## Abstract

We present a mathematical study of two-dimensional electrostatic and electromagnetic shielding by a cage of conducting wires (the so-called ‘Faraday cage effect’). Taking the limit as the number of wires in the cage tends to infinity, we use the asymptotic method of multiple scales to derive continuum models for the shielding, involving homogenized boundary conditions on an effective cage boundary. We show how the resulting models depend on key cage parameters such as the size and shape of the wires, and, in the electromagnetic case, on the frequency and polarization of the incident field. In the electromagnetic case, there are resonance effects, whereby at frequencies close to the natural frequencies of the equivalent solid shell, the presence of the cage actually amplifies the incident field, rather than shielding it. By appropriately modifying the continuum model, we calculate the modified resonant frequencies, and their associated peak amplitudes. We discuss applications to radiation containment in microwave ovens and acoustic scattering by perforated shells.

## 1. Introduction

The Faraday cage effect is the phenomenon whereby electric fields and electromagnetic waves can be blocked by a wire mesh. The effect was demonstrated experimentally by Faraday in 1836 [1], was familiar to Maxwell [2], and its practical application in isolating electrical systems and circuits is well known to modern-day engineers and physicists alike. However, somewhat surprisingly there does not seem to be a widely known mathematical analysis quantifying the effectiveness of the shielding as a function of the basic cage properties (e.g. the geometry of the cage, and the thickness, shape and spacing of the wires in the mesh from which it is constructed). The recent publication [3] provided such an analysis for the two-dimensional electrostatic problem where the cage is a ring of *M* equally spaced circular wires of small radius *r*≪*L*/*M* (here *L* is a typical macro-lengthscale of the cage, e.g. the circumference of the ring of wires) held at a common constant potential, which can be formulated as a Dirichlet problem for the Laplace equation. It was found in [3] that the shielding effect of such a Faraday cage is surprisingly weak: as the number of wires *M* tends to infinity the magnitude of the field inside the cage in general decays at best only inverse linearly in *M*, rather than exponentially, as one might infer from certain treatments of the Faraday cage effect in the physics literature (e.g. [4, §7-5]).

One of the key tools used by Chapman *et al.* [3] to study the Faraday cage effect in the regime of large *M* was a continuum model, in which the shielding effect of the discrete wires is replaced by a homogenized boundary condition on an infinitesimally thin interface between the ‘inside’ and ‘outside’ of the cage. Such boundary conditions can be derived by matching asymptotic expansions of the field away from the mesh with expansions in a boundary layer close to the mesh, where a multiple scales approximation can be applied (cf. [3, §5] and appendix C, and the closely related work in [5–8]).

This paper extends the analysis of Chapman *et al.* [3] in a number of significant ways. Firstly, we explain how the homogenized boundary condition of Chapman *et al.* [3] generalizes to arbitrary wire shapes (not necessarily circular). Secondly, we investigate the ‘thick-wire’ regime in which *r*≪*L*/*M* and is in general ill-posed for *resonance*, where the presence of the cage actually *amplifies* the incident field, rather than shielding it. For the Dirichlet problem, such resonance effects are the strongest in the ‘thick-wire’ regime in which

We conclude this introduction with some comments on related literature. Firstly, we acknowledge that there is already a substantial literature concerning the rigorous analysis of homogenization procedures for potential and scattering problems involving thin, rapidly varying interfaces. While we do not attempt a comprehensive review, we note in particular the works [5–7,9–16], which consider problems closely related (but different) to those studied here. Many of these studies adopt a similar multiple scale-based approaches to ours, albeit from a slightly more rigorous point of view, and some (e.g. [10]) derive higher order asymptotic approximations than those considered here. What sets our work apart from this literature is that we are concerned less with formulating high-order approximations and proving rigorous error estimates and more with understanding the qualitative and quantitative behaviour of the leading-order homogenized approximations—in particular, their shielding performance—something which to date does not appear to have been studied systematically. Secondly, we note that the two-dimensional cage problems we consider can be attacked by direct numerical simulation, at least for *M* relatively small—indeed we shall compare our asymptotic results with two different numerical methods in §4. In the small wire regime, one can obtain approximate numerical solutions particularly efficiently, if the wires are modelled as simple point sources. Such an approach to the electrostatic problem is described in [3, §6], where the associated amplitudes of the point sources are found by an energy minimization procedure. We also mention [17], which treats the wave problem for a circular cage of small equally spaced wires using the so-called ‘Foldy method’ from multiple scattering theory, in which the geometrical assumptions permit a semi-analytical solution for the associated point source amplitudes in terms of the discrete Fourier transform. This method appears to be closely related to the lowest-order version of the Mikhlin-type numerical method used by Chapman *et al.* [3], higher order versions of which shall be our main source of numerical approximations for the circular wire case. The analysis of Martin [17] does not cover the regime

## 2. Problem formulation

Let *Ω*_{−} be a bounded simply connected open subset of the plane with smooth boundary *Γ*=∂*Ω*_{−} and let *x*,*y*)-plane with the complex *z*-plane, *z*=*x*+*iy*. We consider a ‘cage’ of *M* non-intersecting wires *Γ* with constant separation^{1} (measured with respect to arc length along *Γ*)
*Γ*| is the total length of *Γ*; for an illustration, see figure 1*a*. We set

The electrostatic problem is formulated as follows. Given a compactly supported source function *f*, we seek a real-valued potential *ϕ*(*z*) satisfying
*ϕ* taking an unknown (and in general non-zero) constant value on the wires. For completeness, we also consider the Neumann problem in which (2.2) is replaced by
*ν* denotes a unit normal vector on ∂*K*_{j}, and *o*(1) in (2.3). While not having any obvious electrostatic application, this could represent a model for inviscid incompressible fluid flow due to a source in the presence of a cage of impermeable wires.

The time-harmonic electromagnetic problem can be formulated in terms of two complex-valued scalar fields, representing the out-of-plane components of the electric and magnetic fields, respectively, both of which satisfy the Helmholtz equation
*f*, where *k*>0 is the (non-dimensional) wavenumber. (Incident plane waves can also be considered.) The out-of-plane component of the electric field (TE mode) satisfies the Dirichlet boundary condition (2.2) and the out-of-plane component of the magnetic field (TM mode) satisfies the Neumann boundary condition (2.4). At infinity, both fields are assumed to satisfy an outgoing radiation condition. These two problems also model the analogous acoustic scattering problems with sound-soft and sound-hard boundary conditions, respectively.

The goal of this paper is to determine the leading-order asymptotic solution behaviour of the above problems as the number of wires *M* tends to infinity, equivalently, as the wire separation *ε* tends to zero. For the wave problem we shall assume throughout that *ε*→0, so that the wavelength is comparable to the macro-dimensions of the cage and much longer than the inter-wire separation. We also need to specify how the wire size, shape and orientation should vary as *ε*→0. In particular, in order that the wires remain disjoint as *ε*→0 (so that the wires form a ‘cage’ and not a solid shell), the wire radii must in general decrease in proportion to *ε* (or faster).

We consider two different models, defining a reference wire shape either in local Cartesian coordinates aligned with *Γ*, or in local curvilinear coordinates that conform to *Γ*. Since *Γ* is smooth there is no difference between these models at leading order, but the distinction affects higher order corrections (due to the curvature of *Γ*) that will enter some of our calculations. To make the definitions specific, we must introduce some further notation.

Close to *Γ* we can change from Cartesian coordinates (*x*,*y*) to orthogonal curvilinear coordinates (*n*,*s*), such that *n* is the distance from (*x*,*y*) to the closest point on *Γ* (positive/negative *n* representing points inside *Ω*_{+} and *Ω*_{−}, respectively), and *s* is arc length along *Γ* to this closest point measured counterclockwise from some reference point on *Γ*. Given a reference point *z*_{j} on *Γ* with curvilinear coordinates (0,*s*_{j}), we define local curvilinear coordinates *z*_{j}. Explicitly, *θ*_{j} is the counter-clockwise angle from the positive *x*-axis to the outward normal vector to *Γ* at *z*_{j}. To convert between these coordinate systems, there exists a diffeomorphism *F*_{j}:(−*n*_{j},*n*_{j})×(−*ε*/2,*ε*/2)→*U*_{j}, where *U*_{j} is an open neighbourhood of *z*_{j} and *n*_{j}>0 is a constant, such that

We are now ready to specify the wire geometries and their dependence on *ε*. For both models, we assume a fixed reference wire shape *K*; a compact subset of the plane for which the smallest closed disc containing *K* has radius one and is centred at the origin (figure 1*c*).

In Model 1, we define a wire *K*_{j} of radius *r*>0 centred at *z*_{j} by the formula *K*_{j}=*rK* in the *z*-coordinates gives
*K*_{j}=*rK* but interpreted in the *z*-coordinates gives

Examples are illustrated in figure 2. The rationale for considering both wire models is that Model 1 is the more natural from a physical point of view as the wire shape is independent of *r* in the original Cartesian coordinate system, whereas Model 2 is simpler from a mathematical point of view as the wire shape is independent of *r* in the curvilinear coordinates in which we derive our homogenized boundary conditions (see §3). In many aspects of our analysis, the two models produce the same results. But for some problems requiring higher order boundary layer expansions, they may produce different results.

In order that the wires remain disjoint as *ε*→0, we assume that the wire radius *r* satisfies
*ε*→0. For example, *K* is the unit disc (cf. figure 2*a*) and the case of tangential line segments (cf. figure 2*c*). An exceptional case where no such *Γ*, when *K* is the interval [−1,1] (cf. figure 2*b*). Note in particular that a fixed value for *δ* corresponds to the wires taking up a fixed total fraction of the length of *Γ*, as the number of wires is increased.

Our aim is to describe both qualitatively and quantitatively how the asymptotic solution behaviour of the boundary value problems as *ε*→0 depends on the reference wire shape *K*, the scaling parameter *δ* and in the electromagnetic case the wavenumber *k*. In doing so, we generalize the analysis of Chapman *et al.* [3], which considered only the electrostatic case, with circular wires and the small wire regime *δ*≪1.

## 3. Homogenized boundary conditions

In the limit *ε*→0, we look for outer approximations in *Ω*_{±} of the form
*f* and *k* are *Ω*_{±}, with *Γ*, by matching with an appropriate boundary layer solution in a region of width *Γ* in which a multiple scales approximation can be applied.

We first note that in the curvilinear coordinates (*n*,*s*) the Laplacian is [18, (6.2.4)]
*κ*=*κ*(*s*) is the local (signed) curvature of *Γ* at the point (0,*s*), defined with respect to a counterclockwise parametrization. We introduce boundary layer variables (*N*,*S*) via (*n*,*s*)=(*εN*,*εS*). The inner limits of the outer solutions correct to *n* replaced by *εN* and re-expanding, giving
*N*>0 and *N*<0, respectively.

In the boundary layer, we look for a solution in multiple-scales form
*Φ*(*N*,*S*;*s*) is assumed to be 1-periodic in the fast tangential variable *S*. To determine the equation satisfied by *Φ*(*N*,*S*;*s*), we replace ∂/∂*n* by *ε*^{−1}∂/∂*N* and ∂/∂*s* by *ε*^{−1}∂/∂*S*+∂/∂*s* in (3.2) and expand. The leading-order result, for both the electrostatic and the wave problems (assuming *b*). Periodicity requires

A more detailed derivation of this boundary-layer problem is given in appendix A, where we also continue the expansion to *Γ* and its distorting effect on the wire shape in the (*N*,*S*) coordinates (shown by *b*). This distortion can be neglected in the leading-order problem above (and does not arise in Model 2); consequently, we leave these awkward details to the appendices.

### (a) Dirichlet boundary conditions

In the case of Dirichlet boundary conditions, the leading-order behaviour of the boundary layer solution *Φ*(*N*,*S*;*s*) with linear behaviour as *Φ*^{±}(*N*,*S*) satisfy the following canonical cell problems (cf. figure 1*b*):

For any given reference wire shape *K* and scaled radius *δ*, one must solve (3.8)–(3.11), either analytically or numerically, to determine the far-field constants *σ*_{±} and *τ*_{±}; some specific examples are studied in appendix B. We note that if *K* is symmetric in *ξ* (so that the scaled wire *N*, cf. figure 1) then
*δ*≪1 the scaled wire *K*-dependent constant *a*_{0} satisfies *ψ* is the unique solution of Laplace's equation in *ψ*=0 on ∂*K* and *K*, *c*(*K*), by *K* the unit disc, *a*_{0}=0; for *K* a line segment of length 2,

Having extracted the far-field constants *σ*_{±},*τ*_{±} from the solutions of (3.8)–(3.11), matching the linear behaviour of (3.7) with that of (3.3) gives
*σ*_{±},*τ*_{±}, which depend on the size of *δ* (e.g. figure 9). There are essentially three different regimes to consider.

#### (i) Thick wires ( δ = O ( 1 ) )

If *δ* is strictly *σ*_{±},*τ*_{±} are *Γ*. At

#### (ii) Thin wires (*δ*≪1)

If *δ*≪1 then *σ*_{±},*τ*_{±}≫1 (cf. (3.14)). In particular, there is a distinguished scaling in which *δ* to be exponentially small with respect to 1/*ε*, i.e. *c*>0. (This is essentially the same scaling as that considered in [9,11,12] in a related context.) Suppose that we are in this regime, with *δ*∼*Ae*^{−c/ε}, then *ϕ*_{0} is continuous across *Γ* (i.e. *ϕ*_{0}+*εϕ*_{1} is also continuous across *Γ* and satisfies a similar condition

where *α* in terms of *δ* as
*a*_{0}=0) agrees with the effective boundary condition derived in [3, §C]. Note that (3.22) is valid for the two-term approximation *ϕ*_{0}+*εϕ*_{1}; hence in this distinguished scaling, the boundary condition derived in [3, §C] gives the solution correct to *δ* is not particularly small. We also note, however, that as *δ* increases, there may (depending on the value of *a*_{0}) come a point at which *α* blows up to infinity; precisely, this occurs at the critical value *α* is negative and the resulting outer problem may be ill-posed (see later). But of course for such large values of *δ* we are outside of this ‘thin-wire’ regime and the conditions (3.18)–(3.20) should be used instead of (3.22).

### (b) Neumann boundary conditions

In the case of Neumann boundary conditions, the requirement of linearity as *Ψ*(*N*,*S*) satisfies the canonical cell problem
*λ* is determined as part of the solution. This problem also appears elsewhere in acoustics and fluid flow; it is sometimes referred to as a ‘blockage problem’, and the constant *λ* as a ‘blockage coefficient’ [20–22]. Example solutions for *Ψ*(*N*,*S*) and *λ* are given in appendix B.

Matching linear terms between (3.3) and (3.24) gives that
*Γ*. Matching constant terms then gives

As in the Dirichlet case, to interpret (3.30) we must consider the magnitude of *λ*, which depends on both *K* and *δ*. The interesting limit in which *λ* is large is now not *δ*→0, but rather *δ* for which *K* a disc.) When *λ*≫1. We consider separately the cases *λ*≫1/*ε*.

#### (iv) Large gaps ( δ max − δ = O ( 1 ) )

In this case *ϕ*_{0} and its normal derivative are continuous across *Γ*. Hence the leading-order outer solution is just the free field solution of (2.1) or (2.5), and there is no shielding.

#### (v) Small gaps ( δ max − δ ≪ 1 )

In this case *λ*≫1. We first consider the case *c*>0 (see appendix B). Matching the constant terms then gives
*ϕ*_{0}] =

For completeness, we quote the higher order matching conditions, obtained using the results in appendix A

Rather than embarking on a detailed study of different cases, we concentrate on the case that is perhaps of most interest for this small-gap situation; namely, when the wires form a perforated shell around *Γ* (cf. figure 2*c*). This corresponds to tangential line segments (i.e. *K*=[−*i*,*i*]) under Model 2, for which we find

#### (vi) Very small gaps ( δ max − δ ≪ 1 )

In the case that *B*_{1}(*s*)=0. Thus, (3.29) gives
*Γ*. Continuing the expansion for the perforated shell, and supposing

## 4. Shielding performance of Faraday cages

Having derived homogenized boundary conditions for the leading-order outer approximations, we now consider their shielding performance in the context of the boundary-value problems introduced in §2, concentrating on the case when the source function *f* is compactly supported outside of the cage, in *D*∩*Ω*_{+}. For the Laplace problems, the measure of good shielding is that ∇*ϕ* should be small inside the cage interior *Ω*_{−} (since the physical field of interest is the gradient of the potential). For the Helmholtz problems, we require *ϕ* itself to be small in *Ω*_{−}.

We shall illustrate our general results using explicit solutions for the special case where *Γ* is the unit circle and the external forcing is due to a point source of unit strength located at a point *z*_{0} outside the cage (|*z*_{0}|>1). Explicitly, *f*=−*δ*_{z0}, where *δ*_{z0} represents a delta function supported at *z*_{0}. For this example, we express solutions in standard polar coordinates (*ρ*,*θ*) centred at the cage centre, with *θ*=0 corresponding to the direction of the source. We compare the homogenized solutions with numerical solutions to the full problem in the case of disc-shaped or line-segment wires (using Model 1 to define the wire geometry). For disc-shaped wires, these are computed using the same method as [3, appendix A]; the solution is expressed as a truncated sum of radially symmetric solutions to the Laplace or Helmholtz equation centred on the wire centres *z*_{j}; the coefficients in the expansion are determined by a least-squares fit to the boundary conditions at discrete points on the wires. For Laplace problems, solutions for line-segment wires can be computed using a similar method (by conformal mapping; cf. [23]), although our results for this case are computed with a boundary integral equation method using `SingularIntegralEquations.jl`, a Julia package for solving singular integral equations implementing the spectral method of [24].

### (a) Laplace equation with Dirichlet boundary conditions on wires

In the case of thin wires (*δ*≪1), the *δ*<*e*^{−a0}/2*π*.

For *Γ* the unit circle and *f*=−*δ*_{z0}, the leading-order solution inside the cage is
*α*≫1, in which case |∇*ϕ*^{−}(0)|∼1/(*απ*|*z*_{0}|). Recalling the definition of *α* in (3.23), the field inside the cage scales inverse linearly in *M* and logarithmically in *r*, as discussed in [3].

In the case of thick wires (*Γ* is a closed curve one deduces that
*Ω*_{−} is the *τ*_{+} (not *σ*_{+}, *σ*_{−} or *τ*_{−}) appears in this condition for the leading-order interior solution. The field in *Ω*_{−} is therefore *ε*→0.

For *Γ* the unit circle and *f*=−*δ*_{z0}, the leading-order solution inside the cage is

In figure 3, we show the excellent agreement between these approximations and the result of numerical calculations. Note that (4.4) and (4.9) are consistent, since *τ*_{+}∼1/*εα* as *δ*→0.

### (b) Helmholtz equation with Dirichlet boundary conditions on wires

In the thin wire case, the analysis is similar to that for the Laplace case, with

For *Γ* the unit circle and *f*=−*δ*_{z0}, the leading-order solution inside the cage is
*α*≫1.

In the thick wire case, at first glance the analysis appears similar to the Laplace case, with the *k* for which *k*^{2} is a Dirichlet eigenvalue of −∇^{2} on *Ω*_{−}, at which one cannot infer from (4.5) that *Ω*_{−} is again provided by the

For *Γ* the unit circle and *f*=−*δ*_{z0}, the leading-order non-resonant solution inside the cage is
*e*_{m} are as above. In particular,

When one compares the approximations (4.12) and (4.14) with numerical simulations for fixed *k* away from resonance, one observes similar behaviour to that in figure 3, i.e. (4.12) is accurate for small *δ* and (4.14) for larger *δ*. However, interesting new behaviour become apparent when one fixes *δ* and varies the wavenumber *k*. Two plots of this type are presented in figure 4. One finds that close to resonant wavenumbers the numerical solution is strongly peaked, and the amplitude |*ϕ*(0)| can actually exceed that of the free-field solution; that is, the cage *amplifies* the field rather than shielding from it. This amplification is clear in the near-resonant field plots in figure 5.

Returning to figure 4, we note that the position of the peak amplitude is in general slightly shifted from the exact resonance. For sufficiently small *δ* (cf. figure 4*a*), the peaks are captured correctly by the ‘thin-wire’ asymptotic result. But for larger *δ*, the position and height of the peak are not predicted correctly (cf. figure 4*b*). Unfortunately, the ‘thick-wire’ approximation (4.13) cannot capture the peaks either—the *δ*.

### (c) Resonance effects

Close to resonant wavenumbers, our thick-wire

To examine the behaviour close to resonance, let *k*_{*}>0 is a resonant wavenumber with real-valued eigenmode *ψ* satisfying *Ω*_{−} and *ψ*=0 on *Γ*, and *k*_{*}; more generally, we would have a superposition of eigenmodes). Expanding (2.5) with *C*_{0} to be determined. By (3.20), the next-order interior problem is
*k*_{*}. Since the associated homogeneous problem has a non-zero solution, *ψ*, there is a solvability condition to be satisfied, following from the identity
*I*_{3} is for later use), the solvability condition arising from (4.20) is that
*C*_{0} of the *C*_{0} blows up to infinity. This represents a shift in the position of the apparent resonance from the original value *k*_{*} to the perturbed value *K* (through *σ*_{−}) and on the cage geometry *Γ* (through *I*_{1} and *I*_{2}). Furthermore, we note that the sign of the shift is given by the sign of −*σ*_{−}. For line segment wires parallel to *Γ*, *σ*_{−} is positive for all *δ* at which *σ*_{−} (and hence the shift) changes sign. For circular wires, this occurs at *δ*≈0.12 (cf. figure 9).

The true solution is not actually infinite at the shifted value

The leading-order interior problem for *C*_{−1} is to be determined. This large interior solution causes a change to the leading-order exterior problem, for which the boundary condition (from (4.26)) becomes
*Ω*_{+} with *Γ* and (2.3) at infinity, and *Ω*_{+} with *Γ* and

The *C*_{0} replaced with *C*_{−1}. This holds identically, given the definition of *C*_{−1} remains undetermined at this order. Writing the solution to (4.31) and (4.32) as
*Ω*_{+} with *Γ*, and *C*_{0} is arbitrary, the *C*_{0} terms cancel due to (4.23), and the term proportional to *Γ* is a closed loop. Noting that *I*_{4} and *I*_{7} are in general complex, whereas *I*_{1}, *I*_{3}, *I*_{5} and *I*_{6} are real, the condition (4.36) determines *C*_{−1} with

where

From (4.38), it follows that the maximum of |*C*_{−1}| is |*A*| at

The good agreement between these predictions and the result of numerical calculations is shown in figures 5–7. The insets in figure 6 demonstrates that the shape of the amplitude variation with wavenumber near the resonance is well captured, and figure 7 demonstrates how the position and amplitude at the peak vary with *ε*. We emphasize that as the number of wires increases, the resonant response occurs closer to the unperturbed resonant modes of *Ω*_{−}, over an increasingly narrow band of wavenumbers, but with an increasingly large amplitude.

### (d) Neumann solutions and resonance effects

For the equivalent problems satisfying Neumann conditions on the wires, we have seen in §3 that there is in general much weaker shielding than for Dirichlet conditions. Unless the gaps between the wires are small, the leading-order homogenized solution does not notice the wires at all, and even for small gaps the homogenized wires provide a jump condition on *Γ* that does not necessarily lead to a weak field inside the cage. Only in the case of ‘very small gaps’ is there a significant shielding effect. Although this is not the main focus of our study (requiring very small gaps largely defeats the idea of a Faraday cage), we touch briefly on this very small gap case because of its analogy to the Dirichlet problems above. In particular, we focus on the perforated shell introduced in §3, for which the homogenized boundary conditions are (3.37) and (3.38), which depend on

For the Laplace problem, the *Γ*. The interior solution *Γ*. This determines the constant *Γ*. The correction, which controls the size of |∇*ϕ*(0)|, is

For the Helmholtz problem, the *Γ*. Away from resonance, the interior solution is *k* is close to a resonant wavenumber *k*_{*} for which there is a non-zero solution *ψ* to *Ω*_{−} with ∂*ψ*/∂*n*=0 on *Γ*. The resonant case can be analysed in an equivalent fashion to the Dirichlet problem. Without giving the details, we find that the wavenumber is shifted to *I*_{1} and *I*_{2} are as defined in (4.21) for the relevant eigenfunction, while the peak amplitude at the origin is

## 5. Discussion and conclusion

We have derived homogenized boundary conditions for various instances of the two-dimensional Faraday cage problem, helping to quantify the effect of a wire mesh on electrostatic and electromagnetic shielding in the limit as the number of wires tends to infinity. We have given an overview in §3 of the different leading-order behaviour that can occur depending on the scaled wire size *δ*, extending previous results for the ‘thin-wire’ regime *δ*≪1, and incorporating the effects of finite wire size that in general allow for better shielding. The homogenized conditions help to clarify how the wire geometry affects the shielding behaviour, through the solution of cell problems and extraction of far-field constants. This allows us to make some general comments on the shielding efficiency of different wires. For brevity, we focus our discussion mainly on the case of Dirichlet boundary conditions.

In the Dirichlet case, we showed that when the exterior wave field is *ε*=|*Γ*|/*M* and *τ*_{+} encodes the wire geometry. For thin wires, we established the approximation (3.14) for *τ*_{+}, which indicates that the logarithmic capacity of the wires (controlled by their size and shape) is the key property governing shielding. For thicker wires, the orientation of the wires also becomes important, and the parameter *τ*_{+} can become small when the gap between wires is small. In this regime, the relationship between the gap thickness (expressed as a fraction of the length of *Γ*) and the size of *τ*_{+} is strongly dependent on the wire shape. For example, *τ*_{+}=0.01 is achieved with a gap thickness of approximately 0.22 for tangential line segments, but as much as 0.54 for circular wires, and 0.61 for square wires. (For perpendicular line segments, the gap thickness is always 1, but a wire length of 2*δ*≈1.12 is required to achieve a correspondingly small value of *τ*_{+}).

We also derived a model for resonance effects in Faraday cages, showing how the incident exterior wave field can be amplified by the presence of the cage in a narrow range of wavenumbers close to (but not centred on) the resonant wavenumbers for the corresponding solid shell. The analysis showed that at its peak this resonance gives rise to a wave field

A similar analysis applies for a source inside the cage, when it is desired to shield the exterior region (as for a microwave oven, for example). In that case, for the ‘thick-wire’ regime, away from resonance the interior solution is *c* can be followed, with the same result except that (4.38) gives the amplitude of the *τ*_{+}*I*_{7} in (4.39) is replaced with

Although our homogenized boundary conditions were derived for smooth cages *Γ*, applying the resulting models to non-smooth geometries appears to give reasonable results, at least in terms of computing resonance shifts and amplitudes. As an example of both this, and the interior source, we consider a cage of circular wires arranged on a unit square, with a point source located inside the cage at *z*=−0.5. Numerical solutions illustrating the resonance effects are shown in figure 8.

The unperturbed resonances for this problem are *k*_{*}=(*π*/2)(*l*^{2}+*m*^{2})^{1/2}, *I*_{4} and *I*_{8}. For the first resonance (*l*=*m*=1), numerical solution of the relevant exterior problem for `MPSpack` software package, which implements the non-polynomial finite-element method of Barnett & Betcke [25]) gives *I*_{4}≈3.00−16.02*i*, while

Our analysis of the Neumann problem shows that, as one might expect, Neumann wires shield much less effectively than Dirichlet wires of the same size and shape. For the acoustic problem, this implies that it is very difficult to shield noise using a mesh-like structure made of sound-hard material unless the gaps are very small. The implication for the electromagnetic problem is that a cage of parallel wires may provide reasonable shielding of waves whose electric field is polarized parallel to the wire axes, but will not shield waves whose electric field is polarized perpendicular to the wires axes. This effect is the basis of many polarizing filters, and explains, at least intuitively, why the mesh in the doors of microwave ovens is made of a criss-cross wire pattern or a perforated sheet, rather than from parallel wires aligned in a single direction. In principle, homogenized boundary conditions for cage problems in the full three-dimensional electromagnetic case could be derived using the techniques used in this paper, but we leave this for future work.

## Data accessibility

This manuscript has no data.

## Authors' contributions

Both authors contributed equally in formulating, carrying out and writing up the results of this research. The final version has been approved by both authors for publication.

## Competing interests

We have no competing interests.

## Funding

I.J.H. is supported by a Marie Curie FP7 Career Integration Grant within the 7th European Union Framework Programme.

## Acknowledgements

The authors gratefully acknowledge helpful discussions with Nick Trefethen, Jon Chapman, Mikael Slevinsky and Alex Barnett.

## Appendix A. Higher order boundary-layer expansions

In this section, we outline the derivation of the two-term boundary-layer expansion

For Model 1, the curvature of *Γ* complicates matters somewhat. *A priori* the domain is *N*,*S*) coordinates, and the boundary conditions are to be imposed on *ε* and the local curvature of *Γ*. This is undesirable, and it is preferable to solve cell problems on the fixed cell domain *Φ*(*N*,*S*;*s*), as in the main text, the change from

To do this, note that the relationship *N*,*S*) as
*ε*→0, where *κ*=*κ*(*s*) is the local curvature of *Γ*. We suppose for definiteness that the boundary of the reference wire shape, ∂*K*, is smooth and is given by *W*(*ξ*,*η*)=0 in Cartesian coordinates (*ξ*,*η*) (cf. figure 1*c*). The boundary *W*(*N*/*δ*,*S*/*δ*)=0 (this is the actual wire boundary under Model 2), while *Φ*(*N*,*S*)=0 on *κ*(*s*) *d*(*N*,*S*) is the normal perturbation of *ν*=(*ν*_{N},*ν*_{S})=∇*W*/|∇*W*| is the outward unit normal to

A more involved calculation shows that the Neumann condition ∂*Φ*/∂*ν*(*N*,*S*)=0 on *ν*^{⊥}=(−*ν*_{S},*ν*_{N}) is the (counterclockwise) unit tangent vector on *N*,*S*). (The normal to

As a concrete example, consider the circular disc, when *W*(*ξ*,*η*)=*ξ*^{2}+*η*^{2}−1. Parametrizing *ϑ*∈[0,2*π*) gives

To summarize, the boundary-layer problems for Model 1 are given by (A 2) with periodic boundary conditions (3.6), one of (A 4) or (A 6), and matching conditions as *d* and

**(a) Dirichlet problem**

For the Dirichlet problem, the leading-order solution has the general form
*Φ*^{+} and *Φ*^{−} are the canonical solutions defined earlier in (3.8)–(3.11).

The *Φ*^{±}. The problem for *K* is symmetric in *ξ*, then from (3.12) it follows that

The problem for *K* is symmetric in *ξ*, then

The far-field behaviour as

**(b) Neumann problem**

For the Neumann problem, the *Ψ* is the solution of the canonical problem defined in (3.25)–(3.28).

The *Ψ*(*N*,*S*), and *K* is symmetric in *ξ*, then *Ψ*(*N*,*S*)=−*Ψ*(−*N*,*S*), so that

The constant *Ψ*, by integrating (A 21) over

The problem for *K* is symmetric in *ξ* then

Finally, the problem for *K* is symmetric in *ξ* then

The far-field behaviour as

## Appendix B. Cell problem solutions

In this section, we present numerical and analytical solutions to the leading-order boundary layer cell problems for disc-shaped, perpendicular/tangential line segments and square wires.

**(a) Dirichlet problems**

Example solutions for the Dirichlet cell problem (3.8)–(3.11) are shown in figure 9, along with plots of the corresponding far-field constants *σ*=*σ*^{±} and *τ*=*τ*^{±}.

The solutions for circular wires in figure 9*a* are calculated numerically using linear finite elements, with the constants *σ* and *τ* found from a linear fit of the far-field behaviour. For small wires, *δ*→0, we recall the asymptotic behaviour (3.13) and (3.14). For *Φ*^{+}=0 for *X*<0 (so *τ*=0), while a numerical solution gives *σ*≈−0.44 for the constant as *σ* can be found by solving a perturbation problem numerically, from which we obtain *τ* is exponentially small. The solutions for the square wire in figure 9*c* were also computed numerically; we note that in this case

Solutions for the two arrangements of line segments can be found analytically by conformal mapping. For the wires arranged perpendicular to *Γ*, we obtain
*δ*→0 (so that in particular *τ*∼(1/*π*)*e*^{−2πδ} as *b*.

For the wires arranged tangentially along *Γ*, we obtain
*δ*→0 (so

**(b) Neumann problems**

Example solutions for the Neumann cell problem (3.25)–(3.28) are shown in figure 10.

The circular wire case is again calculated numerically, although the asymptotic behaviour for small and large circles provides a good fit over the whole range of *δ*. For *δ*→0, the solution away from the wire can be written approximately as
*λ*∼*πδ*^{2} as *δ*→0. (The strength of the singularity here is again determined by matching to an inner region close to the wire, as in [3, §B], where *Ψ*∼ℜ{*Z*+*δ*^{2}/*Z*}). We remark that the analysis in [22] provides a more refined approximation *λ*∼(*πδ*^{2})/(1−(*πδ*)^{2}/3), which is also plotted in figure 10. For

For line segments arranged perpendicular to *Γ*, the wire has no impact on the solution, which is simply *Ψ*(*N*,*S*)=*N*, so *λ*=0. For line segments arranged tangentially along *Γ*, conformal mapping yields
*δ*→0, and

## Footnotes

↵1 We assume that lengths have been non-dimensionalized relative to a suitable macro-lengthscale (e.g. the radius of the smallest circle containing

*Γ*) so that*ε*is a non-dimensional parameter.

- Received January 25, 2016.
- Accepted April 5, 2016.

- © 2016 The Authors.

Published by the Royal Society under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted use, provided the original author and source are credited.