}}^{*}$ leads to

$$0={\int }_{{\mathsf{Y}}^{*}}H\mathrm{\Delta }Q\mathrm{d}\mathit{y}\hspace{0.17em}=-{\int }_{{\mathsf{Y}}^{*}}\mathbf{\nabla }H\cdot (\mathbf{\nabla }q+\mathbf{\nabla }H)\mathrm{d}\mathit{y}\hspace{0.17em}+({\alpha }^2+{\beta }^2)(y^{*}-\frac{e_1}{h})+o(y^{*}).$$

Since ${\int }_{{\mathsf{Y}}^{*}}\mathbf{\nabla }H\cdot \mathbf{\nabla }H\mathrm{d}\mathit{y}\hspace{0.17em}=({\alpha }^2+{\beta }^2)(y^{*}-\frac{e_1}{h})$, and passing to the limit y* → +∞, we finally find

$$0\le {\int }_{\mathsf{Y}}\mathbf{\nabla }q\cdot \mathbf{\nabla }q\hspace{0.17em}\mathrm{d}\mathit{y}\hspace{0.17em}=(\mathcal{B}+\frac{e_1}{h})({\alpha }^2+{\beta }^2)-2\mathcal{A}\alpha \beta ,\mathrm{\forall }(\alpha ,\beta )\in {\mathbb{R}}^2.$$

We deduce that, for a ≥ e₁, $0\le (\mathcal{B}+\frac{a}{h})({\alpha }^2+{\beta }^2)-2\mathcal{A}\alpha \beta$, $\mathrm{\forall }(\alpha ,\beta )\in {\mathbb{R}}^2$. ▪

Transmission conditions have been implemented numerically using a = e₁, that is, restoring the actual thickness of the array. The actual problem and the unified problem involving (5.1) have been implemented numerically in a consistent manner. In both cases, the numerical method combines a finite-element scheme in space and a finite-difference scheme in time (FEM-BEM C++ code XLiFE++; see https://uma.ensta-paris.fr/soft/XLiFE++). P2 finite elements are used on a triangular mesh along with a centred order 2 Newmark scheme in time, unconditionally stable without numerical damping [15].

(b). Temporal evolution of the field in the cage

We consider now the two-dimensional cage shown in figure 8; the linear array is composed of 10 circular wires of radius e = 0.1 (hence e₁ = e) and spacing h = 1, in x₂ ∈ (0, 10). The bottom boundary Γ⁻ at x₁ = −5 is associated with a Dirichlet boundary condition and radiation boundary conditions are imposed on Γ⁺ at x₁ = 10. Eventually, the lateral boundaries at x₂ = 0 and x₂ = 10 are associated with a Neumann boundary condition.

Figure 8.

Snapshots at short times—the p(x, t) solution of the actual problem (a) and of the p^ap(x, t) solution to the effective problem ruled by (5.1) with a = e (b); both p and p^ap are computed numerically. (Online version in colour.)

Remark 5.2. —

Note that, in (3.1) the Dirichlet condition was chosen on all boundaries for ease of writing. This choice is however arbitrary and it does not affect the model, whose essential ingredients are the transmission conditions; in addition, the analysis does not apply near the ends of the array where a specific analysis is required, which is outside the scope of the present study [16]. In our simulations, Neumann boundary conditions on the lateral walls of the cavity avoid very small amplitudes within the cavity, which would fall within numerical errors.

We use a source S exciting a range of frequencies. This includes the first three resonances of the cavity, namely S(x, t) = g(x) s(t), with $g(\mathbf{\text{x}})={\displaystyle 1/\pi (0.1{)}^2}$ within the disc centred at (x₁ = 1, x₂ = 5) and of radius 0.1, g(x) = 0 elsewhere. Next, $s(t)={\text{e}}^{-\alpha t^2}\mathrm{sin\hspace{0.17em}}({\omega }_{\text{s}}t)$ with α = 5 × 10⁻³ and ω_s = 1 (figure 9). For the actual problem on p, a refined mesh (Δh = 10⁻²) is used in the vicinity of the wires to ensure that the near-evanescent field of the order of e is well resolved; far from the wires, a coarser mesh (Δh = 0.2) ensures at least 16 points per wavelength up to ω ∼ 2. For the effective problem ruled by (5.1) with a = e, we have checked that a constant mesh step Δh = 0.2 is sufficient to produce converged solutions. In time, we use the same time step for both problems Δt = 4 × 10⁻² to facilitate the comparison of the solutions.

Figure 9.

The source s(t) (a) and its Fourier transform $\hat{s}(\omega )$ (b); the red arrows show the resonance frequencies of the cage. (Online version in colour.)

To begin with, we report in figures 8 and 10 snapshots of the fields p and p^ap, both being computed numerically. Three regimes are exemplified:

• —The short times: the source emits, part of the wave train hits the array and it is essentially reflected. The amplitude inside the cavity is weak as the shielding is efficient for most of the frequencies. However, we see in figure 8 that the field is not zero in the cavity both in the actual and effective problems. The cavity is little by little filled in.
• —The intermediate times: the source is still emitting but with weaker amplitude. Meanwhile, the energy within the cavity has increased. At these intermediate times, the amplitudes inside and outside the cavity are comparable.
• —The long times: the source does not emit anymore and the waves issued from the source, directly or after reflection on the array, have left the calculation domain. Now the cavity releases energy very slowly; ‘very slow’ being related to the weak radiative damping of the cavity. The agreement between both solutions is good at short times. More interestingly, the solutions keep on agreeing at intermediate and long times.

Figure 10.

Same representation as in figure 8 at intermediate times (a) and at long times (b). (Online version in colour.)

More quantitatively, we report in figure 11a the time variations of the quantity P(t) (resp. P^ap(t)) being the spatial average of p(x, t) (resp. p^ap(x, t)) over the upper half-part of the cavity avoiding the near field of the wires in the actual geometry; we choose the domain Ω = {(x₁, x₂) ∈ (0, 10) × ( − 2.5, − 2e)}, hence

$$P(t)={\int }_{\mathrm{\Omega }}p(\mathbf{\text{x}},t)\text{d}\mathbf{\text{x}}\text{and}{P}^{\text{ap}}(t)={\int }_{\mathrm{\Omega }}{p}^{\text{ap}}(\mathbf{\text{x}},t)\hspace{0.17em}\text{d}\mathbf{\text{x}}.$$ 5.5

The spectral content $\hat{P}(\omega )$ makes the first three resonance frequencies appear. The quantitative agreement is excellent for kh ∈ (0, 1), with a relative error less than 1%. It reaches 10% at the second resonance. Eventually, although the trends around the third resonance are reasonably captured, the error reaches 100% very locally for kh ∈ (1.77, 1.83) owing to a small shift between the actual resonance and that predicted by the unified model.

Figure 11.

(a) Time variation of P(t) (solid blue line) and of P^ap(t) (dashed black line) for t ∈ (0, 400). (b) Corresponding spectra $\hat{P}(\omega )$ and $\widehat{P^{\text{ap}}}(\omega )$; the dotted grey line shows the spectrum of the error | p^ap − p|. (Online version in colour.)

(c). Energetic aspects: stability of the model in time

In this section, we move on to energetic aspects of the effective model and we address two different questions. Firstly, we have said that (5.1) ensure stability of the model for a = e as used in the numerics reported in the previous section; we shall see that a ≥ a_c with a_c ≤ e is sufficient in a specific geometry, that is, when $(\mathcal{A},\mathcal{B})$ are known. If a < a_c numerical instabilities in time are fostered, resulting in a blow-up of the solution p^ap(x, t). Next, the stability is associated with a positive interfacial energy ${\mathcal{E}}_{\mathrm{\Gamma }}$ whose link with the actual energy is inspected.

(i). Stability of the model with enlarged transmission conditions

In lemma 5.1, we have established a criterion of stability of the enlarged transmission conditions (5.1); it is sufficient that a ≥ e to ensure the stability. The criterion can be refined for given $(\mathcal{A},\mathcal{B})$; as ${\mathcal{E}}_{\mathrm{\Gamma }}$ in (5.2) is convex for $(\mathcal{B}+\frac{a}{h})>0$, ${\mathcal{E}}_{\mathrm{\Gamma }}\ge 0$ if

$$a\ge a_c=h(|\mathcal{A}|-\mathcal{B}).{\hspace{0.17em}}^7$$ 5.6

We report in figure 12a the resulting stability diagram in the plane (a, e) for wires being discs or squares. In both cases, $(\mathcal{A},\mathcal{B})$ are a function of e = e₁ = e₂ only (the radius of the disc or the half-length of the square). It can be seen that a_c ∼ e for squares while a_c is significantly lower than e for discs (in both cases, a_c ≤ e). While the value of a is incidental in the harmonic regime, it is expected that a < a_c produces numerical instability in the transient regime. We have used the transmission conditions (5.1) for various a around a_c; for our wires with radius e = 0.1, the critical value of a is a_c = 0.030. A typical instability observed numerically by means of the blow-up in time of p^ap(x, t) is illustrated in figure 12b; we have reported |P^ap(t)| from (5.5) against t for a stable formulation a > a_c (dashed black line, the solution is bounded) and for an unstable formulation a < a_c (solid blue line, the solution blows up exponentially). In the inset for the unstable formulation, ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)<0$ diverges exponentially to −∞, being roughly compensated by the exponential growth to +∞ of the energy in the volume ${\mathcal{E}}^{\text{ap}}(t;a)$. ‘Roughly’ means that the conservation of the energy (5.2) is not satisfied in our numerics when the fields have too large amplitudes. Note that the stability of the numerical solution in the close vicinity of a = a_c depends on the mesh and time step; a complete analysis of this problem, which is outside the scope of the present work, requires the discrete numerical energy associated with the numerical scheme to be identified.

Figure 12.

Numerical instability in the effective problem ruled by unstable conditions (5.1) when a < a_c. (a) Stability diagram for discs and squares. Blue lines show a_c(e) from (5.6) for wires of extent e. The red symbols show an estimate of a_c determined numerically as the lowest value of a producing a stable solution. (b) Exponential growth in the time of |P^ap(t)|, (5.5), for an unstable formulation (a = 0.015 < a_c, solid blue lines) and for a stable formulation (a = 0.033 > a_c, dashed black lines). The inset shows ${\mathcal{E}}^{\text{ap}}(t;a)$ diverging to +∞ and ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)$ diverging to −∞ for a = 0.015. (Online version in colour.)

(ii). Meaning of the interfacial energy ${\mathcal{E}}_{\mathrm{\Gamma }}$

It is one thing to know that a > a_c ensures the stability as it guarantees ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)>0$, but it is quite another to interpret ${\mathcal{E}}_{\mathrm{\Gamma }}$. Heuristically, ${\mathcal{E}}_{\mathrm{\Gamma }}$ approximates the sum of two actual energetic contributions : (i) that of the evanescent field being approximated by static fields (provided by the elementary problems) and (ii) that of the propagating field being approximated by linear fields (the loadings in the elementary problems). Up to now we have used a = e; hence, ${\mathcal{E}}_{\mathrm{\Gamma }}={\mathcal{E}}_{\mathrm{\Gamma }}(t;e)$ and we begin with this case. We have computed the actual and the effective energies

$$\{\begin{array}{ll}{\mathcal{E}}_{{\mathrm{\Omega }}_b}(t)=\frac{1}{2}{\int }_{{\mathrm{\Omega }}_b}({\mathit{u}}^2+p^2)\mathrm{d}\mathbf{\text{x}}\hspace{0.17em},& {\mathrm{\Omega }}_b=\{(x_1,x_2)\in (-b,b)\times \mathrm{\Gamma }\},\\ {{\mathcal{E}}^{\text{ap}}}_{{\mathrm{\Omega }}_b}(t)={\mathcal{E}}_{\mathrm{\Gamma }}(t;e)+\frac{1}{2}{\int }_{{\mathrm{\Omega }}_b\setminus {\mathrm{\Gamma }}_e}(({\mathit{u}}^{\text{ap}}{)}^2+(p^{\text{ap}}{)}^2)\mathrm{d}\mathbf{\text{x}}\hspace{0.17em},& {\mathrm{\Gamma }}_e=\{(x_1,x_2)\in (-e,e)\times \mathrm{\Gamma }\},\end{array}$$

for b ∈ (e, 5e). The results are reported in figure 13. Figure 13a shows the fields p(x, t₀) and p^ap(x, t₀) in the vicinity of the wires (t₀ = 19); the contribution to p of the evanescent near-field is visible, which is not reproduced by p^ap, by construction (the effect of the evanescent field has been encapsulated in the transmission conditions). As b increases, the domain Ω_b over which the energy is calculated increases. In the actual problem, ${\mathcal{E}}_{{\mathrm{\Omega }}_b}$ increases accordingly, being supplied by the evanescent field and by the propagating field. By contrast, in the effective problem, ${{\mathcal{E}}^{\text{ap}}}_{{\mathrm{\Omega }}_b}$ increases, being supplied by the propagating field only, as ${\mathcal{E}}_{\mathrm{\Gamma }}$ approximates the whole energy of the evanescent field. We expect the two energies to coincide (up to the error of the model) when b is large enough so that the actual energy also contains the whole energy of the evanescent field. The results in figure 13 support this scenario. We observe that ${{\mathcal{E}}^{\text{ap}}}_{{\mathrm{\Omega }}_b}>{\mathcal{E}}_{{\mathrm{\Omega }}_b}$ up to b ≃ 2.5e; afterwards, the two energies coincide up to a relative error (about 5%) attributable to the error of the model.

Figure 13.

Energies in the actual and effective problems. (a) A zoom of p(x, t₀) and p^ap(x, t₀) in {(x₁, x₂) ∈ ( − 1.5, 1.5) × ( − 3, 3)} (t₀ = 19). (b) The difference between the time-averaged energies ${{\mathcal{E}}^{\text{ap}}}_{{\mathrm{\Omega }}_b}$ and ${\mathcal{E}}_{{\mathrm{\Omega }}_b}$ against b and time variations of the energies for ***** b = e = 0.1 and **** b = 5e = 0.5. (Online version in colour.)

We now move to the influence of a, the enlargement of the interface. We have computed ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)$ for a ∈ (0.10, 0.25), which we compare with its counterpart in the actual problem

$${\mathcal{E}}_{{\mathrm{\Omega }}_a}(t)=\frac{1}{2}{\int }_{{\mathrm{\Omega }}_a}({\mathit{u}}^2+p^2)\hspace{0.17em}\mathrm{d}\mathbf{\text{x}}\hspace{0.17em},{\mathrm{\Omega }}_a=\{(x_1,x_2)\in (-a,a)\times \mathrm{\Gamma }\}.$$

From what we have seen from the previous representation, we could expect that ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)\simeq {\mathcal{E}}_{{\mathrm{\Omega }}_a}(t)$ as soon as a > 2.5 e, that is, as soon as ${\mathcal{E}}_{{\mathrm{\Omega }}_a}$ contains the whole contribution of the evanescent field. This is not the case, as illustrated in figures 14 and 15; the reason relies this time on the contribution of the propagating field. For every a, ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)$ approximates the energetic contribution of the evanescent field but it also has to approximate that of the propagating field. The propagating field is approximated by its expansions, a constant at zero order and a linear function at first order; hence, the approximation becomes cruder as a increases. This demonstrates that the ability of ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)$ to resemble ${\mathcal{E}}_{{\mathrm{\Omega }}_a}$ is a compromise between the actual extension of the evanescent field and the validity of a linear approximation of the propagating field. From figure 14, it can be seen that, for small a, ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)>{\mathcal{E}}_{{\mathrm{\Omega }}_a}$ as the contribution of the evanescent field is incomplete in ${\mathcal{E}}_{{\mathrm{\Omega }}_a}$; conversely, for large a, ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)<{\mathcal{E}}_{{\mathrm{\Omega }}_a}$, indicating that ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)$ underestimates the contribution of the propagating field. The compromise is obtained for a ∼ 1.6 e, although it can be seen from figure 15 that the evanescent field is still strong at x₁ = 1.6 e.

Figure 14.

Relative difference between the time-averaged interface effective energy ${\mathcal{E}}_{\mathrm{\Gamma }}(a)$ and the time-averaged actual energy ${\mathcal{E}}_{{\mathrm{\Omega }}_a}$ against a; time variations of ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)$ (dashed black lines) and $\mathcal{E}(t;a)$ (solid blue lines) for ***** a = 0.1, **** a = 0.15 and *** a = 0.25. (Online version in colour.)

Figure 15.

Effective fields p^ap(x, t₀) (t₀ = 19) in the vicinity of Γ for a = 0.1, a = 0.16 and a = 0.25 (a) and corresponding profiles of p (solid blue lines) and p^ap (dashed black lines) at x₁ = a (b). (Online version in colour.)

Note that, if the comparison of ${\mathcal{E}}_{\mathrm{\Gamma }}(t;a)$ and ${\mathcal{E}}_{{\mathrm{\Omega }}_a}(t)$ is enlightening on the meaning of the interfacial energy, there is no guarantee that the value of a minimizing their difference coincides with the value of a minimizing the error between p and p^ap in the far field. (In the temporal regime, we do not find a clear minimum of this error as a varies; in the harmonic regime, we have observed a minimum in the scattering coefficients for a ≃ 1.5e.⁸)

6. Concluding remarks and perspectives

We have shown both theoretically and numerically that the transmission conditions derived in [10,11] for an array of Dirichlet wires in free space apply without additional work when the array delimits a resonant cavity. These conditions, called unified conditions in the present study, have several advantages: (i) they are valid at any frequency and for any size of the wires and (ii) in their enlarged version they guarantee stability in the transient regime. Point (i) is linked to the fact that the unified conditions avoid an iterative resolution of the asymptotic problems, which is at the origin of the problem stressed in [7]. (The zero-order problem is ill-posed at the resonance frequencies of the closed cavity.) Point (ii) is linked to the fact that the effective problem in its enlarged version is associated with a positive interfacial energy, which prevents numerical instabilities.

More generally, we can remark that the construction of a unified, or unique, problem gathering the results of the asymptotic analysis up to first order (or higher orders) follows the construction of the models of continuum mechanics or continuum physics. As for these classical models, the properties of new homogenized models can be analysed in all generality, independently of the specific context in which they will be used, as the sources and the surrounding boundaries. By contrast, an iterative resolution focuses on a specific solution for given sources and surrounding boundaries. As such it reduces the range of applicability of the homogenization, as the work has to be done for each new problem. In addition, as in the example reported in this study, it can lead to unnecessary complications.

Finally, we stress the importance of establishing effective models in the time domain: they enable us to check some good energetic properties, namely the conservation of a positive energy in the bulk and a good balance of the fluxes of energy. (We have given an illustration of the blow-up of the numerical solution in time when these properties are not satisfied.)

Appendix A. Step-by-step derivation of the resonant cases

In this appendix, we use the ansatz (3.5). We shall use the following result: any harmonic function q(y) in $\mathsf{Y}$ satisfying a homogeneous Dirichlet condition on $\mathrm{\partial }\hat{\mathsf{Y}}$ and with no exponential growth when y₁ goes to ±∞ can be written as

$$q(\mathit{y})=a{Q}^{-}(\mathit{y})+b{Q}^{+}(\mathit{y}),$$ A 1

where Q^± are defined in (2.3) and (2.4).

Remark A.1. —

For the sake of conciseness, the proof of the existence and uniqueness of Q⁻ and Q⁺ is not written here (e.g. [8]). It is based on the application of the Lax–Milgram lemma together with a Hardy equality [19, lemma 2.5.7]. A crucial point is that there is no harmonic function in $\mathsf{Y}$ satisfying a homogeneous Dirichlet condition on $\mathrm{\partial }\hat{\mathsf{Y}}$ and behaving like a constant as y₁ goes to both ±∞.

(a) The near-resonant case (2i): κ₁ ≠ κ

In that case, we make the ansatz (3.5) with m₀ = 0. Inserting the formal series into equation (3.1) and separating the different powers of ε leads to a collection of equations for the near- and far-field terms. These equations are linked with the matching conditions that enforce the far- and near-field expansions to coincide in some intermediate area (e.g.[10,12,20]). In the present case, they can be written as

$$p_i(0^{\pm },x_2)=\underset{y_1\to \pm \mathrm{\infty }}{lim}\left(\sum _{m=1}^{i-m_0}\frac{(y_1{)}^m}{m!}\frac{{\mathrm{\partial }}^m{p}_{i-m}}{\mathrm{\partial }{x}_1^m}(0^{\pm },x_2)-q_i(\mathit{y},x_2)\right)i\ge -1.$$ A 2

By convention, the previous sum is empty (and vanishes) if i − m₀ < 1. First, the leading-order far-field term p₀ and near-field term q₀ (1-periodic w.r.t. y₂) are solutions to

$$\{\begin{array}{l}\mathrm{\Delta }{p}_0+K_n^{*}\hspace{0.17em}{p}_0=S\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{\pm },\\ \frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}=i\sqrt{K_n^{*}}\hspace{0.17em}{p}_0\mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},{\mathrm{p}}_0=0\mathrm{on\hspace{0.17em}}(\mathrm{\partial }{\mathrm{\Omega }}^{-}\cup \mathrm{\partial }{\mathrm{\Omega }}^{+})\setminus ({\mathrm{\Gamma }}^{+}\cup \mathrm{\Gamma }),\end{array}\{\begin{array}{ll}\mathrm{\Delta }{q}_0=0& \mathrm{in\hspace{0.17em}}\mathsf{Y},\\ q_0=0& \mathrm{on\hspace{0.17em}}\mathrm{\partial }\hat{\mathsf{Y}},\end{array}$$ A 3

complemented (and linked) by the zero-order matching conditions (A 2) (i = 0). In (A 3, right) $\mathsf{Y}$ denotes the periodicity cell $\mathsf{Y}=\{(y_1,y_2)\in \mathbb{R}\times (-\frac{1}{2},\frac{1}{2})\setminus \overline{\hat{\mathsf{Y}}}\}.$ Using (1), q₀ can be written as q₀(y, x₂) = a(x₂)Q⁻(y) + b(x₂)Q⁺(y), and the matching condition indicates that q₀ tends to constants as y₁ goes to ±∞. From remark A.1, we get a(x₂) = b(x₂) = 0; hence, p₀(0^±, x₂) = 0. These conditions complement the problem (A 3, left), and the solution reads

$$p_0=P_n^{*}\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+}\mathrm{and}{\mathrm{p}}_0={\alpha }_1{\mathrm{p}}_n^{*}\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},$$ A 4

where P*_n is the unique solution to (3.2, left) and p_n* is defined in (3.8). It remains to determine α₁. In fact, the value of α₁ is imposed by the solvability condition associated with the far-field problem of order 1 in Ω⁻. More specifically, we have

$$\{\begin{array}{ll}\mathrm{\Delta }{p}_1+K_n^{*}{p}_1=-{\kappa }_1{\alpha }_1{p}_n^{*}& \mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_1=0& \mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },\end{array}\{\begin{array}{l}\mathrm{\Delta }{q}_1=0\mathrm{in\hspace{0.17em}}\mathsf{Y},\\ q_1=0\hspace{0.17em}\mathrm{on\hspace{0.17em}}\mathrm{\partial }\hat{\mathsf{Y}},\end{array}$$ A 5

with q₁ 1-periodic w.r.t. y₂ and the first-order matching conditions (A 2) (i = 1). As previously, we look for q₁(y, x₂) = a(x₂)Q⁻(y) + b(x₂)Q⁺(y). Identifying in the matching conditions the parts linear in y₁ and the constants, we get a(x₂) = ∂p₀/∂x₁(0⁻, x₂), b(x₂) = ∂p₀/∂x₁(0⁺, x₂), hence $p_1(0^{-},x_2)=-\mathcal{B}a(x_2)+\mathcal{A}b(x_2)$, $p_1(0^{+},x_2)=-\mathcal{A}a(x_2)+\mathcal{B}b(x_2)$, and using (A 4)

$$\{\begin{array}{l}{q}_1(\mathit{y},x_2)={\alpha }_1\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)Q^{-}(\mathit{y})+\frac{\mathrm{\partial }{P}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)Q^{+}(\mathit{y}),\\ p_1(0^{-},x_2)=\mathcal{A}\hspace{0.17em}\frac{\mathrm{\partial }{P}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)-{\alpha }_1\mathcal{B}\hspace{0.17em}\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2).\end{array}$$

Finally, integrating over Ω⁻ (A 5, left) after multiplication by p_n* and integrating by part, we see that problem (A 5, left) is solvable if and only if

$${\alpha }_1({\kappa }_1+\mathcal{B}\hspace{0.17em}{\int }_{\mathrm{\Gamma }}{(\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2))}^2\mathrm{d}{x}_2\hspace{0.17em})=\mathcal{A}{\int }_{\mathrm{\Gamma }}\frac{\mathrm{\partial }{P}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)\mathrm{d}{x}_2.$$

In view of the definition (3.9) of κ and since by assumption κ₁ ≠ κ, the previous equation entirely defines α₁ in (A 4).

(b) The on-resonance case (2ii) : κ₁ = κ

In that case, we still make the ansatz (3.5) but we found m₀ = −1 (reflecting the presence of a strong resonant effect).

(i) The leading order: i = −1)

As previously, the far-field term p₋₁ and near-field term q₋₁ (1-periodic w.r.t. y₂) are solutions to

$$\{\begin{array}{l}\mathrm{\Delta }{p}_{-1}+K_n^{*}\hspace{0.17em}{p}_{-1}=0\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{\pm },\\ \frac{\mathrm{\partial }{p}_{-1}}{\mathrm{\partial }{x}_1}=i\sqrt{K_n^{*}}\hspace{0.17em}{p}_{-1}\mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},{\mathrm{p}}_{-1}=0\hspace{0.17em}\mathrm{on\hspace{0.17em}}(\mathrm{\partial }{\mathrm{\Omega }}^{-}\cup \mathrm{\partial }{\mathrm{\Omega }}^{+})\setminus ({\mathrm{\Gamma }}^{+}\cup \mathrm{\Gamma }),\end{array}\hspace{0.17em}\{\begin{array}{ll}\mathrm{\Delta }{q}_{-1}=0& \mathrm{in\hspace{0.17em}}\mathsf{Y},\\ q_{-1}=0& \mathrm{on\hspace{0.17em}}\mathrm{\partial }\hat{\mathsf{Y}},\end{array}$$ A 6

complemented by the matching conditions (A 2) for i = −1. As previously, we deduce from the near-field equation (A 6, right) that q₋₁(y, x₂) = a(x₂)Q⁻ (y) + b(x₂)Q⁺(y); hence, again, as (A 2) imposes that q₋₁ tends to a constant as y₁ goes to ±∞, we obtain that a(x₂) = b(x₂) = 0, and therefore q₋₁ = 0. Consequently p₋₁(0^±, x₂) = 0. Together with (A 6, left), we deduce that

$$p_{-1}=0\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\mathrm{and}{\mathrm{p}}_{-1}={\alpha }_2{\mathrm{p}}_n^{*}\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},$$ A 7

where p_n* is defined in (3.8), the constant α₂ being undetermined at this stage. Expectedly, in the absence of a source term at order i = −1, the field outside the cavity is zero at this order.

(ii) The order i = 0)

The analysis of the leading order provides neither the leading-order asymptotic for p^ε in Ω⁺ nor the constant α₂. Therefore, we need to analyse p₀ and q₀ (1-periodic w.r.t. y₂) solutions to

$$\{\begin{array}{l}\mathrm{\Delta }{p}_0+K_n^{*}\hspace{0.17em}{p}_0=S\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},\\ \frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}=i\sqrt{K_n^{*}}{p}_0\hspace{0.17em}\mathrm{on\hspace{0.17em}}{\mathrm{\Gamma }}^{+},{\mathrm{p}}_0=0\hspace{0.17em}\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{+}\setminus ({\mathrm{\Gamma }}^{+}\cup \mathrm{\Gamma }),\end{array}\{\begin{array}{l}\mathrm{\Delta }{p}_0+K_n^{*}\hspace{0.17em}{p}_0=-\kappa {\alpha }_2{p}_n^{*}\hspace{0.17em}\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_0=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },\end{array}$$ A 8

and

$$\mathrm{\Delta }{q}_0=0\mathrm{in\hspace{0.17em}}\mathsf{Y},{\mathrm{q}}_0=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }\hat{\mathsf{Y}},$$ A 9

together with the zero-order matching conditions (A 2) and using (A 7). Looking for q₀(y, x₂) = a(x₂)Q⁻(y) + b(x₂)Q⁺(y) and identifying in the matching conditions the parts linear in y₁ and the constants, we get $a(x_2)={\alpha }_2\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)$ and b(x₂) = 0, hence

$$q_0(\mathit{y},x_2)={\alpha }_2\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2)Q^{-}(\mathit{y}),\hspace{0.17em}{p}_0(0^{+},x_2)=-\mathcal{A}\hspace{0.17em}{\alpha }_2\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2),\hspace{0.17em}{p}_0(0^{-},x_2)=-\mathcal{B}\hspace{0.17em}{\alpha }_2\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2).$$ A 10

The above relations provide p⁰(0^±, x₂), which complement the problems (A 8) set in Ω⁺ and Ω⁻; at this stage, they are disconnected. In Ω⁺, p₀ solution to (A 8, left) along with (A 10, centre) reads

$$p_0(\mathit{x})={\alpha }_2\mathcal{A}{P}^{+}(\mathit{x})+P_n^{*}(\mathit{x})\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{+},$$ A 11

where P⁺ and P*_n are the (unique) solutions to (3.14, left) and (3.2, left). It then provides the asymptotic behaviour of p^ε in Ω⁺ (up to the definition of α₂). In Ω⁻, since K*_n is a resonance frequency, the far-field problem (A 8, right)–(A 10, right) is not well-posed (it has a kernel of dimension 1 spanned by p_n*); however, it is solvable. Indeed, multiplying (A 8, right) by p_n* and integrating over Ω⁻ gives

$${\alpha }_2(\kappa +\mathcal{B}\hspace{0.17em}{\int }_{\mathrm{\Gamma }}{(\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}(0,x_2))}^2\mathrm{d}{x}_2\hspace{0.17em})=0,$$

and the above equality is fulfilled for any α₂ by definition of κ (see (3.9)). It follows that the p₀ solution to (A 8, right)–(A 10, right) reads

$$p_0(\mathit{x})={\alpha }_2\mathcal{B}{P}^{-}(\mathit{x})+\beta p_n^{*}(\mathit{x})\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},$$ A 12

where P⁻ is the unique solution to (3.14, right) and p_n* is the real-valued eigenvector associated with K*_n (for the cavity problem (3.2, right)). In (A 12), α₂, which reaches back from order i = −1 (see (A 7)), and β are constants that remain to be determined.

(ii) The order i = 1

Here, we shall determine the constant α₂ that we need to parametrize locally the resonance curve. To do so, it is enough to investigate the solvability of the far-field problem for p₁ in Ω⁻ and the near-field problem for q₁ (1-periodic w.r.t. y₂)); namely,

$$\{\begin{array}{l}\mathrm{\Delta }{p}_1+K_n^{*}\hspace{0.17em}{p}_1=-\kappa p_0-{\kappa }_2{\alpha }_2{p}_n^{*}\mathrm{in\hspace{0.17em}}{\mathrm{\Omega }}^{-},\\ p_0={\alpha }_2\mathcal{B}{P}^{-}+\beta p_n^{*},\hspace{0.17em}{p}_1=0,\hspace{0.17em}\mathrm{on\hspace{0.17em}}\mathrm{\partial }{\mathrm{\Omega }}^{-}\setminus \mathrm{\Gamma },\end{array}\{\begin{array}{l}\mathrm{\Delta }{q}_1=-2{\alpha }_2\frac{{\mathrm{\partial }}^2{p}_n^{*}}{\mathrm{\partial }{x}_1\mathrm{\partial }{x}_2}(0,x_2)\hspace{0.17em}\frac{\mathrm{\partial }{Q}^{-}}{\mathrm{\partial }{y}_2}\hspace{0.17em}\mathrm{in\hspace{0.17em}}\mathsf{Y},\\ q_1=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }\hat{\mathsf{Y}}.\end{array}$$ A 13

We have used the forms of p₋₁ in (A 7), of p₀ in (A 12) and of q₀ in (A 10). The two problems are complemented by the matching conditions (A 2) at order −1 (i = −1) with ∂p₋₁/∂x₁(0^±, x₂) = 0. We define the Q, 1-periodic w.r.t. y₂, unique solution to

$$\mathrm{\Delta }Q=-2\frac{\mathrm{\partial }{Q}^{-}}{\mathrm{\partial }{y}_2}\mathrm{in\hspace{0.17em}}\mathsf{Y},\mathrm{Q}=0\mathrm{on\hspace{0.17em}}\mathrm{\partial }\hat{\mathsf{Y}},\underset{{\mathrm{y}}_1\to \pm \mathrm{\infty }}{lim}\mathbf{\nabla }\mathrm{Q}=\mathbf{0}.$$ A 14

(Here, again, see remark A.1; the well-posedness of (A 14) results from the Lax–Milgram lemma.) As ${\tilde{q}}_1=(q_1-{\alpha }_2(\mathrm{\partial }{p}_n^{*}/\mathrm{\partial }{x}_1\mathrm{\partial }{x}_2)(0,x_2)Q)$ satisfies $\mathrm{\Delta }{\tilde{q}}_1=0$ with ${\tilde{q}}_1=0$ on $\mathrm{\partial }\hat{\mathsf{Y}}$, we look for q₁ of the form

$$q_1(\mathit{y},x_2)=a(x_2)Q^{-}(\mathit{y})+b(x_2)Q^{+}(\mathit{y})+{\alpha }_2\frac{{\mathrm{\partial }}^2{p}_n^{*}}{\mathrm{\partial }{x}_1\mathrm{\partial }{x}_2}(0,x_2)Q(\mathit{y}).$$

Inserting q₁ in (A 13, right), and identifying the parts linear in y₁ and the constants, we obtain that

$$q_1(\mathit{y},x_2)=\frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}(0^{-},x_2)Q^{-}(\mathit{y})+\frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}(0^{+},x_2)Q^{+}(\mathit{y})+{\alpha }_2\frac{{\mathrm{\partial }}^2{p}_n^{*}}{\mathrm{\partial }{x}_1\mathrm{\partial }{x}_2}(0,x_2)Q(\mathit{y})$$

and

$$p_1(0^{-},x_2)=-\mathcal{B}\frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}(0^{-},x_2)+\mathcal{A}\frac{\mathrm{\partial }{p}_0}{\mathrm{\partial }{x}_1}(0^{+},x_2).$$

Eventually, using p₀ given by (A 11) and (A 12), we end up with

$$p_1(0^{-},x_2)={\alpha }_2{((\mathcal{A}{)}^2\frac{\mathrm{\partial }{P}^{+}}{\mathrm{\partial }{x}_1}-(\mathcal{B}{)}^2\frac{\mathrm{\partial }{P}^{-}}{\mathrm{\partial }{x}_1})}_{|x_1=0}+{(\mathcal{A}\frac{\mathrm{\partial }{P}_n^{*}}{\mathrm{\partial }{x}_1}-\mathcal{B}\beta \frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1})}_{|x_1=0},$$

which complements problem (A 13, left). It is now sufficient to multiply (A 13, left) by p_n* and to integrate over Ω⁻ to see that problem (A 13, left) is solvable if and only if ${\alpha }_2{\kappa }_2={\int }_{\mathrm{\Gamma }}{p}_1\mathrm{\partial }{p}_n^{*}/\mathrm{\partial }{x}_1(0,x_2)\mathrm{d}{x}_2\hspace{0.17em}\hspace{0.17em}-\kappa {\int }_{{\mathrm{\Omega }}^{-}}{p}_0{p}_n^{*}\mathrm{d}\mathit{x}\hspace{0.17em},$ resulting in

$${\alpha }_2[{\kappa }_2-(\mathcal{A}{)}^2{\int }_{\mathrm{\Gamma }}{\left(\frac{\mathrm{\partial }{P}^{+}}{\mathrm{\partial }{x}_1}\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}\right)}_{|x_1=0}\mathrm{d}{x}_2\hspace{0.17em}+(\mathcal{B}{)}^2{\int }_{\mathrm{\Gamma }}{\left(\frac{\mathrm{\partial }{P}^{-}}{\mathrm{\partial }{x}_1}\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}\right)}_{|x_1=0}\mathrm{d}{x}_2\hspace{0.17em}]=\mathcal{A}{\int }_{\mathrm{\Gamma }}{\left(\frac{\mathrm{\partial }{P}_n^{*}}{\mathrm{\partial }{x}_1}\frac{\mathrm{\partial }{p}_n^{*}}{\mathrm{\partial }{x}_1}\right)}_{|x_1=0}\mathrm{d}{x}_2\hspace{0.17em},$$ A 15

as mentioned in (3.12) (with (3.11) and (3.13)). In (A 15), note that the term in β has cancelled; also, we have used that P⁻ is orthogonal to p_n*. Also, we have used that ${\displaystyle \underset{y_1\to \pm \mathrm{\infty }}{lim}Q=0}$ for wires symmetric w.r.t. y₂, as Q is an odd function of y₂; the result (A 15) on α₂ holds for any wire shape.

Appendix B. The elementary problem

Here, we report additional information on the effective parameters $(\mathcal{A},\mathcal{B})$ issued from the elementary solution Q⁺ to (2.3): figure 16b shows an example of the fields Q⁺(y) for a wire being a disc or a square, and figure 16a shows the variations of $(\mathcal{A},\mathcal{B})$ against e/h; values of $(\mathcal{A},\mathcal{B})$ used for squares (in the harmonic regime) and for discs (in the transient regime) throughout the paper are given in (B 1).

B 1

Figure 16.

(a) Effective parameters $(\mathcal{A},\mathcal{B})$ against e/h for squares (solid lines) and for discs (dashed lines); the dotted grey line shows ${\mathcal{A}}_0=-(1/2\pi )\log \mathrm{sin\hspace{0.17em}}(2\pi e/h)$. (b) Examples of solution Q⁺ to the elementary problem (2.3). (Online version in colour.)

Appendix C. The thin wire approximation

The thin wire approximation presented in [7] allows us to capture the leakage of the cavity at the dominant order. The reason for this is that the amplitude within the cavity is of order ε, including at the resonance frequency. (This is already visible in figure 5 for e = 0.01 h, where the amplitude ${\mathcal{P}}_{\text{max}}$ does not overcome 0.1.) This is obtained by choosing a scaling e/h = O(e^−(c/ε)) (with c a constant), which means that the wire thickness e vanishes exponentially faster than the array spacing h when ε goes to zero. In this case, it is found that

$$p^{\text{tw}}(0,x_2)=h{\mathcal{A}}_0(\frac{\mathrm{\partial }{p}^{\text{tw}}}{\mathrm{\partial }{x}_1}(0^{+},x_2)-\frac{\mathrm{\partial }{p}^{\text{tw}}}{\mathrm{\partial }{x}_1}(0^{-},x_2)),$$ C 1

with p^tw continuous. Expectedly, the above condition is consistent with the formulation (2.5) when $\mathcal{A}=\mathcal{B}={\mathcal{A}}_0$. For the simple scattering problem considered in §4, the solution reads as in (4.1) with

$${\mathcal{R}}^{\text{ap}}=-\frac{\overline{z}}{z},{\mathcal{P}}^{\text{ap}}=-\frac{2\text{i}k{\text{c}}_{\theta }h{\mathcal{A}}_0}{z},z=1+k{\text{c}}_{\theta }h{\mathcal{A}}_0(\cos (k{\text{c}}_{\theta }D)-i\mathrm{sin\hspace{0.17em}}(k{\text{c}}_{\theta }D)),$$ C 2

and ${\mathcal{A}}_0=-(1/2\pi )\log (\mathrm{sin\hspace{0.17em}}(2\pi e/h))$, which is consistent with (4.1) and (4.2) in the considered limit. We report the resonance curves in figure 17 as in figure 5. The thin wire model appears to be more limited, its accuracy depending on how close $(\mathcal{A},\mathcal{B})$ are to their asymptotic value ${\mathcal{A}}_0$.

Figure 17.

Validity of the thin wire approximation; same representation as in figure 5. Solid blue lines, direct numerics; dashed black lines; unified model with (4.1) and (4.2); and dotted green lines, thin wire model with (4.1); see (C 2). The panels in (b) show a zoom of the panels in (a). (Online version in colour.)

Data accessibility

The numerical results of the direct problem in the harmonic regime were obtained using a standard multimodal method as presented in Marigo JJ, Maurel A. 2017 An interface model for homogenization of acoustic metafilms. World Scientific handbook of metamaterials and plasmonics. Volume 2: Elastic, acoustic, and seismic metamaterials (eds R Craster, S Guenneau), pp. 599–645. Singapore: World Scientific. The numerical results in the transient regime were obtained using the FEM-BEM C++ code XLiFE++; see also [15].

Competing interests

We declare we have no competing interests.

Authors' contributions

B.D, J-F.M. and E.L. realized the numerics in the transient regime. A.M. and K.P. realized the numerics in the harmonic regime. All authors participated in the modelling and analysis, contributed to the paper, gave final approval for publication and agree to be held accountable for the work performed herein.

Funding

K.P. is grateful for the support of the Agence de l’Innovation de Défense (AID) from the Direction Générale de l’Armement (DGA), under grant no. 2019 65 0070.