F Feppon

H Ammari

SUBWAVELENGTH RESONANT ACOUSTIC SCATTERING IN FAST TIME-MODULATED MEDIA

Keywords: Acoustic scattering, time-modulated metamaterial, subwavelength resonances, high-contrast media, Sturm-Liouville eigenmodes AMS Subject classifications. 35B34, 35B40, 45M05, 35L05, 35B10

Cet article propose une analyse mathématique de la diffraction des onde acoustiques induite par une inclusion dont le paramètre physique de densité (i) présente un fort contraste avec le milieu extérieur permettant l'apparition de résonances sous-longueur d'onde et (ii) est modulé en temps, avec une fréquence de modulation largement supérieure à celle de l'onde incidente. Notre analyse montre qu'en général, l'effet de la modulation rapide est moyenné dans le temps et que le système se comporte comme un résonateur non modulé avec une densité effective apparente. Cependant, lorsqu'un réglage exceptionnel de la modulation permet une concordance entre une ou plusieurs valeurs propres temporelles d'un problème de Sturm-Liouville et des valeurs propres spatiales d'un problème de Neumann, l'onde incidente à basse fréquence devient soudainement capable d'exciter des modes à haute fréquence dans le résonateur. Ce phénomène permet la génération d'ondes transmises transportant des composantes hautes fréquences dans le champs lointain, ainsi que l'existence de modes sortant à croissance exponentielle en temps. Notre analyse permet de supposer que de tels systèmes modulés en temps pourraient servir de dispositif de rayonnement spontané ou de générateur d'harmoniques élevées.

Introduction

Time-modulated metamaterials [START_REF] Zangeneh-Nejad | Active times for acoustic metamaterials[END_REF][START_REF] Caloz | Spacetime metamaterials-part i: general concepts[END_REF][START_REF] Caloz | Ten applications of metamaterials[END_REF] have been receiving a lot of attention for the promising applications they offer in the design of wave devices. Modulating in time the physical parameters of a medium allows indeed to perform a variety of wave operations, such as temporal dispersion or frequency conversion [START_REF] Wu | Serrodyne frequency translation using time-modulated metasurfaces[END_REF][START_REF] Ramaccia | Temporal multilayer structures for designing higher-order transfer functions using time-varying metamaterials[END_REF][START_REF] Ramaccia | Propagation and scattering effects in temporal metastructures[END_REF][START_REF] Preble | Changing the colour of light in a silicon resonator[END_REF][START_REF] Gao | Theoretical model for manipulating light distribution in the time domain by using a spatially homogenous dynamic medium[END_REF], signal amplification [START_REF] Galiffi | Broadband nonreciprocal amplification in luminal metamaterials[END_REF][START_REF] Pacheco-Peña | Temporal metamaterials with gain and loss[END_REF], signal compression [START_REF] Chen | Enhanced acoustic sensing through wave compression and pressure amplification in anisotropic metamaterials[END_REF], spacetime cloaking [START_REF] Mccall | A spacetime cloak, or a history editor[END_REF], and nonreciprocal propagation [START_REF] Chen | Nonreciprocal wave propagation in a continuumbased metamaterial with space-time modulated resonators[END_REF][START_REF] Palermo | Surface wave non-reciprocity via time-modulated metamaterials[END_REF]. On the other hand, subwavelength resonators are known to be ideal constituents of spatial metamaterials or metasurfaces [START_REF] Liu | Locally resonant sonic materials[END_REF][START_REF] Lemoult | Wave propagation control at the deep subwavelength scale in metamaterials[END_REF][START_REF] Ma | Acoustic metamaterials: From local resonances to broad horizons[END_REF]: these can be achieved through high contrast inclusions which have the property of resonating with incident wavelengths much larger than the size of the resonators, thereby allowing to manipulate waves at subwavelength scales [START_REF] Schweizer | Resonance Meets Homogenization[END_REF][START_REF] Ma | Acoustic metamaterials: From local resonances to broad horizons[END_REF][START_REF] Cosmo | Acoustic metamaterials based on local resonances: homogenization, optimization and applications[END_REF][START_REF] Sheng | Locally resonant sonic materials[END_REF]. High-contrast metamaterials obtained by filling a homogeneous medium with many of such inclusions enable to achieve a variety of applications in photonics and phononics, such as negative index scattering [START_REF] Ammari | Double-negative acoustic metamaterials[END_REF], spatial cloaking [START_REF] Ammari | Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance[END_REF][START_REF] Kohn | A variational perspective on cloaking by anomalous localized resonance[END_REF], superfocusing [START_REF] Lanoy | Subwavelength focusing in bubbly media using broadband time reversal[END_REF][START_REF] Ammari | Sub-wavelength focusing of acoustic waves in bubbly media[END_REF], guiding [START_REF] Orazbayev | Quantitative robustness analysis of topological edge modes in C6 and valley-Hall metamaterial waveguides[END_REF][START_REF] Ammari | A High-Frequency Homogenization Approach Near the Dirac Points in Bubbly Honeycomb Crystals[END_REF][START_REF] Ammari | Topologically protected edge modes in one-dimensional chains of subwavelength resonators[END_REF], sensing [START_REF] Chen | Enhanced acoustic sensing through wave compression and pressure amplification in anisotropic metamaterials[END_REF][START_REF] Ammari | High-order exceptional points and enhanced sensing in subwavelength resonator arrays[END_REF] or superresolution imaging [START_REF] Pendry | Negative Refraction Makes a Perfect Lens[END_REF][START_REF] Kaina | Negative refractive index and acoustic superlens from multiple scattering in single negative metamaterials[END_REF][START_REF] Ammari | Super-resolution in high-contrast media[END_REF]. Quite many works have been proposed for modelling and understanding time-modulated media from physical or numerical studies, e.g. [START_REF] Caloz | Spacetime metamaterials-part ii: theory and applications[END_REF][START_REF] Caloz | Spacetime metamaterials-part i: general concepts[END_REF][START_REF] Caloz | Towards space-time metamaterials[END_REF][START_REF] Engheta | Metamaterials with high degrees of freedom: space, time, and more[END_REF][START_REF] Galiffi | Photonics of time-varying media[END_REF][START_REF] Galiffi | Broadband nonreciprocal amplification in luminal metamaterials[END_REF][START_REF] Huang | Modulating image oscillations in focusing by a metamaterial lens: Time-dependent green's function approach[END_REF][START_REF] Koutserimpas | Zero refractive index in time-floquet acoustic metamaterials[END_REF][START_REF] Pacheco-Peña | Temporal metamaterials with gain and loss[END_REF][START_REF] Pacheco-Peña | Effective medium concept in temporal metamaterials[END_REF][START_REF] Ptitcyn | Time-modulated meta-atoms[END_REF][START_REF] Ramaccia | Temporal multilayer structures for designing higher-order transfer functions using time-varying metamaterials[END_REF][START_REF] Ramaccia | Propagation and scattering effects in metastructures based on temporal metamaterials[END_REF][START_REF] Zangeneh-Nejad | Active times for acoustic metamaterials[END_REF][START_REF] Preble | Conversion of a signal wavelength in a dynamically tuned resonator[END_REF][START_REF] Yanik | Dynamic photonic structures: stopping, storage, and time reversal of light[END_REF]. There exist, however and to the best of our knowledge, very few works proposing a systematic mathematical analysis of wave propagation in temporal media. We can mention the work of Koutserimpas and Fleury [START_REF] Koutserimpas | Electromagnetic Waves in a Time Periodic Medium With Step-Varying Refractive Index[END_REF][START_REF] Koutserimpas | Zero refractive index in time-floquet acoustic metamaterials[END_REF] who propose an approach for understanding wave amplification, in periodic media only. Quite recently, Ammari and Hiltunen [START_REF] Ammari | Time-dependent high-contrast subwavelength resonators[END_REF] proposed an analysis for predicting the arising of subwavelength resonances in high contrast time-modulated phononic crystals, with the motivation of achieving non-reciprocal wave propagation [START_REF] Ammari | Non-reciprocal wave propagation in space-time modulated media[END_REF]. The authors assumed the frequency of the modulation to be small-of the same order of the propagating wave frequency-leading to band gaps asymmetric with respect to the origin, ω-gaps and folding effects. Still, the rigorous characterization of the resonances in this regime is delicate, and the derivation proposed in this work remains formal. In this case also, the authors assume that the medium is extended periodically and infinitely in the three space directions, which allows for simplifications thanks to the Floquet transform on the coordinate variables. Apart from these contributions, we are not aware of any work investigating wave scattering in an open medium due to a time-modulated obstacle.

In this work, we propose a rigorous mathematical analysis of wave scattering induced by a subwavelength resonator subjected to a fast periodic time-modulation of one of the physical parameters. By fast, we mean that the time modulation has a significantly larger frequency compared to the incident frequency. Although a fast time-modulation may be challenging to realize in a practical physical system, this setting is mathematically more amenable to a rigorous analysis than to the case where the time modulation has a frequency of the same order of the incident wave. Furthermore, it leads to interesting physical effects. More specifically, we consider the following acoustic scattering problem in a three-dimensional homogeneous medium, with a time-modulated and highly contrasted resonator D:

                           1 κ 0 ∂ 2 u ∂t 2 - 1 ρ 0 ∆u = 0 in R × R 3 \D, 1 κ r ∂ 2 u ∂t 2 - 1 ρ(t)ρ r ∆u = 0 in R × D, 1 ρ 0 ∂u ∂n + = 1 ρ r ρ(t) ∂u ∂n - on R × ∂D, u| + = u| -on R × ∂D, u -u in is outgoing, (1.1)
where n is the outward normal to D. The parameters ρ 0 and κ 0 denote respectively the density and bulk modulus of the homogeneous background medium. The resonator D is a simply connected smooth bounded domain (with C ∞ boundary); it is characterized by its bulk modulus κ r and a time-modulated density ρ r ρ(t) with reference density ρ r , where all these physical parameters are positive: ρ 0 > 0, ρ r > 0, ρ(t) > 0 for all t ∈ R and κ 0 > 0, κ r > 0.

The modulation t → ρ(t) is a T -periodic function whose frequency is denoted by Ω = 2π T .

We assume that u in is an incident plane wave, which is a time-harmonic solution to the wave equation in the free-space R 3 propagating with a given frequency ω > 0:

u in (t, x) = ûin (x)e -iωt , (1.2)
where ûin is solution to the Helmholtz equation:

∆ + ω 2 v 2 0 ûin = 0 in R 3 . (1.3)
In what follows, we denote by v 0 and v r the wave speeds

v 0 := κ 0 ρ 0 , v r := κ r ρ r ,
and by δ the ratio between the densities of both media:

δ := ρ r ρ 0 .
We consider the scattering problem (1.1) in the subwavelength and high contrast regimes:

ω → 0 and δ → 0.

(

On the other hand, the frequency Ω of the modulation ρ(t) is kept constant, so that Ω is large compared to the incident frequency (ω ≪ Ω), which corresponds to the fast modulation setting. In Section 4.3, we show that the solution u(t, x) to (1.1) is fully determined by considering a suitable outgoing radiation condition, where "outgoing" means that u -u in should be a function of t -|x|/v 0 at infinity. When there is no time-modulation, i.e. ρ(t) = 1, the system (1.1) models the scattering of sound waves by a small bubble in water. It is then known that the high contrast regime δ → 0 leads to the arising of subwavelength or the so-called Minnaert resonances [START_REF] Minnaert | On musical air-bubbles and the sounds of running water[END_REF][START_REF] Ammari | Minnaert resonances for acoustic waves in bubbly media[END_REF], corresponding to a strong amplification of the scattered field for incident frequencies close to the resonances. In the present study, we consider, for the simplicity of the derivation, only the case where only the density of the medium is modulated. The modulation of the parameter κ r could be taken into account up to a small adaptation described in Remark 4.1.

Given this context, the purpose of this article is to show the arising of a special kind of subwavelength resonances under some exceptional-yet feasible-conditions on the time modulation ρ(t). In general, the modulation has no effect at first order and the system behaves exactly as in the static case (Section 5.1.3). However, when a set of explicit conditions are fulfilled, a strong coupling between the low frequency incident field and the high frequency modulation arises, leading the resonator D to generate pulsed scattered waves with high frequency components (we illustrate this phenomenon on Figure 1). Furthermore, we find the existence of modes exponentially growing in time, which suggests that such a tuned system could serve as a spontaneously radiating device or for high harmonic generation [START_REF] Steinmeyer | A review of ultrafast optics and optoelectronics[END_REF][START_REF] Paul | Polarization-independent active metamaterial for high-frequency terahertz modulation[END_REF][START_REF] Chen | A metamaterial solid-state terahertz phase modulator[END_REF]. A possible application of acoustic metamaterials is the design of acoustic insulators, due to their ability to convert large (audible) wavelengths into small (inaudible) ones, which could then be more easily absorbed by classical materials adapted to small wavelengths. If these properties are expected to arise in time-modulated metamaterials [START_REF] Pacheco-Peña | Temporal metamaterials with gain and loss[END_REF][START_REF] Galiffi | Broadband nonreciprocal amplification in luminal metamaterials[END_REF][START_REF] Koutserimpas | Electromagnetic Waves in a Time Periodic Medium With Step-Varying Refractive Index[END_REF], our work is the first, to the best of our knowledge, to propose a rigorous mathematical analysis of such phenomena in open systems, and high-frequency time-modulated metamaterials. A significant advance is also to account for modulations ρ which can be arbitrarily rough (we assume only 1/ρ ∈ L ∞ per ((0, T ))), while [START_REF] Ammari | Time-dependent high-contrast subwavelength resonators[END_REF] assumed that ρ has a finite number of Fourier modes.

The conditions under which the exceptional resonant coupling arise can be formulated in terms of two eigenvalue problems naturally associated to (1.1). First, let us denote by 0 = λ 0 < λ 1 ≤ λ 2 ≤ . . . and by (ϕ l ) l∈N the eigenvalues and eigenfunctions of the Laplace operator with Neumann boundary conditions on ∂D:

   -∆ϕ l = λ l ϕ l in D, ∂ϕ l ∂n = 0 on ∂D, l ∈ N. (1.5) u in (t, x) D ρ(t)
u s (t, x)

Figure 1. High frequency coupling between a low frequency incident field and a subwavelength resonator D modulated by a high-frequency time modulation ρ(t). Upon a strong coupling between the resonator and the modulation (namely when Λ ̸ = {(0, 0)}, where Λ is given by (1.7)), the scattered field u s = u -u in carries high frequency components in the far field.

Second, consider the Sturm-Liouville eigenvalue problem associated to the modulation 1/ρ(t) with eigenvalues 0 = µ 0 < µ 1 ≤ µ 2 ≤ . . . and eigenfunctions (p m ) m∈N :

   - d 2 dt 2 p m (t) = µ m ρ(t) p m (t), p m is T -periodic, m ∈ N. (1.6)
Let us finally denote by Λ the set of indices associated to the eigenvalues λ l and µ m which are proportional by a factor 1/v 2 r : Λ := (m, l)

∈ N × N | λ l = µ m v 2 r . (1.7)
Since λ 0 = µ 0 /v 2 r = 0 (these eigenvalues are associated to constant functions in D and on (0, T )), the set Λ is not empty and contains the tuple (0, 0). It is reasonable to believe that generically, the coincidence of further eigenvalues is exceptional and that in most situations, Λ = {(0, 0)}. In that case, we show that the tuple (0, 0) is associated to the existence of a single subwavelength resonant mode that is approximately constant in time and space in the resonator D, which generates scattered waves devoid of high frequency components at first order.

However, under an appropriate tuning of the modulation ρ(t), the set Λ may contain a non-trivial pair of indices (m, l) ̸ = (0, 0). This can be achieved, for instance, upon a magnification of ρ(t) by a suitable constant multiplicative factor, since this would result in the same magnification for the eigenvalues (µ m ) m∈N . Then, this configuration leads to the arising of as many additional subwavelength resonant modes as the size of the set Λ, which are approximately equal to linear combinations of the functions (p m (t)ϕ l (x)) (l,m)∈Λ on R × D. More precisely, the result of Proposition 5.3 shows that there are 2#Λ resonant frequencies whose leading order asymptotics are given by

ω ± i (δ) ∼ ±v r δ 1 2 λ 1 2 i , 1 ≤ i ≤ #Λ, (1.8
) where #Λ denotes the number of elements of the set Λ, and where (λ i ) 1≤i≤#Λ are the eigenvalues of the following generalized eigenvalue problem with eigenvectors (a i ) 1≤i≤#Λ :

M a i + λ i Ga i = 0. The matrix M ≃ (M ml,m ′ l ′ ) (m,l),(m ′ ,l ′ )∈Λ
is determined by the Dirichlet-to-Neumann operator of the (timemodulated) acoustic scattering problem (defined in (5.11)). The matrix G = diag(γ m ) 1≤m≤#Λ is a diagonal matrix of coefficients (γ m ) 1≤m≤#Λ (equation (5.10)). In the exceptional coupling case Λ ̸ = {(0, 0)}, both matrices have no distinguished signs, so that the eigenvalues λ i are complex in all generality. Then, we find in Proposition 5.5 that at least in the case where D is the disk, one of the complex square roots ±λ i in (1.8) must have a strictly positive imaginary part, which is associated to the existence of some exponentially growing outgoing mode (i.e. a non-trivial solution to (1.1) with u in = 0).

Besides the analysis of these resonant phenomena, we identify in Corollary 5.1 a leading order approximation for the far field pattern of the scattered wave. After suitable rescalings and a Foldy-Lax approximation argument inspired from [START_REF] Ammari | Effective medium theory for acoustic waves in bubbly fluids near minnaert resonant frequency[END_REF][START_REF] Feppon | Homogenization of sound-absorbing and high-contrast acoustic metamaterials in subcritical regimes[END_REF], this allows us to discuss, at least at a formal level, the arising of an effective medium for the temporal metamaterial which would be constituted of many tiny copies of such time-modulated resonator D.

The paper outlines as follows. After describing our notation conventions in Section 2, our work starts in Section 3 with the introduction of a novel mathematical method for the study of subwavelength resonances in open media. Our approach relies on a variational formulation of (1.1) and the Dirichlet-to-Neumann operator associated to the Helmholtz equation in exterior domains. We expose our method on the unmodulated version of (1.1) in which ρ(t) = 1, which allows us to verify that we retrieve all the results of [START_REF] Ammari | Minnaert resonances for acoustic waves in bubbly media[END_REF][START_REF] Ammari | Wave interaction with subwavelength resonators[END_REF][START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF][START_REF] Ammari | Effective medium theory for acoustic waves in bubbly fluids near minnaert resonant frequency[END_REF][START_REF] Feppon | Homogenization of sound-absorbing and high-contrast acoustic metamaterials in subcritical regimes[END_REF] concerned with this setting, namely the leading asymptotic of the subwavelength resonant frequencies, point scatterer approximations, and a formal derivation of an effective medium theory for a system constituted of many small subwavelength resonators. Our novel approach leads to slightly simpler derivations because we do not rely on a layer potential representation, but is also more flexible in the sense that it could easily be applied in other dimensions (d = 1 or d = 2, left for a future work), and to the time-modulated case which is the object of the remainder of the paper.

We then focus on the scattering problem (1.1) with the time-dependent coefficient ρ(t). The formulation of an associated outgoing radiation condition for such time-dependent problem is non-standard and requires a particular study, which is the object of Section 4. We formulate an outgoing radiation condition which ensures the existence and uniqueness of a quasi-periodic, outgoing solution to the scattering problem (1.1). We define the Dirichlet-to-Neumann operator associated with such radiating solutions and we describe its main properties. We mention the existence of an associated time-modulated fundamental solution and of an associated layer potential theory which has resemblances with retarded potentials [START_REF] Sayas | Retarded Potentials and Time Domain Boundary Integral Equations[END_REF], and which enables us to compute far field asymptotic expansions for time-modulated outgoing waves.

In Section 5, we establish the arising of subwavelength resonances and we identify the leading order asymptotics of resonant frequencies for the time-modulated system (1.1), using the Dirichlet-to-Neumann approach introduced in Section 3. We show that there are exactly as many (complex) subwavelength resonant frequencies as twice the size of the set Λ. Assuming that the modulation is tuned in such a way that Λ = {(0, 0), (l, m)} for some (l, m) ̸ = (0, 0), we show how the nature of these resonances can be determined by the sign of the parameter γ m and a matrix (M lm,l ′ m ′ ) (l,m),(l ′ ,m ′ )∈Λ of coefficients determined by the Dirichlet-to-Neumann operator. Then, we derive a far field pattern approximation of the scattered field in the strong coupling tuning for which Λ = {(0, 0), (l, m)}.

In the conclusive Section 6, we discuss the arising of an effective medium theory for a temporal metamaterial that would be constituted of many small identical time-modulated resonators. We find that under some assumptions on the size s and number N of the resonators, the effective medium is governed by an integral equation of the second-kind with a direction and time-dependent kernel, and we identify under which scaling regimes exceptional effective properties are expected to occur. In the critical regime where the product sN of resonators converges to a positive constant, higher order reflections involve high frequency scattered waves which cannot be easily described by an effective medium theory. Following [START_REF] Ammari | Transient anomaly imaging by the acoustic radiation force[END_REF], we content ourselves to provide a homogenized equation for the low frequency truncation of the total wave field.

A final Appendix A gathers the definition and properties of the Bloch and Floquet transforms of tempered distributions, which would be the appropriate setting for studying time-modulated wave scattering induced by not necessarily time harmonic incident fields. These transforms are usually defined on L 2 spaces and are applied in the spatial domain to solution fields decaying at infinity. This setting is too restrictive for treating periodicity in the time-domain, hence our motivation for extending the Bloch and Floquet transform on the larger space of tempered distributions.

General setting and notation conventions

In all what follows, D ⊂ R 3 is a simply connected, smooth bounded domain. Its characteristic function is written 1 D , whose values are defined by

1 D (x) = 1 if x ∈ D, 0 if x ∈ R 3 \D.
(2.1)

Functional spaces

We write H s (D) and H s (∂D) the usual space of complex-valued functions with Sobolev exponent s ∈ R. For a given Hilbert space W , we denote by H s per ((0, T ), W ) the space of functions (t, x) → u(t, x) which are T -periodic with respect to the variable t (u(t + T, •) = u(t, •)) with u(t, •) ∈ W for almost every t ∈ R, and such that n∈Z

(1 + n 2 ) s 2 ||û n || 2 W < +∞,
where || • || W is the norm of W and (û n ) n∈Z denote the Fourier trigonometric coefficients of u:

u(t, x) = n∈Z ûn (x)e -inΩt with ûn (x) := 1 T T 0 u(t, x)e inΩt dt.
In a similar manner, we consider L ∞ per ((0, T ), W ) (resp. C ∞ per ((0, T ), W )) the space of bounded (resp. smooth) periodic functions with values in W . The dual space of a Hilbert space W is denoted by W ′ .

Throughout the paper, we consider the Hilbert space H := L 2 per ((0, T ), L 2 (D)), equipped with the inner product induced by the modulation ρ:

⟨u, v⟩ H := 1 T T 0 D 1 ρ(t) u(t, x)v(t, x) dx dt, (2.2)
and the Hilbert space

V := H 1 per ((0, T ), L 2 (D)) ∩ L 2 per ((0, T ), H 1 (D)) (2.3) equipped with the inner product ⟨u, v⟩ V := 1 T T 0 D u dx dt T 0 D v dx dt + T 0 D ∇u • ∇v + 1 ρ(t) ∂ t u∂ t v dx dt. (2.4)
We denote by ||u|| H := ⟨u, u⟩

1 2
H and ||u| V := ⟨u, u⟩

1 2
V the associated norms.

Layer potentials

For a given real number k ∈ R, S k D and K k * D denote respectively the single layer potential and the adjoint of the Neumann-Poincaré operator on D: for any ϕ ∈ H -1 2 (∂D),

S k D [ϕ](x) := ∂D Γ k (x -y)ϕ(y) dσ(y), x ∈ R 3 , (2.5)
K k * D [ϕ](x) := ∂D ∇ x Γ k (x -y) • n(x)ϕ(y) dσ(y), x ∈ ∂D, (2.6)
where

Γ k (x) := - e ik|x| 4π|x| (2.7)
is the fundamental solution to the Helmholtz equation and dσ is the surface measure of ∂D and the purely imaginary number is denoted by (a straight) i. Since D is assumed to be a domain with C ∞ boundary, and in particular of regularity C 1,α for some α > 0,

S k D maps H -1/2 (∂D) → H 1 2 (∂D) and K k * D : H -1 2 (∂D) → H -1 2 (∂D) is a compact operator. We recall that when k 2 is not a Dirichlet eigenvalue of D, S k D is invertible and its inverse is denoted by (S k D ) -1 : H 1 2 (∂D) → H -1 2 (
∂D) (see for instance [START_REF] Ammari | Layer Potential Techniques in Spectral Analysis[END_REF]). Due to the smoothness of k → Γ k , the applications k → S

-∆Φ = 0 in R 3 \D, Φ = 1 on ∂D, Φ(x) = O(|x| -1 ) as |x| → +∞, (2.8)
and we recall the definition of the capacity of D:

cap(D) := - ∂D ∂Φ ∂n dσ, (2.9)
where n is the outward normal to D. We denote by |D| the volume of D.

Remark 2.1. Throughout the paper, we use the same notation S k D to refer to both the single layer potential (2.5) defined on the whole space and the boundary integral operator

S k D : H -1 2 (∂D) → H 1 2 (∂D)
, the use of one or the other being determined by the context. However, the notation (S k D ) -1 is obviously reserved for the inverse of the latter.

A Dirichlet-to-Neumann approach to subwavelength resonances in the unmodulated case

In this section, we introduce a novel approach for analyzing subwavelength resonances based on the weak formulation of the wave scattering problem and the Dirichlet-to-Neumann operator. We illustrate this method by retrieving the results established with layer potential methods in [START_REF] Ammari | Minnaert resonances for acoustic waves in bubbly media[END_REF][START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF] for the static version of (1.1) in which the modulation is kept constant; namely ρ(t) = 1 for all t ∈ R. One advantage of the proposed Dirichletto-Neumann approach is that it can be rather easily extended to the time-modulated system (1.1) considered in the subsequent Sections 4 and 5. This Dirichlet-to-Neumann approach has also been successfully applied to study Minnaert resonances of one-dimensional (non-modulated) high contrast systems in [START_REF] Feppon | Subwavelength resonances in one-dimensional high-contrast acoustic media[END_REF].

Since we assume the time harmonic regime (1.2) for the incident field, the solution to (1.1) is itself timeharmonic and is given by

u(t, x) = û(x)e -iωt , (3.1)
where û solves the following system of coupled Helmholtz equations:

                                 ∆û + ω 2 v 2 0 û = 0 in R 3 \D, ∆û + ω 2 v 2 r û = 0 in D, u| + = u| -on ∂D, ∂ û ∂n - = δ ∂ û ∂n + on ∂D, ∂ |x| - iω v 0 (û -ûin ) = O(|x| -2 ) as |x| → +∞. (3.2)
The last equality is the outgoing Sommerfeld radiation condition, which ensures the uniqueness of the solution and that the scattered wave is outgoing [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]. The analysis of the amplitude response to (3.2) in the high-contrast regime δ → 0 by using integral representations of the solution is now quite established [START_REF] Ammari | Minnaert resonances for acoustic waves in bubbly media[END_REF][START_REF] Ammari | Mimicking the active cochlea with a fluid-coupled array of subwavelength Hopf resonators[END_REF][START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF]. The object of this section is to propose a slightly simpler characterization of the subwavelength resonances, by reformulating (3.2) in the domain D in terms of the Dirichlet-to-Neumann map associated to the Helmholtz equation in R 3 \D. This approach turns out to be quite flexible, and it can be generalized naturally for the analysis of the time-modulated system (1.1). This section is organized in four parts. We first recall the definition of the Dirichlet-to-Neumann map in Section 3.1 and we rewrite the scattering problem (3.2) in terms of this operator. We then provide an explicit characterization of the subwavelength resonances in Section 3.2 based on a variational formulation posed in the bounded domain D, and we compute their leading order asymptotic. Then, we explain how to retrieve a modal decomposition and a point scatterer approximation formula in Section 3.3. Sections 3.2 and 3.3 thus retrieve the results derived with layer potentials in [START_REF] Ammari | Minnaert resonances for acoustic waves in bubbly media[END_REF][START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF]. Finally, we outline in Section 3.4 the Foldy-Lax approximation argument which allows to derive an effective medium theory for a heterogeneous medium filled with many rescaled copies of such resonators, referring to [START_REF] Ammari | Effective medium theory for acoustic waves in bubbly fluids near minnaert resonant frequency[END_REF][START_REF] Feppon | Homogenization of sound-absorbing and high-contrast acoustic metamaterials in subcritical regimes[END_REF] for a more complete analysis.

Formulation of the scattering problem in terms of the Dirichlet-to-Neumann map

We start by recalling the definition of the Dirichlet-to-Neumann map [START_REF] Colton | Integral Equation Methods in Scattering Theory[END_REF][START_REF] Kress | Linear integral equations[END_REF][START_REF] Nédélec | Acoustic and electromagnetic equations: integral representations for harmonic problems[END_REF][START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF].

T k f := ∂w f ∂n + where      ∆w f + k 2 w f = 0 in R d \D, w f = f on ∂D, ∂ |x| -ik w f (x) = O(|x| -2 ) as |x| → +∞. (3.3)
Throughout this section, we assume that k = ω/v 0 with ω → 0. We have the following well-known result regarding the analyticity of T ω v 0 with respect to ω in a neighborhood of zero, in three dimensions (see e.g. [START_REF] Ammari | Full low-frequency asymptotics for the reduced wave equation[END_REF][START_REF] Ammari | Mathematical and Computational Methods in Photonics and Phononics[END_REF][START_REF] Ammari | Polarization and moment tensors: with applications to inverse problems and effective medium theory[END_REF], and also the proof of Proposition 4.6 below for the analyticity on the whole real line).

Proposition 3.1. The operator T ω v 0 : H 1 2 (∂D) → H -1
2 (∂D) is analytic with respect to ω ∈ C in a neighborhood of zero. In other words, there exist operators (T n ) n∈N such that T ω v 0 can be decomposed as the following convergent power series in the space of operators H 1 2 (∂D) → H -1 2 (∂D):

T ω v 0 = +∞ n=0 ω n v n 0 T n . (3.4)
Furthermore, the leading order asymptotic of T ω v 0 reads, as ω → 0:

T ω v 0 [ϕ] = T 0 [ϕ] + iω 4πv 0 ∂D ∂Φ ∂n ϕ dσ ∂Φ ∂n + O(ω 2 ),
where Φ is the solution to the exterior problem (2.8).

Proof. It is well-known that the Dirichlet-to-Neumann map can be expressed in terms of the single layer potential and the Neumann-Poincaré operator (see [START_REF] Ammari | Mathematical and Computational Methods in Photonics and Phononics[END_REF][START_REF] Ammari | Polarization and moment tensors: with applications to inverse problems and effective medium theory[END_REF]):

T ω v 0 = 1 2 I + K ω v 0 (S ω v 0 D ) -1 . (3.5)
The analyticity of T

T ω v 0 = 1 2 I + K * D + O(ω 2 ) S -1 D - ω v 0 S -1 D S D,1 S -1 D + O(ω 2 ) = T 0 - ω v 0 T 0 S D,1 S -1 D + O(ω 2 ),
where S D,1 is the operator S D,1 [ϕ] := -i 4π ∂D ϕ dσ (see e.g [START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF]Proposition 3.1]). Therefore, we find that

T ω v 0 [ϕ] = T 0 [ϕ] + iω 4πv 0 ∂D S -1 D [ϕ] dσ T 0 [1 D ] + O(ω 2 ).
The result follows since S -1 D is self-adjoint and S -1

D [1 ∂D ] = ∂Φ ∂n . □
Classically, the scattering problem (3.2) can be rewritten in terms of T ω v 0 as a PDE posed on the bounded domain D:

       ∆û + ω 2 v 2 r û = 0 in D, ∂ û ∂n = δT ω v 0 [û -ûin ] + δ ∂ ûin ∂n on ∂D. (3.6)
It can equivalently be rewritten as the following variational formulation: find û ∈ H 1 (D) such that for any v ∈ H 1 (D),

D ∇û • ∇v - ω 2 v 2 r ûv dx -δ ∂D T ω v 0 [û]v dσ = δ ∂D ∂ ûin ∂n -T ω v 0 [û in ] v dσ. (3.7)
The system (3.6) fully determines û inside D, while the value of û outside D can be obtained by solving an exterior Helmholtz problem. In the regime ω → 0, we have for instance the representation

û(x) = ûin (x) + S ω v 0 D [(S ω v 0 D ) -1 [û |∂D -ûin|∂D ]](x), ∀x ∈ R 3 \D, (3.8)
where the trace û|∂D is determined by (3.6).

The goal of this section is to analyse the arising of the resonances of (3.6), which are defined as the poles of the scattering operator ûin → û. Definition 3.2. We call subwavelength "resonance" a complex frequency ω ≡ ω(δ) ∈ C satisfying ω(δ) → 0 as δ → 0 and such that (3.6) admits a non-zero solution u(ω, δ) ∈ H 1 (D) for a zero right-hand side ûin = 0:

       ∆u(ω, δ) + ω 2 v 2 r u(ω, δ) = 0 in D, ∂u(ω, δ) ∂n = δT ω v 0 [u(ω, δ)] on ∂D.
(3.9) Equation (3.9) is an instance of a nonlinear eigenvalue problem [START_REF] Mennicken | Root functions, eigenvectors, associated vectors and the inverse of a holomorphic operator function[END_REF] in ω ∈ C. Setting δ = 0 in (3.6), we see that ω = 0 is a resonant frequency associated to the constant function u(0, 0) := 1 D , where 1 D is the characteristic function (2.1) of D. By using a continuation argument as in [START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF] or Gohberg-Sigal theory, [START_REF] Ammari | Mathematical and Computational Methods in Photonics and Phononics[END_REF], we find that it is possible to construct two resonant frequencies ω + (δ) and ω -(δ) of order O(δ 1 2 ) upon a suitable perturbation u(ω, δ) of u(0, 0). The problem of characterizing the resonances is therefore reduced to the analysis of the perturbation and the splitting of the nonlinear eigenvalue ω = 0 with respect to the parameter δ = 0.

It is possible, thanks to a simple calculation, to predict the leading order of the resonance and its connection with the capacity of D (assumed here to have a single connected component for simplicity), based on an argument inspired from [START_REF] Schweizer | Resonance Meets Homogenization[END_REF]. Setting v = 1 D in (3.7) for a zero right-hand side ûin ≡ 0 and û ≡ u(ω, δ) ≃ u(0, 0) = 1 D , we obtain in the regime δ, ω → 0:

0 ≃ - ω 2 v 2 r |D| -δ ∂D T ω v 0 [1 ∂D ] dσ ≃ - ω 2 v 2 r |D| -δ ∂D T 0 [1 ∂D ] dσ. (3.10)
Since by the definition (3.3) of T 0 , it holds

cap(D) = - ∂D T 0 [1 ∂D ] dσ,
we can predict from (3.10) that there are two resonant frequencies ω + (δ) and ω -(δ) satisfying

ω ± (δ) ≃ ±v r cap(D) |D| δ 1 2 ,
which is the result derived in [START_REF] Ammari | Minnaert resonances for acoustic waves in bubbly media[END_REF][START_REF] Ammari | Mimicking the active cochlea with a fluid-coupled array of subwavelength Hopf resonators[END_REF][START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF]. In the next subsections, we make this argument more systematic and we express the leading order asymptotic of the solution û to (3.6) for a positive real frequency ω close to the resonant value ω + (δ).

Characterization of the resonances

In what follows, we introduce the bilinear form a ω,δ defined for u, v ∈ H 1 (D) by

a ω,δ (u, v) := D ∇u • ∇v dx + D u dx D v dx - ω 2 v 2 r D uv dx -δ ∂D T ω v 0 [u]v dσ.
The bilinear form a ω,δ is obtained by adding the rank-one bilinear form (u, v) → D u dx D v dx to the left-hand side of (3.7). It is an analytic perturbation in ω and δ of the bilinear form a 0,0 defined by

a 0,0 (u, v) := D ∇u • ∇v dx + D u dx D v dx,
which is continuous coercive on H 1 (D) owing to the Poincaré-Wirtinger inequality.

From standard perturbation theory, it is clear that a ω,δ remains coercive for sufficiently small complex values of δ, ω. Hence, for any right hand-side f ∈ H 1 (D) ′ , there exists a unique Lax-Milgram solution u f (ω, δ) to the problem

a ω,δ (u f (ω, δ), v) = ⟨f, v⟩ H 1 (D) ′ ,H 1 (D) , for all v ∈ H 1 (D), (3.11
) which is analytic in ω and δ (see e.g. [START_REF] Ern | Theory and Practice of Finite Elements[END_REF]). In the context of (3.7), we shall consider

⟨f, v⟩ H 1 (D) ′ ,H 1 (D) := δ ∂D ∂ ûin ∂n -T ω v 0 [û in ] v dσ, for all v ∈ H 1 (D).
We now show that subwavelength resonant frequencies ω ± (δ) are the root of a single well-determined scalar equation.

Lemma 3.1. Let ω ∈ C and δ ∈ R belong to a neighborhood of zero. For any f ∈ H 1 (D) ′ , the variational problem

find u ∈ H 1 (D) such that ∀v ∈ H 1 (D), D ∇u • ∇v - ω 2 v 2 r uv dx -δ ∂D T ω v 0 [u]v dσ = ⟨f, v⟩ H 1 (D) ′ ,H 1 (D) , (3.12)
admits a unique solution u if and only if

D u 1 (ω, δ) dx ̸ = 1, (3.13)
where u 1 (ω, δ) ∈ H 1 (D) is the unique solution to the variational problem

a ω,δ (u 1 (ω, δ), v) = ⟨1 D , v⟩ H 1 (D) ′ ,H 1 (D) = D v dx, for all v ∈ H 1 (D). (3.14)
When (3.13) is satisfied, the solution to (3.12) reads

u ≡ u(ω, δ) = u f (ω, δ) + D u f (ω, δ) dx 1 -D u 1 (ω, δ) dx u 1 (ω, δ), (3.15)
where u f (ω, δ) is the solution to (3.11). When (3.13) is not satisfied, then u 1 (ω, δ) is a non-zero solution to (3.12) with f = 0 and ω ≡ ω(δ) is a subwavelength resonant frequency.

Proof. Clearly, (3.12) is equivalent to

a ω,δ (u, v) - D u dx D v dx = ⟨f, v⟩ H 1 (D) ′ ,H 1 (D) ⇔ a ω,δ (u, v) = a ω,δ (u f (ω, δ), v) + D u dx a ω,δ (u 1 (ω, δ), v), which implies u = u f (ω, δ) + D u dx u 1 (ω, δ).
Integrating on D, this equation has a solution given by (3.15) if and only if (3.13) is satisfied. □

In other words, subwavelength resonances ω ≡ ω(δ) are characterized by the equation

D u 1 (ω, δ) dx = 1, (3.16)
where u 1 (ω, δ) is the solution to (3.14), which also reads in strong form

       -∆u 1 (ω, δ) - ω 2 v 2 r u 1 (ω, δ) + D u 1 (ω, δ) dx 1 D = 1 D in D, ∂u 1 (ω, δ) ∂n = δT ω v 0 [u 1 (ω, δ)] on ∂D.
(3.17)

The existence of the complex resonant frequencies ω ≡ ω(δ) such that (3.16) is then guaranteed by applying the analytic implicit function theorem (see [61, Chapter 0]) to (3.16) as in [START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF]. In order to obtain an asymptotic expansion of ω(δ), we start by computing the leading order term in the asymptotic expansion of u 1 (ω, δ) as ω → 0 and δ → 0.

Proposition 3.2. The solution u 1 (ω, δ) to (3.17) has the following asymptotic behavior as ω, δ → 0:

u 1 (ω, δ) = 1 |D| + ω 2 v 2 r |D| 2 - cap(D) |D| 3 δ + iωδ cap(D) 2 4πv 0 |D| 3 1 D + 1 - iω cap(D) 4πv 0 δ u 0,1 + O(ω 2 + δ) 2 , (3.18)
where the remainder is estimated with the H 1 (D)-norm. The function u 0,1 is the unique solution to the following Laplace problem:

                 -∆ u 0,1 = cap(D) |D| 2 in D, ∂ u 0,1 ∂n = 1 |D| ∂Φ ∂n on ∂D, D u 0,1 dx = 0, (3.19)
where Φ is the solution to (2.8).

Proof. Since u 1 (ω, δ) is analytic in ω and δ, there exist functions (u p,k ) p,k≥0 such that the following series is convergent in H 1 (D):

u 1 (ω, δ) = +∞ p,k=0 ω p δ k u p,k . (3.20)
Inserting (3.4) into (3.14) and identifying powers of δ and ω, we obtain the following cascade of equations characterizing the functions (u p,k ) p,k≥0 :

         -∆u p,k + D u p,k dx 1 D = 1 v 2 r u p-2,k + 1 D δ p=0 δ k=0 in D, ∂u p,k ∂n = p n=0 1 v n 0 T n [u p-n,k-1 ],
where we assume that u p,k = 0 for negative indices p and k. It is easily obtained by induction that

u 2p,0 = 1 D v 2p r |D| p+1
and u 2p+1,0 = 0, for any p ≥ 0.

Then, for p = 0 and k = 1, we find that u 0,1 satisfies, for any v ∈ H 1 (D):

D ∇u 0,1 • ∇v dx + D u 0,1 dx D v dx = ∂D T 0 1 D |D| v dσ = 1 |D| ∂D ∂Φ ∂n v dσ. Setting v = 1 D , we find that D u 0,1 dx = -cap(D) |D| 2 and u 0,1 = - cap(D) |D| 3 1 D + u 0,1 ,
where u 0,1 is a function satisfying

D ∇ u 0,1 • ∇v dx = cap(D) |D| 2 D v dx + 1 |D| ∂D ∂Φ ∂n v dσ, ∀v ∈ H 1 (D),
and D u 0,1 dx = 0. Integrating by parts, we find that u 0,1 solves the Laplace problem (3.19).

Let us finally compute u 1,1 , that is the solution to

       -∆u 1,1 + D u 1,1 dx 1 D = 0 in D, ∂u 1,1 ∂n = T 0 [u 1,0 ] + 1 v 0 T 1 [u 0,0 ] = 1 v 0 |D| T 1 [1 D ].
Using the result of Proposition 3.1, we find

T 1 [1 D ] = -i cap(D)
4π ∂Φ ∂n , and then

u 1,1 = - i cap(D) 4πv 0 u 0,1 = i cap(D) 2 4πv 0 |D| 3 1 D - i cap(D) 4πv 0 u 0,1 .
Substituting these results in (3.20) yields the expansion (3.18). □ Remark 3.1. The compatibility condition required for the Neumann boundary value problem (3.19) to have a solution is satisfied due to (2.9).

Integrating (3.18) over D, we obtain the following asymptotic expansion for the equation (3.16) characterizing the resonant frequencies:

0 = D u 1 (ω, δ) dx -1 = ω 2 v 2 r |D| - cap(D) |D| 2 δ + iωδ cap(D) 2 4πv 0 |D| 2 + O(ω 2 + δ) 2 .
This equation has two solutions ω + (δ) and ω -(δ) which are analytic functions of δ 1 2 and whose leading asymptotic expansions are given by

ω ± (δ) = ± v r cap(D) 1 2 |D| 1 2 δ 1 2 - iv 2 r cap(D) 2 8πv 0 |D| δ + O(δ 3
2 ).

(3.21)

The asymptotic (3.21) of the subwavelength resonant frequencies coincide with the result derived from layer potential representations in [START_REF] Ammari | Minnaert resonances for acoustic waves in bubbly media[END_REF][START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF].

Modal decomposition and point scatterer approximation

We now show how a modal decomposition and a point-scatterer approximation can be derived for the solution û to (3.2) based on this Dirichlet-to-Neumann approach, which we shall reproduce later in Section 5.2 on the time-modulated system (1.1). In this part, we assume that the resonator D is centered around the origin.

We first need an asymptotic expansion of the solution u f (ω, δ) to the variational problem (3.11), which reads in strong form

       -∆u f (ω, δ) + D u f (ω, δ) dx 1 D - ω 2 v 2 r u f (ω, δ) = 0 in D, ∂u f (ω, δ) ∂n -δT ω v 0 [u f (ω, δ)] = δ ∂ ûin ∂n -δT ω v 0 [û in ] on ∂D. (3.22) Proposition 3.3. The solution u f (ω, δ) to (3.22
) admits the following leading order asymptotic expansion:

u f (ω, δ) = δ cap(D) |D| ûin (0)1 D -δ ûin (0)|D| u 0,1 + O(ωδ), (3.23)
where u 0,1 is the solution to (3.19).

Proof. Since the incident field solves the homogeneous Helmholtz equation (1.3) in R 3 , its Fourier transform is supported in the sphere of radius ω/v 0 . Therefore, there exists a function α :

S 2 → C such that ûin (x) = S 2 α(θ)e i ω v 0 θ•x dσ(θ).
From this expression, it is clear that we can expand the incident field in powers of ω as follows:

ûin (x) = +∞ p=0 S 2 α(θ) i p ω p p!v p 0 (θ • x) p dσ(θ) ≡ +∞ p=0 ω p u in,p (x),
where we remark that the functions u in,p satisfy

ω p u in,p (x) = 1 p! ∇ p ûin (0) • x p , ∇û in (x) = p≥1 ω p ∇u in,p (x).
Substituting ûin into (3.22), we write an ansatz u f (ω, δ) = p≥0,k≥1 ω p δ k v p,k which yields the following cascade of equations for (v p,k ) p≥0,k≥1 :

         -∆v p,k + D v p,k dx 1 D = 1 v 2 r v p-2,k in D, ∂v p,k ∂n = p n=0 1 v n 0 T n [v p-n,k-1 ] + ∇u in,p • n - p n=0 1 v n 0 T n [u in,p-n ] δ k=1 on ∂D,
where we assume that v p,k = 0 for k ≤ 0 and p < 0 by convention. Therefore, we find by using an integration by parts on D for k = 1 and p = 0 that

D v 0,1 dx = - ∂D ûin (0) ∂Φ ∂n dσ = cap(D)û in (0),
and hence v 0,1 can be decomposed as

v 0,1 = cap(D) |D| ûin (0)1 D + ûin (0) v 0,1 ,
with v 0,1 being the unique solution to

               -∆ v 0,1 = - cap(D) |D| 1 D in D, ∂ v 0,1 ∂n = - ∂Φ ∂n on ∂D, D v 0,1 = 0.
Remembering the definition (3.19), we find that v 0,1 = -|D| u 0,1 and the result follows. □

Coming back to the formula (3.15), we obtain the following modal approximation of the solution û inside the domain D. ), the solution û to (3.12) has the following asymptotic expansion in H 1 (D):

û ≡ û(ω, δ) = - ûin (0) ω 2 ω 2 M -1 + iω cap(D) 4πv0 (1 + O(δ 1 2 ))1 D ,
where

ω M := v r cap(D) |D| δ 1 2 . (3.24)
Proof. Inserting (3.18) and (3.23) into (3.15), we obtain:

û = O(δ) - δu in (0) cap(D) + O(δω) ω 2 v 2 r -cap(D) |D| δ + iωδ cap(D) 2 4πv0|D| + O((ω 2 + δ) 2 ) 1 D + O(ω 2 + δ) ,
which yields the result after using [START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF]Lemma 4.3] for bounding the error induced by approximating the denominator. □

Inserting this result in the representation (3.8) outside the resonator D, we obtain that D behaves in the far field as a point source with a resonant amplification coefficient. The following result was obtained in [START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF]Proposition 5.5] and in a slightly different form in [START_REF] Ammari | Minnaert resonances for acoustic waves in bubbly media[END_REF].

Corollary 3.2. The following far field approximation holds for the scattered field in the regime where ω is real and ω = O(δ

1 2 ): û(x) -ûin (x) = ûin (0) ω 2 ω 2 M -1 + iω cap(D) 4πv0 cap(D)(1 + O(δ 1 2 ) + O(|x| -1 ))Γ ω v 0 (x) as |x| → +∞, (3.25)
where we recall that Γ ω v 0 is the fundamental solution (2.7) to the Helmholtz equation in the background medium with wave number ω/v 0 .

Proof. Let us recall the following far field approximation for the single layer potential in the low frequency regime:

S ω v 0 D [ϕ](x) = ∂D ϕ dσ (1 + O(ω) + O(|x| -1 ))Γ ω v 0 (x).
The result follows from (3.8) with

ϕ = (S ω v 0 D ) -1 [û |∂D -ûin|∂D ] = - ûin (0) ω 2 ω 2 M -1 + iω cap(D) 4πv0 (1 + O(δ 1 2 ))(S D ) -1 [1 D ] + O(1). □ Remark 3.2. Since ω M = v r cap(D) |D| δ 1 2 = κ 1 2 r ρ 1 2 r cap(D) |D| ρ 1 2 r ρ 1 2 0 = κ 1 2 r ρ 1 2 0 cap(D) |D| ,
the leading order approximation of the scattered field (3.25) does not depend on the particular value of the density parameter ρ r , albeit this approximation is valid only when ρ r ≫ ρ 0 and κ r = O(ρ r ).

y 1 y 2 y N y 3 y i D s D N,s = ∪ N i=1 (y i + sD) U ûin ûN,s Figure 2.
Scattering of an incident wave ûin by a cloud of N copies of a high-contrast subwavelength resonator D rescaled by a small size factor s > 0 and located around centers (y i ) 1≤i≤N . The highly-contrasted medium is denoted by D N,s and is filling a bounded domain U .

Dilute regime and effective medium theory for high-contrast metamaterial

The point scatterer approximation (3.25) is a key ingredient for deriving an effective medium theory for a medium constituted of many small copies of the resonator D. In this part, we sketch the derivation of the homogenized equation for a medium containing N resonators rescaled by a factor s, based on [START_REF] Ammari | Effective medium theory for acoustic waves in bubbly fluids near minnaert resonant frequency[END_REF]. Our motivation is later to adapt this procedure in Section 5.2 in order to obtain, under some conditions, an effective medium theory for the time-modulated system. The procedure outlined in this article remains formal and is based on the well-established Foldy-Lax approximation of the effective medium [START_REF] Foldy | The multiple scattering of waves. I. General theory of isotropic scattering by randomly distributed scatterers[END_REF][START_REF] Figari | The point interaction approximation for diffusion in regions with many small holes[END_REF][START_REF] Ammari | Effective medium theory for acoustic waves in bubbly fluids near minnaert resonant frequency[END_REF]; we refer the reader to our recent work [START_REF] Feppon | Homogenization of sound-absorbing and high-contrast acoustic metamaterials in subcritical regimes[END_REF] for a rigorous justification with quantitative error estimates.

The setting is depicted on Figure 2: we consider a set D N,s of N copies of the inclusions D, rescaled by a small factor s > 0 and translated in the vicinity of centers (y i ) 1≤i≤N :

D N,s = 1≤i≤N (y i + sD).
The centers (y i ) 1≤i≤N are randomly and independently distributed in a bounded domain U ⊂ R 3 according to a probability density V dx (this density satisfies V ∈ L ∞ (U ), V ≥ 0 and U V dx = 1). In particular, the law of large number asserts that we have the convergence of the empirical measure in the sense of distributions:

1 N N i=1 δ yi → V dx as N → +∞.
We denote by ûN,s the solution to the scattering problem in the medium D N,s :

                                 ∆û N,s + ω 2 v 2 0 ûN,s = 0 in R 3 \D N,s , ∆û N,s + ω 2 v 2 r ûN,s = 0 in D N,s , ûN,s | + = u N,s | -on ∂D N,s , ∂ ûN,s ∂n - = δ ∂ ûN,s ∂n + on ∂D N,s , ∂ |x| - iω v 0 (û N,s -ûin ) = O(|x| -2 ) as |x| → +∞. (3.26) Recalling that cap(sD) = s cap(D), |sD| = s 3 |D|, (3.27)
we find that the subwavelength resonant frequency ω M (equation (3.24)) scales as

ω M = v r cap(D) |D| δ 1 2
s .

Hence, we assume that there exists some constant α > 0 such that the size factor s scales as

s ∼ αδ 1 2 ,
so that ω M converges to a determined frequency that is of order one. Then, the Foldy-Lax approximation states that in the medium D N,s , if the centers y i are "sufficiently far" from one another, then the resonators behave as a system of N distant point sources. Furthermore, the field scattered at the point y i should match the contribution of the other point sources according to the law given by point scatterer approximation formula (3.25). Using the rescaling (3.27), we expect therefore u N,s (y i ) ≃ z N,i where (z

N,i ) 1≤i≤N satisfies z N,i -ûin (y i ) = j̸ =i z N,j ω 2 ω 2 M -1 s cap(D)Γ ω v 0 (y i -y j ), 1 ≤ i ≤ N.
The latter equation is an algebraic system of N equations for the N unknowns (z N,i ) 1≤i≤N , called Foldy-Lax system [START_REF] Figari | Remarks on the point interaction approximation[END_REF][START_REF] Ammari | The equivalent media generated by bubbles of high contrasts: Volumetric metamaterials and metasurfaces[END_REF]. Using a law of large number result from [START_REF] Feppon | Analysis of a Monte-Carlo Nystrom method[END_REF], we can show the convergence of the solution (z N,i ) 1≤i≤N as N → +∞ to the values of the solution (û(y i )) 1≤i≤N of the integral equation

û(y) -sN cap(D) ω 2 ω 2 M -1 U Γ ω v 0 (y -y ′ )û(y ′ )V (y ′ ) dy ′ = ûin (y).
Left-multiplying this equation by the operator ∆ + ω 2 /v 2 0 , we finally obtain the following equation for the homogenized wave field û in the effective medium U :

           ∆û +   ω 2 v 2 0 -sN cap(D) ω 2 ω 2 M -1 V 1 U   û = 0 in R 3 , ∂ |x| - iω v 0 (û -ûin )(x) = O(|x| -2 ) as |x| → +∞.
(3.28)

Observe that the resonant denominator ω 2 /ω 2 M -1 can take negative or positive values depending on whether ω < ω M or ω > ω M . The heterogeneous medium behaves then respectively as a highly dispersive or a highly dissipative medium. We refer the reader to [START_REF] Feppon | Homogenization of sound-absorbing and high-contrast acoustic metamaterials in subcritical regimes[END_REF] for a formal analysis and a statement of assumptions on s → 0, N → +∞ and δ → 0 for which there is a rigorous convergence result ûN,s → û.

Outgoing waves in a periodically time-modulated medium

In the remainder of this paper, we consider the time-modulated scattering problem (1.1) with a non-constant time-modulation ρ ∈ L ∞ per ((0, T ), R). The goal of this section is to give a precise sense to "outgoing" for the scattered wave generated by the time-modulated resonator D. Inspired from the Floquet-Bloch decomposition [START_REF] Kuchment | Floquet Theory for Partial Differential Equations[END_REF][START_REF] Allaire | Quantum Transport[END_REF][START_REF] Ammari | Time-dependent high-contrast subwavelength resonators[END_REF], it is natural to seek a solution to (1.1) of the form

u(t, x) = e -iωt û(t, x), (4.1)
where the function (t, x) → û(t, x) is T -periodic in time. The ansatz (4.1) generalizes the time-harmonic regime (3.1) to the periodically modulated case; this form is peculiar to the time-harmonic assumption (1.2) on the incident field but it would generalize to arbitrary incident waves by using an appropriate Floquet-Bloch decomposition in time (see the Appendix A for a mathematical setting and Appendix A.4 for a formal justification of the ansatz (4.1) based on the Bloch transform). Inserting (4.1) into (1.1) leads us to consider the following time-dependent Helmholtz equation for û(t, x):

                                 1 v 2 0 (-iω + ∂ t ) 2 û(t, x) -∆û(t, x) = 0, (t, x) ∈ R × R 3 \D, 1 v 2 r (-iω + ∂ t ) 2 û(t, x) - 1 ρ(t) ∆û(t, x) = 0, (t, x) ∈ R × D, 1 ρ(t) ∂ û(t, x) ∂n - = δ ∂ û(t, x) ∂n + , (t, x) ∈ R × ∂D, û|+ (t, x) = û|-(t, x), (t, x) ∈ R × ∂D, t → û(t, x) is T -periodic, e -iωt (û(t, x) -ûin (t, x)) is outgoing, (4.2)
where we now need to clarify the meaning of the word "outgoing" in the last line of (4.2).

This section is organized as follows. Section 4.1 starts by stating the set of assumptions on the geometry of the resonator D and on the modulation ρ(t) which are considered in our analysis. In Section 4.2, we show that a suitable definition of "outgoing" for (4.2) is to require that all the Fourier modes of the T -periodic scattered wave be outgoing (in the usual sense), which translates into the following radiation condition for the modulated amplitude û(t, x):

∂ |x| - iω v 0 + 1 v 0 ∂ t u(t, x) = O(|x| -2 ) as |x| → +∞. (4.3)
This outgoing radiation condition is associated to a time-periodic Dirichlet-to-Neumann operator T ω per , for which we state the main properties. Relying on Fredholm's theory, we propose a well-posedness theory for the timemodulated wave equation (4.2) subjected to the radiation condition (4.3) in Section 4.3, in the regime where both the incident frequency ω and the contrast parameter δ > 0 are small. Finally, the last Section 4.4 defines periodic layer potentials S ω D,per and K ω * D,per from a suitable T -periodic Green function Γ ω per (t, x) satisfying the outgoing radiation condition (4.3). We prove a number of properties which we use in the subsequent Section 5.2.

Assumptions on the geometry of the resonator D and on the modulation ρ

In order to establish the well-posedness of the problem (4.2), we assume a set of conditions on the geometry of the resonator D and on the modulation ρ(t).

Nontrapping assumption on D and explicit high frequency bounds

The remainder of our analysis requires frequency explicit bounds on the exterior Dirichlet-to-Neumann map T k of the Helmholtz equation on the domain R 3 \D (Definition 3.1). We therefore assume the following hypothesis which brings substantial simplifications.

(H1) The resonator D is a smooth nontrapping set, more precisely R 3 \D is nontrapping in the sense of [25, Definition 1.1]:

• D has C ∞ boundary;
• for any R > sup x∈D |x|, there exists a T (R) < ∞ such that all the billiard trajectories (in the sense of Melrose-Sjöstrand [START_REF] Melrose | Singularities of boundary value problems. ii[END_REF])

that start in (R d \D) ∩ B R at time zero leave (R d \D) ∩ B R by time T (R).
The "nontrapping" condition physically states that incident rays illuminating and reflecting on D according to the law of geometric optics exit any bounded set in finite time, which is the case for instance if D is convex. This property implies the following convenient result obtained from resolvent estimates in [START_REF] Baskin | Sharp high-frequency estimates for the helmholtz equation and applications to boundary integral equations[END_REF][START_REF] Galkowski | Optimal constants in nontrapping resolvent estimates and applications in numerical analysis[END_REF].

     -k 2 -∆ w f,k = 0 in R 3 \D, w f,k = f on ∂D, ∂ |x| -ik w f,k = O(|x| -2 ) as |x| → +∞, (4.4) satisfies the bound ||∇w f,k || L 2 (B(0,R)\D) + |k| ||w f,k || L 2 (B(0,R)\D) ≤ C||f || H 1 2 (∂D) , (4.5)
with a constant C independent of k ∈ R and f . This result also entails that the Dirichlet-to-Neumann map T k of (3.3) is linearly bounded with respect to the frequency k ∈ R (see [START_REF] Chandler-Wilde | High-frequency bounds for the helmholtz equation under parabolic trapping and applications in numerical analysis[END_REF]Lemma 4.2]). Proposition 4.2. Assume (H1). The Dirichlet-to-Neumann map T k of Definition 3.1 satisfies the following frequency dependent bound:

||T k [f ]|| H -1 2 (∂D) ≤ C(|k| ||f || L 2 (∂D) + ||f || H 1 2 (∂D)
), (4.6)

for a constant C > 0 independent of f ∈ H 1 2 (
∂D) and k.

Note that the constant C > 0 is dependent on R > 0 since solutions to the Helmholtz equation (4.4) do not belong to H 1 (R 3 \D).

Non-coincidence of Dirichlet eigenvalues with the harmonic frequencies nΩ/v 0

We consider the following non-degeneracy condition in Section 4.4 when examining the invertibility of a periodic single layer potential:

(H2) inf λ∈σ dir (-∆),n∈Z nΩ v0 - √ λ > 0 where σ dir (-∆) is the Dirichlet spectrum of -∆.
This assumption implies in particular that the single layer potential

S ω+nΩ v 0 D : H -1 2 (∂D) → H 1 2 (
∂D) is an invertible operator for any n ∈ Z and ω sufficiently small. We now list the assumptions on the modulation ρ which we consider in all the remainder of this paper, and in particular in our well-posedness analysis of the time-modulated reduced wave equation (4.2). First, we assume that the modulation is T -periodic (ρ(t + T ) = ρ(t) for almost any t ∈ R), and is such that 1 ρ is a bounded function, positively bounded from below:

1 ρ ∈ L ∞ per ((0, T ), R) and 1 ρ(t)
> c for a.e. t ∈ R, for a constant c > 0.

In addition, we make some hypothesis on the kernel of the time-modulated wave operator with periodic and Neumann boundary conditions on (0, T ) × ∂D. Let us first introduce an analytic family of relevant Sturm-Liouville eigenvalues and eigenfunctions which extend those of the problem (1.6).

     --iω + d dt 2 p m (t; ω) = µ m (ω) ρ(t) p m (t; ω), p m (•; ω) is T -periodic, (4.7)
and which satisfy the following properties:

(i) the eigenvalues (µ m (ω)) are non-negative and ordered increasingly at ω = 0:

0 = µ 0 (0) < µ 1 (0) ≤ µ 2 (0) . . . ;
(ii) the sequence (p m (•; ω)) m∈N is an orthonormal basis of H 1 per ((0, T )) with respect to the weighted inner product induced by 1/ρ:

1 T T 0 1 ρ(t) p m (t; ω)p m ′ (t; ω) dt = δ mm ′ for any m, m ′ ∈ N; (4.8)
(iii) the eigenvalues (µ m (ω)) and the orthonormal sequence (p m (•; ω)) are analytic functions of ω.

Proof. This proposition is the consequence of classical perturbation theory for the analytic self-adjoint operator ω → -(-iω + d/ dt), see [START_REF] Kato | Perturbation Theory for Linear Operators[END_REF][START_REF] Baumgärtel | Analytic perturbation theory for matrices and operators[END_REF].

□
v ∈ H 1 per ((0, T ), H 1 (D)) to the problem              1 v 2 r ∂ tt - 1 ρ(t) ∆ v = 0, (t, x) ∈ R × D, 1 ρ(t) ∂v ∂n = 0, (t, x) ∈ R × ∂D, t → v(t, x) is T -periodic,
is the subspace generated by functions of the form

v(t, x) = p m (t)ϕ l (x), with (m, l) ∈ Λ,
where Λ is the set defined by (1.7), that is the set of tuples (m, l) associated to a Sturm-Liouville eigenvalue µ m and a Neumann eigenvalue satisfying (H3) The set Λ defined by (1.7) is finite and all the eigenvalues µ m or λ l with (m, l) ∈ Λ are simple. Furthermore, there exists a constant c > 0 such that inf

λ l = µ m /v 2 r . Proof. Decomposing v(t, x) := m∈N p m (t)u m (x), we find that      - µ m v 2 r u m -∆u m =
(m,l)∈N×N\Λ, |ω|≤c 1 - µ m (ω) λ l v 2 r > 0, (4.9)
where µ m (ω) is the Sturm-Liouville eigenvalue of (4.7).

The bound (4.9) requires that the Sturm-Liouville eigenvalues (µ m (ω)) and the Neumann eigenvalues (λ l (ω)) l∈N remain well separated for (m, l) / ∈ Λ and for |ω| < c. For instance, there are no subsequences (µ mn ) n∈N and (λ ln ) n∈N such that the difference |µ mn /v 2 r -λ ln | becomes arbitrarily small for subsequences of integers l n , m n → +∞. We expect that this assumption of non coincidence of more than a finite number of eigenvalues is generically satisfied: in fact, in general, we expect that Λ is reduced to the singleton {(0, 0)}.

Remark 4.1. It is possible to adapt the above hypotheses in order to take into account a time-modulated bulkmodulus κ r ≡ κ r κ(t) with κ(t + T ) = κ(t). The Sturm-Liouville eigenvalue problem (4.7) would need to be changed into

   --iω + d dt 1 κ(t) -iω + d dt p m (t; ω) = µ m (ω) ρ(t) p m (t; ω), p m (•; ω) is T -periodic.

Time-dependent Dirichlet-to-Neumann map and outgoing periodic waves

In this section, we identify a suitable outgoing radiation condition for the reduced T -periodic wave equation making it well-posed (Section 4.2.1), and we establish a number of properties for the resulting Dirichlet-to-Neumann map (in Section 4.2.2).

A time-periodic outgoing radiation condition

For a T -periodic boundary datum f ∈ L 2 per ((0, T ), H 1 2
(∂D)) and ω ∈ (-Ω/2, Ω/2), we consider the following time-dependent Helmholtz problem in the exterior domain R 3 \D:

         1 v 2 0 (-iω + ∂ t ) 2 w f (t, x) -∆w f (t, x) = 0, (t, x) ∈ R × R 3 \D, w f (t, x; ω) = f (t, x), (t, x) ∈ R × ∂D, t → w f (t, x; ω) is T -periodic. (4.10)
We denote by ( fn ) n∈Z the Fourier coefficients of f , namely

f (t, x) = n∈Z fn (x)e -inΩt , with ||f || 2 L 2 per ((0,T ),H 1 2 (∂D)) = n∈Z || fn || 2 H 1 2 (∂D)
< +∞.

The following lemma shows that one can discriminate an outgoing solution to (4.10) by requiring all the Fourier modes of w f to be outgoing.

Lemma 4.1. Assume (H1). For any f ∈ L 2 per ((0, T ), H 1 2 (∂D)), there exists a unique solution

w f ∈ H 1 per ((0, T ), L 2 loc (R 3 \D)) ∩ L 2 per ((0, T ), H 1 loc (R 3 \D)) to (4.10) whose Fourier expansion reads w f (t, x) = n∈Z ŵn (x)e -inΩt , (4.11)
and which is determined by the following cascade of Helmholtz equations for the coefficients ŵn :

             - 1 v 2 0 (ω + nΩ) 2 ŵn -∆ ŵn = 0 in R 3 \D, ŵn = fn on ∂D, ∂ |x| - i(ω + nΩ) v 0 ŵn = O(|x| -2
) as |x| → +∞, for any n ∈ Z.

(4.12)

Owing to the Sommerfeld radiation condition, w f is the unique solution such that ŵn (x)e -inΩt e -iωt is outgoing for any n ∈ Z, and it further satisfies the following continuity estimate on any ball B(0, R) of radius R > 0:

||w f || H 1 per ((0,T ),L 2 (B(0,R)) + ||w f || L 2 per ((0,T ),H 1 (B(0,R))) ≤ C||f || L 2 per ((0,T ),H 1 2 (∂D)) , (4.13)
for a constant C > 0 independent of f . Going back to the temporal domain, the cascade of radiation conditions of (4.12) can be rewritten in the following radiation condition for the solution w f (t, x) to (4.10):

∂ |x| - iω v 0 + 1 v 0 ∂ t w f (t, x) = O(|x| -2 ) as |x| → +∞. (4.14)
Proof. The fact that w f is a solution to (4.10) is clear by decomposing w f on its Fourier modes. The continuity estimate (4.13) is a consequence of the assumption (H1) and the bound (4.5), which implies that for any R > 0, there exists a constant C > 0 such that

||w|| 2 H 1 per ((0,T ),L 2 (B(0,R)) = n∈Z (1 + n 2 )|| ŵn || 2 L 2 (B(0,R)) ≤ C n∈Z || fn || 2 H 1 2 (∂D) ≤ C||f || 2 L 2 per ((0,T ),H 1 2 (∂D))
, and similarly, per ((0, T ), H -1 2 (∂D)) defined by

||w|| 2 L 2 per ((0,T ),H 1 (B(0,R)) = n∈Z || ŵn || 2 H 1 (B(0,R)) ≤ C n∈Z || fn || 2 H 1 2 (∂D) ≤ C||f || 2 L 2 per ((0,T ),H 1
T ω per [f ] = ∂w f ∂n , for any f ∈ L 2 per ((0, T ), H 1 2 (∂D)),
where w f is the unique solution to (4.10) equipped with the radiation condition (4.14):

                 1 v 2 0 (-iω + ∂ t ) 2 w f (t, x) -∆w f (t, x) = 0, (t, x) ∈ R × R 3 \D, w f (t, x; ω) = f (t, x), (t, x) ∈ R × ∂D, t → w f (t, x; ω) is T -periodic, ∂ |x| - iω v 0 + 1 v 0 ∂ t w f (x) = O(|x| -2 ) as |x| → +∞. (4.15)
As a direct consequence of the Fourier expansion (4.11), the periodic Dirichlet-to-Neumann map T ω per admits the following trigonometric series expansion for any f ∈ L 2 per ((0, T ), H 1 2 (∂D)):

T ω per [f ] = n∈Z T ω+nΩ v 0 [ fn ]e -inΩt , (4.16)
where ( fn ) n∈Z are the Fourier coefficients of f , and

T ω+nΩ v 0
is the exterior Dirichlet-to-Neumann operator of the Helmholtz equation with wave number (ω + nΩ)/v 0 defined in (3.3).

In what follows, we establish a number of useful properties of the T -periodic Dirichlet-to-Neumann map T ω per . The following property plays an important role in the well-posedness analysis of (4.2) in Section 4.3 below.

Proposition 4.5. The periodic Dirichlet-to-Neumann T ω per is a continuous operator from L 2 ((0, T ),

H 1 2 (∂T )) ∩ H 1 2 ((0, T ), L 2 (∂D)) into its dual: there exists a uniform constant C > 0 such that for any f ∈ L 2 ((0, T ), H 1 2 (∂T ))∩ H 1 2 ((0, T ), L 2 (∂D)), T 0 ∂D T ω per [f ]g dσ dt ≤ C||f || L 2 ((0,T ),H 1 2 (∂D))∩H 1 2 per ((0,T ),L 2 (∂D)) ||g|| H 1 2
per ((0,T ),L 2 (∂D))

.

Proof. Using the Plancherel identity and the bound (4.6), we obtain:

T 0 ∂D T ω per [f ]g dσ dt = T n∈Z ∂D T ω+nΩ v 0 [ fn ]g n dσ ≤ T C n∈Z (|n| || fn || L 2 (∂D) + || fn || H 1 2 (∂D) )||ĝ n || L 2 (∂D) ≤ C(||f || H 1 2 per ((0,T ),L 2 (∂D)) ||g|| H 1 2
per ((0,T ),L 2 (∂D))

+ ||f || L 2 per ((0,T ),H 1 2 (∂D) ||g|| L 2 ((0,T ),L 2 (∂D)) ). □ Remark 4.2. If (H2) holds, then T ω+nΩ v 0
can be rewritten for any ω ∈ R sufficiently small and n ∈ Z as

T ω+nΩ v 0 = 1 2 + K ω+nΩ v 0 * D (S ω+nΩ v 0 D ) -1 .
The next proposition will be needed in the analysis of resonances in Section 5.

Proposition 4.6. Assume (H1). T ω per is an analytic operator for ω in a neighborhood of zero: there exist operators (T k ) k∈N of L 2 per ((0, T ),

H 1 2 (∂D)) → H -1 per ((0, T ), H -1 2 (∂D)) such that T ω per = +∞ k=0 ω k v k 0 T per,k . (4.17)
In fact, in view of (4.16), the operators T per,k are given by

T per,k [f ] = n∈Z T nΩ v 0 k [f n ]e -inΩt ,
where (T

nΩ v 0
k ) k∈N denote the coefficients of the asymptotic expansion

T ω+nΩ v 0 = k∈N ω k v k 0 T nΩ v 0 k
of the Dirichletto-Neumann operator of the Helmholtz equation with wave number (ω + nΩ)/v 0 .

Proof. It is a classical fact that the exterior Dirichlet-to-Neumann map T k is analytic with respect to any wave number k ∈ R in dimension 3; for instance, T k can be expressed in terms of the inverse of the combined field layer potential, see [START_REF] Chandler-Wilde | Numerical-asymptotic boundary integral methods in high-frequency acoustic scattering[END_REF]Equation (2.85)]. This implies the analyticity of T ω+nΩ v 0

, hence of T ω per . □

The first term T per,0 ≡ T 0 per of the asymptotic expansion (4.17) of T ω per exhibits a symmetry property upon time-reversal. Proposition 4.7. Assume (H1) and (H2). Let f, g ∈ L 2 per ((0, T ),

H 1 2 (∂D)). It holds T 0 ∂D T 0 per [f ]g dσ dt = T 0 ∂D f • τ T 0 per [g • τ ] dσ dt, (4.18)
where τ (t) = -t is the time-reversal operation. In particular, T 0 per is a Hermitian operator when restricted to T -periodic functions which are invariant by time-reversal.

Proof. Denote by ( fn ) n∈Z and (ĝ n ) n∈Z the Fourier coefficients of f and g. From (4.16) and (4.17), the following trigonometric expansion holds for T 0 per [f ] ≡ T per,0 [f ]:

T 0 per [f ](x, t) = n∈Z 1 2 I + K nΩ v 0 * D (S nΩ v 0 ) -1 [ fn ](x)e -inΩt , t ∈ R, x ∈ ∂D.
Therefore, the Plancherel and the Calderón identity imply

T 0 ∂D T per,0 [f ]g dσ dt = T n∈Z ∂D 1 2 + K nΩ v 0 * D (S nΩ v 0 D ) -1 [ f n ] g n dσ = T n∈Z ∂D f n (S -nΩ v 0 D ) -1 1 2 + K -nΩ v 0 D [ g n ] dσ = T n∈Z ∂D f n 1 2 + K -nΩ v 0 * D (S -nΩ v 0 D ) -1 [ g n ] dσ = T n∈Z ∂D f -n 1 2 + K nΩ v 0 * D (S nΩ v 0 D ) -1 [ g -n ] dσ = T 0 ∂D f • τ T per,0 [g • τ ] dσ dt,
where we have denoted by K

-nΩ v 0 D the Neumann-Poincaré operator (whose K -nΩ v 0 * D
is the adjoint) and used the fact that ( f-n ) n∈Z and (ĝ -n ) n∈Z are the Fourier coefficients of f • τ and g • τ .

□

We conclude this part by stating a positivity property for T 0 per in the case of a spherical resonator D. Proposition 4.8. Assume (H1) and that D is a ball: D = B(0, R) for some R > 0. Then there exists a constant C > 0 such that for any f ∈ L 2 per ((0, T ),

H 1 2 (∂D)) ∩ H 1 2 per ((0, T ), H 1 2 (∂D)), -ℜ T 0 ∂D T 0 per [f ]f dσ dt ≥ C||f || 2 L 2 ((0,T ),L 2 (∂D)) . (4.19)
Proof. Let us recall the uniform negativity of the real part of the exterior Dirichlet-to-Neumann map T k with wave number k ∈ R on the ball, also known as the capacity operator: [77, Theorem 2.6.4] implies the existence of a constant C > 0 independent of k such that

-ℜ T 0 ∂D T k [ f ] f dσ dt ≥ C|| f || 2 L 2 ((0,T ),L 2 (∂D)) . (4.20)
Using the Plancherel identity, we therefore obtain Based on the properties of the time-periodic Dirichlet-to-Neumann operator T ω per established in the previous Section 4.2.2, we now provide an existence and uniqueness theory for the time-periodic reduced wave equation (4.2). Following Section 3, the system (4.2) can be rewritten as a partial differential equation posed on the bounded domain (0, T ) × D:

-ℜ T 0 ∂D T 0 per [f ]f dσ dt = -T n∈Z ℜ ∂D T 0,n [ fn ] fn dσ ≥ T C n∈Z || fn || 2 L 2 (∂D) ≥ C T 0 ∂D |f (t, x)| 2 dσ(x) dt.
             1 v 2 r (-iω + ∂ t ) 2 û(t, x) - 1 ρ(t) ∆û(t, x) = 0, (t, x) ∈ R × D, 1 ρ(t) ∂ û(t, x) ∂n -δT ω [û(t, x)] = δ ∂ ûin ∂n -T ω per [û in ] , (t, x) ∈ R × ∂D, t → û(t, x) is T -periodic. (4.21)
We recall the definition of the space V = H 1 per ((0, T ), L 2 (D)) ∩ L 2 per ((0, T ), H 1 (D)) (see Section 2.1) and we consider the following variational formulation for (4.21):

find u ∈ V such that ∀v ∈ V, a(u, v) = δ T 0 ∂D ∂ ûin ∂n -T ω per [û in ] v dσ dt, (4.22)
where a(u, v) is defined for any u, v ∈ V by

a(u, v) := T 0 D 1 ρ(t) ∇u • ∇v - 1 v 2 r [(-iω + ∂ t )u] (-iω + ∂ t )v dx dt -δ T 0 ∂D T ω per [u]v dσ dt. (4.23)
In order to prove that (4.21) is well-posed, we resort to Fredholm's theory [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF]. For this purpose, we introduce a 0,0 the bilinear form

a 0,0 (u, v) := T 0 D 1 ρ(t) ∇u • ∇v - 1 v 2 r ∂ t u∂ t v dx dt + (m,l)∈Λ ⟨u, p m ϕ l ⟩ H ⟨v, p m ϕ l ⟩ H , (4.24)
where we recall the definition (2.2) of the inner product on H = L 2 per ((0, T ), L 2 (D)), and where the assumption (H3) ensures that the set Λ is finite.

Remark that any u ∈ V can be decomposed as the convergent sum

u(t, x) = m,l∈N u ml p m (t)ϕ l (x) with ||u|| 2 V = |u 00 | 2 + m,l∈N (µ m + λ l )|u ml | 2 , (4.25)
where we have denoted u ml := ⟨u, p m ϕ l ⟩ H . In what follows, we denote by † : H → H the operator defined by

∀u ∈ H, u † := m,l∈N sign λ l - µ m v 2 r ⟨u, p m ϕ l ⟩ H p m ϕ l with sign(t) := 1 if t ≥ 0, -1 if t < 0.
Obviously, the operator † is an isometry of H and V , and it holds (u † ) † = u for any u ∈ H.

Proposition 4.9. Assume (H3). There exists a uniform constant c > 0 such that the following coercivity inequality holds:

∀u ∈ V, a 0,0 (u, u † ) ≥ c||u|| 2 V . (4.26)
Proof. By the definition of †, it holds

a 0,0 (u, u † ) = (m,l)∈N×N λ l - µ m v 2 r |u ml | 2 + (m,l)∈Λ |u ml | 2 = (m,l)∈N×N\Λ 1 λ l + µ m λ l - µ m v 2 r (λ l + µ m )|u ml | 2 + |u 0,0 | 2 + (m,l)∈Λ\{(0,0)} 1 λ l + µ m (λ l + µ m )|u ml | 2 . Since 1 λ l + µ m ≥ 1 2 max(λ l , µ m ) ≥ min 1 2λ l , 1 2µ m ,
we have the lower bound

1 λ l + µ m λ l - µ m v 2 r ≥ 1 2 inf (m,l)∈N×N\Λ min 1 - µ m λ l v 2 r , 1 v 2 r λ l v 2 r µ m -1 ≥ c,
for a constant c > 0 which is inferred from (H3). Then, we obtain a 0,0 (u, u † ) ≥ min c, min

(m,l)∈Λ\{(0,0)} 1 λ l + µ m , 1   |u 0,0 | 2 + (m,l)∈N×N (λ l + µ m )|u ml | 2   ,
and the result follows from (4.25). □

The property (4.26) entails that the bilinear form a satisfies a Garding's inequality for δ > 0.

Lemma 4.2. Assume (H1) and (H3). For any δ ∈ R in small neighborhood of zero, there exist constants α, β > 0 such that the bilinear form a of (4.23) satisfies the following Garding's inequality:

∀u ∈ V, |a(u, u † )| ≥ α||u|| 2 V -β||u|| 2 H . (4.27)
Proof. First, it is a classical fact that the trace of a function u

∈ V = H 1 per ((0, T ), L 2 (D)) ∩ L 2 per ((0, T ), H 1 (D)) is an element of H 1 2 per ((0, T ), L 2 (∂D)) ∩ L 2 per ((0, T ), H 1 2 (∂D)) (see [69, Theorem 2.1]
). Therefore, owing to the Proposition 4.5, the bilinear form a is continuous on V . Then, one can rewrite a(u, u † ) in terms of a 0,0 (u, u † ) as follows:

ℜ a(u, u † ) = a 0,0 (u, u † ) - (m,l)∈Λ |⟨u, p m ϕ l ⟩ H | 2 - 1 v 2 r T 0 D ω 2 uu † -iωu∂ t u † + iω∂ t uu † dx dt -δ T 0 ∂D T ω per [u]u † dσ dt. (4.28)
This yields (4.27) by using the Haussdorff-Young inequality (|ab| ≤ (ϵa 2 + ϵ -1 b 2 )/2) and the continuity property of T ω per (Proposition 4.5). □

Relying on Fredholm's theory, we obtain the following existence and uniqueness result.

Proposition 4.10. Assume (H1) and (H3) and that δ ∈ R is sufficiently small for (4.27) to hold. For any δ > 0, the wave problem (4.2), admits a unique solution u ∈ H 1 per ((0, T ), L 2 (D)) ∩ L 2 per ((0, T ), H 1 (D)) for any real ω satisfying |ω| < c with c being the constant of (H3).

Proof. Due to the Garding's inequality (4.27), we infer that the operator A : V → V ′ defined by ⟨Au, v⟩ V ′ ,V := a(u, v) is Fredholm. Therefore, (4.23) admits a unique solution in V if and only if A is injective. In order to obtain the injectivity of A, it is enough to show that a(u, v) = 0 for any v ∈ V implies u = 0. Assuming therefore a(u, v) = 0 for any v ∈ V , let us denote by

u(t, x) = n∈Z ûn (x)e -inΩt , t ∈ R, x ∈ D,
the Fourier decomposition of u in L 2 per ((0, T ), H 1 (D)). Taking the imaginary part of a(u, ûn e -in• ) = 0 for a given n ∈ Z yields

0 = -δT ∂D T ω+nΩ v 0 [û n ]û n dσ.
This equality implies from the Rellich's uniqueness theorem [START_REF] Colton | Integral Equation Methods in Scattering Theory[END_REF] that ûn = 0 on ∂D for any n ∈ N, and hence u(t, x) = 0 for any t ∈ R and x ∈ ∂D. Decompose now u(t, x) on the Sturm-Liouville and Neumann eigenfunctions p m (t; ω) and ϕ l (x) defined in (4.7):

u(t, x) = m,l∈Z u ml p m (t; ω)ϕ l (x), (4.29)
where this series is convergent in V . Computing a(u(t, x), p m (•; ω)ϕ l ) = 0, we find that for any (m, l) ∈ N × N,

u ml λ l - µ m (ω) v 2 r = 0.
Therefore, either u ml = 0 or λ l = µ m (ω)/v 2 r , however (H3) implies that the latter equality is possible only for (m, l) ∈ Λ. Consequently, the sum (4.29) is finite and involves only indices (m, l) ∈ Λ. Multiplying (4.29) by p m (t; ω)/ρ and using the orthonormality property (4.8), we obtain u ml ϕ l (x) = 0 for all x ∈ ∂D for all (m, l) ∈ Λ.

Therefore, if u(t, x) is not identically zero on D, then one of the Neumann eigenfunctions ϕ l must satisfy ϕ l = 0 on ∂D. However, this is not possible, because the only function solving the overdetermined problem is the double layer potential and • denotes the jump across ∂D [START_REF] Ammari | Mathematical and Computational Methods in Photonics and Phononics[END_REF]). Consequently, u(t, x) = 0 for any t ∈ R and x ∈ D, which completes the proof. □

       -∆ϕ = λ l ϕ

Time-periodic layer potentials

In this last part, we give the expression of the outgoing fundamental solution to the exterior T -periodic Helmholtz equation (4.15). We then introduce associated periodic layer potentials and we state their main properties.

Outgoing fundamental solution

The radiation condition (4.14) is related to the choice of a particular Green function for (4.15). We define the periodic outgoing fundamental solution Γ ω per (t, x) to be the unique T -periodic distribution solution to the time-dependent Helmholtz equation

1 v 2 0 (-iω + ∂ t ) 2 -∆ Γ ω per (t, x) = n∈Z δ(t -nT )δ(x), (t, x) ∈ R × R 3 , (4.30)
satisfying the radiation condition of (4.15).

Proposition 4.11. The distribution

Γ ω per (t, x) := - e i ω v 0 |x| 4π|x| n∈Z δ |x| v 0 -t -nT = n∈Z Γ ω+nΩ v 0 (x)e -inΩt , (4.31)
is the unique fundamental solution to (4.30) satisfying the outgoing radiation condition

∂ |x| - iω v 0 + 1 v 0 ∂ t Γ ω per (t, x) = O(|x| -2 ) as |x| → +∞.
Proof. Let us solve (4.30) using Fourier series: we represent Γ ω per as

Γ ω per (t, x) = n∈Z Γ ω per,n (x)e -inΩt .
Using the Poisson summation formula, we find

n∈Z δ(t -nT ) = 1 T n∈Z δ t T -n = 1 T n∈Z e 2iπ T nt = 1 T n∈Z e -inΩt .
Inserting into (4.30), we find that Γ ω per,n is solution to the Helmholtz equation

- 1 v 2 0 (ω + nΩ) 2 Γ ω per,n -∆ Γ ω per,n = 1 T δ in R 3 .
The only solution Γ ω per,n (x)e -i(ω+nΩ)t that is outgoing is Ĝω n (x) = -e i ω+nΩ v 0 |x| 4πT |x| . Consequently, using the Poisson summation formula, we find

Γ ω per (t, x) = - e i ω v 0 |x| 4πT |x| n∈Z e inΩ( 1 v 0 |x|-t) = - e i ω v 0 |x| 4πT |x| n∈Z δ 1 T |x| v 0 -t -n = - e i ω v 0 |x| 4π|x| n∈Z δ |x| v 0 -t -nT . □ 4.4.2

. Periodic retarded layer potentials

We now define the single and double layer potential associated to the periodic Green function (4.31). We first define these potentials on functions smooth with respect to time, before extending them to T -periodic distributions. In what follows, let us denote by D ′ per ((0, T ), W ′ ) the space of T -periodic distributions with values in the dual W ′ of a Hilbert space W . We recall that any T ∈ D ′ per ((0, T ), W ′ ) can be decomposed as a trigonometric Fourier expansion T = n∈Z Tn e -inΩt , with Tn ∈ W ′ for any n ∈ Z, where the Fourier coefficients grow at most polynomially: || Tn || W ′ ≤ C|n| p for some constant C > 0 and a polynomial exponent p ∈ N (see [START_REF] Donoghue | Distributions and Fourier transforms[END_REF]). Then, T ∈ D ′ per (W ′ ) defines a distribution of C ∞ per (W ) from the duality pairing

⟨T, ϕ⟩ D ′ per ((0,T ),W ′ ) := n∈Z ⟨ Tn , φn ⟩ W ′ ,W , for any ϕ ∈ C ∞ per ((0, T ), W ) with ϕ(t, x) = n∈Z φn (x)e -inΩt . (4.32)
In particular, if T ∈ L 2 per ((0, T ), W ), it holds

⟨T, ϕ⟩ D ′ per ((0,T ),W ) = 1 T T 0 ⟨T, ϕ⟩ W dt. (4.33)
Accordingly, for a given function ϕ(t, x) ∈ C ∞ per ((0, T ), H -1 2 (∂D)) which is T -periodic in t and for x ∈ ∂D, we pose

S ω D,per [ϕ](t, x) := ⟨Γ ω per (t -•, x -•), ϕ(•, •)⟩ D ′ per (L 2 (∂D)) , (4.34)
which, from the identification (4.32) of the duality pairing and (4.31), leads to the following definition.

Definition 4.2. We denote by S ω D,per :

C ∞ per ((0, T ), H -1 2 (∂D)) → C ∞ per ((0, T ), H 1 loc (R 3
)) the single layer potential defined for ϕ ∈ C ∞ per ((0, T ), H -1 2 (∂D)) by

S ω D,per [ϕ](t, x) := ∂D Γ ω v 0 (x -y)ϕ t - |x -y| v 0 , y dσ(y), (4.35)
where we recall the definition (2.7) for the fundamental solution Γ ω v 0 of the (static) Helmholtz equation.

Remark 4.3. The operator S ω D,per can be called a "retarded" potential, following the terminology of the works [START_REF] Costabel | Time-Dependent Problems with the Boundary Integral Equation Method[END_REF][START_REF] Steinbach | A new approach to space-time boundary integral equations for the wave equation[END_REF][START_REF] Bamberger | Formulation variationnelle pour le calcul de la diffraction d'une onde acoustique par une surface rigide[END_REF][START_REF] Bamberger | Formulation variationnelle espace-temps pour le calcul par potentiel retardé de la diffraction d'une onde acoustique (I)[END_REF][START_REF] Sayas | Retarded Potentials and Time Domain Boundary Integral Equations[END_REF] considering scattering problems for the wave equation in the time-domain.

One could think from the definition (4.35) that ϕ requires some smoothness in the variable t for (4.35) to be well-defined. In fact, this is not the case as the periodic layer potential S ω D,per can be extended as an operator S ω D,per : D ′ per ((0, T ), H -1 2 (∂D)) → D ′ per ((0, T ), H 1 loc (R 3 )). Indeed, we have the following characterization. for any ϕ ∈ D ′ per ((0, T ), H -1 2 (∂D)) with Fourier coefficients ( φn ) n∈Z , where for k ∈ R, S k D is the "usual" single layer potential as defined in (2.5). The operator S ω D,per thus defined is a mapping from D ′ per ((0, T ),

H -1 2 (∂D)) into D ′ per ((0, T ), H 1 loc (R 3 )).
Proof. Expanding ϕ ∈ C ∞ per ((0, T ), H -1 2 (∂D)) in Fourier series, we obtain

S ω D,per [ϕ](t, x) = n∈Z ∂D Γ ω v 0 (x -y)e inΩ |x-y| v 0 -t ϕ n (y) dσ(y) = n∈Z e -inΩt ∂D Γ ω+nΩ v 0 (x -y) ϕ n (y) dσ(y) = n∈Z S ω+nΩ v 0 D [ ϕ n ](x)e -inΩt ,
which shows that the identity (4.36) holds for functions ϕ ∈ C ∞ per ((0, T ), H -1 2 (∂D)). Now, let us assume that ϕ ∈ D ′ per ((0, T ), H -1 2 (∂D)). In order to show that (4.36) defines a periodic distribution of D ′ per ((0, T ), H 1 2 (∂D)), it is sufficient to show that the Fourier coefficients of S ω D,per [ϕ] are polynomially bounded with respect to the norm of H 1 2 (∂D). This holds because there exists a constant C > 0 independent of n such that

|||S ω+nΩ v 0 D ||| H -1 2 (∂D)→H 1 2 (∂D) ≤ Cn 1 2 log n, (4.37)
for a smooth domain D (see [START_REF] Galkowski | Wavenumber-explicit regularity estimates on the acoustic single-and double-layer operators[END_REF]), and the Fourier coefficients ( φn ) n∈Z ) grow also polynomially. Finally, the uniform estimate of (4.5) implies that S ω D,per [ϕ] defines also a distribution of D ′ per ((0, T ), H 1 loc (R 3 )). □ Remark 4.4. The bound (4.37) implies that S ω D,per maps H s per ((0, T ),

H -1 2 (∂D)) into H s-1 2 -ϵ per ((0, T ), H 1 2 (∂D))
for any ϵ > 0.

Neumann-Poincaré operator and jump relations

We define the adjoint of the Neumann-Poincaré operator by a definition similar to (4.34); we pose for any ϕ ∈ C ∞ per ((0, T ), H -1 2 (∂D)):

K ω * D,per [ϕ](t, x) := ⟨n(x) • ∇ x Γ ω per (t -•, x -•), ϕ(•, •)⟩ D ′ per ((0,T ),L 2 (∂D))
, which leads to the following definition. Definition 4.3. We denote by K ω * D,per : C ∞ per ((0, T ), H -1 2 (∂D)) → C ∞ per ((0, T ), H -1 2 (∂D)) the operator defined by

K ω * D,per [ϕ](t, x) := ∂D n(x) • ∇ x Γ ω v 0 (x -y)ϕ t - |x -y| v 0 , y - 1 v 0 x -y |x -y| Γ ω v 0 (x -y)∂ t ϕ t - |x -y| v 0 , y dσ(y), x ∈ ∂D. (4.38)
The first term of (4.38) is analogous to the standard Neumann-Poincaré operator and features a singular kernel. The second part of the integral is non-singular, but features partial derivative with respect to time.

Similarly, K ω * D,per can be extended into an operator

K ω * D,per : D ′ per ((0, T ), H -1 2 (∂D)) → D ′ per ((0, T ), H -1 2 (∂D)).
Lemma 4.4. The operator K ω * D,per can be extended to periodic distributions by setting

K ω * D,per [ϕ](t, x) := n∈Z K ω+nΩ v 0 * D [ ϕ n ](x)e -inΩt , for any T -periodic distribution ϕ ∈ D ′ per ((0, T ), H -1 2 (∂D))
with Fourier coefficients ( φn ) n∈Z , where for k ∈ R, K k * D is the "usual" adjoint of the Neumann-Poincaré operator defined by (2.6). The operator K ω * D,per defines a mapping from D ′ per ((0, T ), H -1 2 (∂D)) into itself.

Proof. The result is obtained similarly as in the proof of Lemma 4.3, using the wave number explicit bound

|||K ω+nΩ v 0 * D ||| H -1 2 (∂D)→H -1 2 (∂D) ≤ Cn 1 4 log n, (4.39)
which holds for smooth domains D (see the appendix of [START_REF] Han | Sharp norm estimates of layer potentials and operators at high frequency[END_REF]). □ Remark 4.5. The bound (4.39) implies that K ω * D,per maps H s per ((0, T ),

H -1 2 (∂D)) into H s-1 4 -ϵ per ((0, T ), H -1 2 (∂D)) for any ϵ > 0.
Due to standard properties of singular kernels [START_REF] Mclean | Strongly elliptic systems and boundary integral equations[END_REF], we have the following result.

Proposition 4.12. The time-periodic single layer potential S ω D,per satisfies the following jump relation:

∂S ω D,per [ϕ] ∂n ± = ± 1 2 I + K ω * D,per [ϕ], for all ϕ ∈ D ′ per ((0, T ), H -1 2 (∂D)),
where I is the identity mapping.

Invertibility of the single layer potential

It seems difficult, if not unfeasible, to show that S ω D,per is invertible as a mapping H s per ((0, T ),

H -1 2 (∂D)) into H s-1 2 -ϵ per ((0, T ), H 1 2 (
∂D)) for some s > 0 and ϵ > 0. However, it is fairly easy to state an invertibility result when considering S ω D,per as an operator acting on T -periodic distributions. Proposition 4.13. Assume (H1) and (H2). For any ω ∈ R in a small neighborhood of zero, the single layer potential S ω D,per is an invertible operator from D ′ per ((0, T ), H -1 2 (∂D)) → D ′ per ((0, T ), H 1 2 (∂D)), whose inverse reads:

(S ω D,per ) -1 [f ] = n∈Z (S ω+nΩ v 0 D ) -1 [ fn ]e -inΩt , (4.40)
where ( fn ) n∈Z denote the Fourier coefficients of a given f ∈ D ′ per ((0, T ), H 1 2 (∂D)).

Proof. Assumption (H2) ensures that S ω+nΩ v 0 D is invertible for any n ∈ N and ω sufficiently small. Owing to the Fourier series expansion (4.36), it is necessary that (4.40) be the inverse of S ω D,per , if this operator is invertible. Therefore, the only point to verify is that (4.40) defines a periodic distribution of D ′ per ((0, T ), H -1 2 (∂D)), i.e. that the coefficients of the trigonometric series (4.40) grow polynomially with respect to n. For this, we need a wave number explicit bound on the operator norm (S

||T ω+nΩ v 0 - [ϕ]|| H -1 2 (∂D) ≤ C inf λ∈σ dir (-∆),n∈Z nΩ+ω v0 2 -λ ||ϕ|| H 1 2 (∂D) , (4.42)
for an independent constant C > 0. Since for ω ∈ R sufficiently small, inf

λ∈σ dir (-∆),n∈Z nΩ + ω v 0 2 -λ = inf λ∈σ dir (-∆),n∈Z nΩ + ω v 0 - √ λ × nΩ + ω v 0 + √ λ ≥ C inf λ∈σ dir (-∆),n∈Z nΩ v 0 - √ λ , (4.43)
the bound (4.42) together with (H2) imply that for ω sufficiently small, the operator norm of T ω+nΩ v 0 -is uniformly bounded in n and ω. Therefore, (4.41) and Proposition 4.2 imply the existence of a constant C > 0 independent of n such that

|||(S ω+nΩ v 0 D ) -1 ||| H 1 2 (∂D)→H -1 2 (∂D) ≤ C|n|, (4.44)
so that (4.40) defines an element of D ′ per ((0, T ), H -1 2 (∂D)). □ Remark 4.6. The wave number explicit bound (4.44) implies that (S ω D,per ) -1 maps H s per ((0, T ),

H 1 2 (∂D)) into H s-1 per ((0, T ), H -1 2 (∂D)).
Finally, it is possible to express the periodic Dirichlet-to-Neumann operator T ω per in terms of the potentials S ω D,per and K ω * D,per , generalizing the identity (3.5). Proposition 4.14. Assume (H1) and (H2). The periodic Dirichlet-to-Neumann map T ω per of (4.15) can be extended to periodic distributions by setting

T ω per [ϕ] := n∈Z T ω+nΩ v 0 [ φn ]e -inΩt ,
for any ϕ ∈ D ′ per ((0, T ), H 1 2 (∂D)). Then, the operator

T ω per : D ′ per ((0, T ), H 1 2 (∂D)) → D ′ per ((0, T ), H -1 2 (∂D))
thus defined admits the following representation:

T ω per = 1 2 I + K ω * D,per (S ω D,per ) -1 .

Subwavelength resonances in fast time-modulated media

This last section is dedicated to the study of scattering resonances in the fast time-modulated medium (1.1), for time-harmonic incident fields (1.2) and in the subwavelength and high-contrast regimes (1.4). Our analysis is based on the Dirichlet-to-Neumann approach introduced in Section 3, where the considered Dirichlet-to-Neumann map is the one associated to outgoing time-periodic waves studied in Section 4. Throughout this part, we assume that the assumptions (H1) to (H3) are fulfilled.

We start in Section 5.1 by characterizing the behavior of the solution operator of the reduced time-modulated wave equation (4.2) for small ω and δ. This allows us to determine the leading order asymptotics of the subwavelength resonant frequencies which arise due to the scattering with the time-modulated resonator D. We find that in the exceptional situation where Λ ̸ = {(0, 0)}, resonant frequencies with positive imaginary parts may exist, leading to the arising of exponentially growing outgoing modes.

Assuming that the modulation is tuned in order to ensure that the exceptional coupling occurs, we determine a modal decomposition for the scattered field in (5.2) which is shown to oscillate like p m (t)ϕ l (x) inside the resonator D. We then identify a far field expansion for the scattered wave.

Asymptotics of the subwavelength resonant frequencies

In what follows, we consider the reformulation of (4.2) in terms of the Dirichlet-to-Neumann map T ω per given by (4.21), or equivalently by the variational formulation (4.22). Following the Definition 3.2, we call "subwavelength resonance" a complex frequency ω ≡ ω(δ) ∈ C satisfying ω(δ) → 0 as δ → 0 and such that (4.21) admits a non-zero solution u(ω, δ) for a zero right-hand side ûin = 0. According to the well-posedness result of Proposition 4.10, such frequencies must necessarily have a non-zero imaginary part.

Characterization of the inverse of the scattering operator

In this part, we characterize the inverse of the solution operator to (4.21), whose poles are the resonant frequencies.

Following Section 3.2, we introduce the bilinear form a ω,δ defined for u, v ∈ V by

a ω,δ (u, v) := a 0,0 (u, v) - 1 v 2 r T 0 D (2iω∂ t + ω 2 )uv dx -δ T 0 D T ω per [u]v dσ,
where we recall the definition (4.24) of the bilinear form a 0,0 and the definition (2.3) of the space V . Since a 0,0 satisfies the coercivity property (4.26), the perturbed bilinear form a ω,δ is also coercive for small ω and δ > 0. By using Fredholm's theory as in Proposition 4.10, we obtain for any f ∈ V ′ the existence and uniqueness of a solution u f (ω, δ) to the variational problem

a ω,δ (u f (ω, δ), v) = ⟨f, v⟩ V ′ ,V , ∀v ∈ V. (5.1)
Let us, in particular, denote by u ml (ω, δ) the unique solution to the problem

a ω,δ (u ml (ω, δ), v) = ⟨p m ϕ l , v⟩ H , (5.2)
where we recall the definition (2.2) of the inner product ⟨•, •⟩ H .

Let us finally recall that the solution û(t, x) to the time-periodic reduced wave problem (4.22) satisfies

a(û, v) = ⟨f, v⟩ V ′ ,V , ∀v ∈ V,
with f being the linear form

⟨f, v⟩ V ′ ,V := δ T 0 ∂D ∂ ûin ∂n -T ω per [û in ] dσ dt. (5.3)
The bilinear form a of (4.23) and a ω,δ are related through the identity

a ω,δ (u(ω, δ), v) = a(u, v) + (m,l)∈Λ ⟨u(ω, δ), p m ϕ l ⟩ H ⟨v, p m ϕ l ⟩ H ,
where the term on the right-hand side is a finite range operator that makes a ω,δ coercive. The next proposition is the counterpart of Lemma 3.1 in the time-modulated setting.

Proposition 5.1. For a given right-hand side f ∈ V ′ , the variational problem

a(u, v) = ⟨f, v⟩ V ′ ,V , for any v ∈ V, (5.4)
admits a unique solution u ∈ V if and only if the linear system

(I -C(ω, δ))x = F (5.5)
has a unique solution x ≡ x(ω, δ) ≡ (x ml (ω, δ)) (m,l)∈Λ , where C(ω, δ) and F are the matrix and column vectors given by

C(ω, δ) := (⟨u m ′ ,l ′ (ω, δ), p m ϕ l ⟩ H ) (m,l),(m ′ ,l ′ )∈Λ×Λ , F := (⟨u f (ω, δ), p m ϕ l ⟩ H ) (m,l)∈Λ . (5.6)
When it is the case, the solution u ≡ u(ω, δ) to (5.4) reads

u(ω, δ) = u f (ω, δ) + (m,l)∈Λ x ml (ω, δ)u ml (ω, δ), (5.7)
with u f (ω, δ) and u ml (ω, δ) defined by (5.1) and (5.2).

Proof. The variational problem (5.4) can be equivalently written as

a(u, v) = a ω,δ (u(ω, δ), v) - (m,l)∈Λ ⟨u(ω, δ), p m ϕ l ⟩ H ⟨v, p m ϕ l ⟩ H = ⟨f, v⟩ V ′ ,V ⇔ a ω,δ (u(ω, δ), v) - (m,l)∈Λ ⟨u(ω, δ), p m ϕ l ⟩ H a ω,δ (u ml (ω, δ), v) = a ω,δ (u f (ω, δ), v) ⇔ u(ω, δ) - (m,l)∈Λ ⟨u(ω, δ), p m ϕ l ⟩ H u ml (ω, δ) = u f (ω, δ).
Integrating against p m ϕ l /ρ, we find that this linear system has a solution if and only if

⟨u(ω, δ), p m ϕ l ⟩ H - (m ′ ,l ′ )∈Λ ⟨u m ′ ,l ′ (ω, δ), p m ϕ l ⟩ H ⟨u(ω, δ), p m ′ ϕ l ′ ⟩ H = ⟨u f (ω, δ), p m ϕ l ⟩ H .
This system is equivalent to (5.5) with x ml (ω, δ) := ⟨u(ω, δ), p m ϕ l ⟩ H and admits (5.7) as a solution. □

Asymptotic expansion of the subwavelength resonances

The Proposition 5.1 reduces the study of the resonances of the infinite dimensional problem (5.4) to the one of the finite-dimensional problem (5.5): the subwavelength resonant frequencies of (5.4) are the complex numbers (ω(δ)) for which the matrix I -C(ω(δ), δ) is not invertible. To characterize these numbers, we write an asymptotic expansion for I -C(ω, δ), which is obtained from an asymptotic expansion of u ml (ω, δ).

Proposition 5.2. The solution u ml (ω, δ) to (5.2) has the following asymptotic expansion as ω, δ → 0:

u ml (ω, δ)(t, x) = p m (t)ϕ l (x) + iωp 1 m (t)ϕ l (x) + ω 2 γ m v 2 r p m (t) + p 2 m (t) ϕ l (x) + δ (m ′ ,l ′ )∈Λ M m ′ l ′ ,ml p m ′ (t)ϕ l ′ (x) + δ w ml (t, x) + O(ω 3 + δω + δ 2 ), (5.8)
where for any (m, l) ∈ Λ:

• p m is the eigenfunction of the Sturm-Liouville problem (1.6), and p 1 m and p 2 m are the unique T -periodic solutions to

             - d 2 p 1 m dt 2 - µ m ρ(t) p 1 m = -2 dp m dt , p 1 m is T -periodic, T 0 1 ρ(t) p 1 m (t)p m (t) dt = 0,              - d 2 p 2 m dt 2 - µ m ρ(t) p 2 m = 2 dp 1 m dt -p m + γ m T p m ρ(t) , p 2 m is T -periodic, T 0 1 ρ(t) p 2 m (t)p m (t) dt = 0, (5.9)
• w ml (t, x) is a function satisfying ⟨ w ml , p m ′ ϕ l ′ ⟩ H = 0 for all (m ′ , l ′ ) ∈ Λ,
• γ m is the real number

γ m := T 0 -2 d dt p 1 m p m + |p m (t)| 2 dt = T 0 - d dt p 1 m 2 + µ m ρ(t) p 1 m 2 + |p m (t)| 2 dt, (5.10)
• M ≡ (M ml,m ′ l ′ ) (m,l),(m ′ ,l ′ )∈Λ×Λ is the real matrix M ml,m ′ l ′ := T 0 ∂D T 0 per [p m ′ ϕ l ′ ]p m (t)ϕ l (x) dt dσ(x), (5.11)
where T 0 per is the time-periodic Dirichlet-to-Neumann map of Definition 4.1 with frequency ω = 0.

Proof. To simplify the notation, let us write w(ω, δ) := u ml (ω, δ) in this proof. The strong form of (5.2) reads

         1 v 2 r ∂ tt w - 1 ρ(t) ∆w + (m ′ ,l ′ )∈Λ ⟨w, p m ′ ϕ l ′ ⟩ H 1 T 1 ρ(t) p m ′ (t)ϕ l ′ (x) - 1 v 2 r (2iω∂ t + ω 2 )w = 1 T 1 ρ(t) p m (t)ϕ l (x) 1 ρ(t) ∂w ∂n -δT ω per [w] = 0.
Since this system is invertible for any values of ω and δ small enough, w ≡ w(ω, δ) is analytic in ω and δ and we can write the asymptotic expansion

w = +∞ p,k=0 ω p δ k w p,k .
Inserting this ansatz in the above equation yields the following cascade of equations determining (w p,k ) p,k≥0 :

                   1 v 2 r ∂ tt w p,k - 1 ρ(t) ∆w p,k + (m ′ ,l ′ )∈Λ ⟨w p,k , p m ′ ϕ l ′ ⟩ H 1 T 1 ρ(t) p m ′ ϕ l ′ = 1 T 1 ρ(t) p m (t)ϕ l (x)δ p=0 δ k=0 + 1 v 2 r (2i∂ t w p-1,k + w p-2,k ), 1 ρ(t) ∂w p,k ∂n = p n=0 1 v n 0 T per,n [w p-n,k-1 ], (5.12)
where we define w p,k = 0 for negative p, k < 0 by convention. Let us now compute the first terms of this cascade of equation. In what follows, we consider two indices (m ′ , l ′ ) ∈ Λ.

• p = 0, k = 0 yields w 0,0 = p m ϕ l ;

• p = 1, k = 0. Integrating (5.12) against v = p m ′ ϕ l ′ yields ⟨w 1,0 , p m ′ ϕ l ′ ⟩ H = 2i v 2 r T 0 D d dt p m (t)ϕ l (x)p m ′ (t)ϕ l ′ (x) dx dt = δ l=l ′ δ m=m ′ 2i v 2 r T 0 d dt p m (t)p m (t) dt = 0.
Therefore, w 1,0 is the solution to

       1 v 2 r ∂ tt w 1,0 - 1 ρ(t) ∆w 1,0 = 2i v 2 r d dt p m (t)ϕ l (x), 1 ρ(t) ∂w 1,0 ∂n = 0.
We deduce that w 1,0 (t,

x) = ip 1 m (t)ϕ l (x) where p 1 m (t) satisfies 1 v 2 r d 2 dt 2 p 1 m + λ l ρ(t) p 1 m = 2 v 2 r d dt p m ⇔ - d 2 dt 2 p 1 m - µ m ρ(t) p 1 m = -2 d dt p m .
According to the Fredholm's alternative, this equation has a unique T -periodic solution p 1 m which satisfies

T 0 1 ρ(t) p 1 m (t)p m (t) dt = 0. • p = 2, k = 0. Integrating (5.12) against v = p m ′ ϕ l ′ yields ⟨w 2,0 , p m ′ ϕ l ′ ⟩ H = 1 v 2 r 2i × i T 0 D d dt p 1 m (t)ϕ l (x)p m ′ (t)ϕ l ′ (x) dx dt + T 0 D p m (t)ϕ l (x)p m ′ (t)ϕ l ′ (x) dx dt = δ m=m ′ δ l=l ′ v 2 r -2 T 0 d dt p 1 m p m dt + T 0 |p m (t)| 2 dt .
Therefore, we obtain w 2,0 = γm v 2 r p m (t)ϕ l (x) + w 2,0 (t, x), where w 2,0 (t, x) is the unique solution to

             1 v 2 r ∂ tt w 2,0 - 1 ρ(t) ∆ w 2,0 = 1 v 2 r -2∂ t p 1 m + p m - γ m T 1 ρ(t) p m ϕ l , 1 ρ(t) ∂ w 2,0 ∂n = 0, ⟨ w 2,0 , p m ′ ϕ l ′ ⟩ H = 0 for all (m ′ , l ′ ) ∈ Λ.
Similarly, we find w 2,0 (t, x) = p 2 m (t)ϕ l (x) with p 2 m (t) being the unique solution to (5.9).

• p = 0, k = 1. Integrating (5.12) against v = p m ′ ϕ l ′ yields ⟨w 0,1 , p m ′ ϕ l ′ ⟩ H = T 0 ∂D T 0 per [p m ϕ l ]p m ′ (t)ϕ l ′ (x) dt dσ(x) = M m ′ l ′ ,ml .
Therefore, w 0,1 = (m ′ ,l ′ )∈Λ M m ′ l ′ ,ml p m ′ ϕ l ′ + w ml , where w ml is the unique solution to

               1 v 2 r ∂ tt w ml - 1 ρ(t) ∆ w ml = - (m ′ ,l ′ )∈Λ M ml,m ′ l ′ 1 T 1 ρ(t) p m ′ ϕ l ′ , 1 ρ(t) ∂ w ml ∂n = T 0 per [p m ϕ l ], ⟨ w ml , p m ′ ϕ l ′ ⟩ H = 0 for all (m ′ , l ′ ) ∈ Λ.
(5.13)

The asymptotic (5.8) follows. □

Inserting this expansion into the definition (5.6) of C(ω, δ) yields the following characterization of the resonances.

Proposition 5.3. The following asymptotic expansion of the matrix C(ω, δ) defined in (5.6) holds as ω, δ → 0:

C(ω, δ) = I + ω 2 v 2 r diag(γ m ) (m,l)∈Λ + δM + O(ω 3 + δω + δ 2 ), (5.14)
where (γ m ) (m,l)∈Λ and the matrix M are given by (5.10) and (5.11). Consequently, the problem (4.2) admits exactly 2#Λ subwavelength resonances (ω ± i (δ)) 1≤i≤#Λ whose leading asymptotic expansions are given by:

ω ± i (δ) ∼ ±v r δ 1 2 λ 1 2 i , (5.15)
where (λ i ) 1≤i≤#Λ are the (complex) eigenvalues of the generalized eigenvalue problem

M a i + λ i Ga i = 0, G = diag(γ m ) (m,l)∈Λ .
(5.16)

Proof. Integrating (5.8) against the function p m (t)ϕ l (x)/ρ(t) yields the following asymptotic expansion for C(ω, δ):

C(ω, δ) ml,m ′ l ′ = δ m=m ′ δ l=l ′ + ω 2 v 2 r γ m δ m=m ′ δ l=l ′ + δM ml,m ′ l ′ + O(ω 3 + δω + δ 2 ),
which coincides with (5.14). Resorting to the generalized Rouché theorem [START_REF] Ammari | Mathematical and Computational Methods in Photonics and Phononics[END_REF] or to the implicit function theorem as in [START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF] (this would require the eigenvalues of T to be simple), we obtain the existence of exactly 2#Λ subwavelength resonances (ω ± i (δ)) 1≤i≤#Λ for which I -C(ω ± i (δ), δ) is not invertible, and these can be approximated at first order by those of the following nonlinear eigenvalue problem in ω: ), which happens close to the resonant frequencies, then (5.14) provides the leading order terms of this asymptotic expansion and the remainder is of order O(δ The matrix M plays a role analogous to the capacity cap(D) involved in (3.18), or to the capacitance matrix in the context of subwavelength resonances induced by several simply connected resonators (see [START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF]). The next proposition lists a few properties of the matrix M . (i) The coefficients (M ml,m ′ l ′ ) (m,l),(m ′ ,l ′ )∈Λ×Λ are real and are also given by

M ml,m ′ l ′ = T n∈Z pm ′ ,n pm,n ∂D T nΩ v 0 [ϕ l ′ ]ϕ l dσ.
(ii) The matrix M is symmetric if ρ is invariant by time reversal: ρ(t) = ρ(-t). If further, D is a ball, then T is symmetric negative definite.

(iii) M 00,00 = -T p 2 0 cap(D) |D| and γ 0 = T p 2 0 where

p 0 := 1 T T 0 1 ρ(t) dt -1 2 .
(5.17)

(iv) For any (m, l) ∈ Λ, M 00,ml = M ml,00 = T |D| -1 2 p 0 pm,0

∂D ∂Φ ∂n ϕ l dσ,
where Φ is the solution to the exterior problem (2.8).

Proof.

(i) Decomposing p m ϕ l into Fourier modes, we find

T 0 per [p m ′ ϕ l ′ ] = n∈Z pm ′ ,n T nΩ v 0 [ϕ l ′ ]e -inΩt ,
and consequently,

M ml,m ′ l ′ = T 0 ∂D T 0 [p m ′ ϕ l ′ ]p m ϕ l dσ dt = T n∈Z pm ′ ,n pm,n ∂D T nΩ v 0 [ϕ l ′ ]ϕ l dσ.
Using the property

T nΩ v 0 [ϕ l ] = T -nΩ v 0 [ϕ l ],
and the fact that p m is real, we find that M ml,m ′ l ′ is real.

(ii) If ρ(t) = ρ(-t), then the simplicity assumption (H3) of the eigenfunction p m implies that p m (t) = p m (-t). The symmetry of M is then a consequence of Proposition 4.7. If further D is the sphere, then the bound of Proposition 4.8 implies the negative definiteness of the matrix M . (iii)-(iv) It is easy to compute M 00,ml for (m, l) ∈ Λ, because it is associated with the constant, normalized Sturm-Liouville and Neumann eigenmodes p 0 and ϕ 0 given by

p 0 (t) = 1 T T 0 1 ρ(t) dt -1 2 , ϕ 0 (x) = 1 D (x) |D| 1 2
.

Hence, p0,n = 1

T T 0 1 ρ(t) dt -1
2 δ 0n and the result follows.

□

In the next subsections, we discuss the nature of the subwavelength resonances depending on the set Λ introduced in (1.7), the constant γ m and the matrix M of (5.10) and (5.11). 5.1.3. Absence of exceptional subwavelength resonances in the generic case Λ = {(0, 0)} In the generic case where the modulation ρ(t) and the resonator D are not "tuned", it can be expected that Λ = {(0, 0)}, i.e. only the zero eigenvalues µ 0 /v 2 r and λ 0 coincide. In this situation, we find that the matrix C(ω, δ) of (5.14) is reduced to a constant, which has the following asymptotic expansion:

C(ω, δ) = 1 + T 1 T T 0 1 ρ(t) dt -1 ω 2 v 2 r -T 1 T T 0 1 ρ(t) dt -1 cap(D) |D| δ + O(ω 3 + δω + δ 2 ).
Computing the zeros of C(ω, δ) -I, we retrieve two subwavelength resonances which coincide at first order with the classical "Minnaert resonance" ω ± (δ) of (3.21):

ω ± 1 (δ) ∼ v r cap(D) |D| δ 1 2 .
This resonant frequency is associated to a resonant mode that is approximately constant inside the resonator D in both time and space. Owing to the properties of the time-periodic Dirichlet-to-Neumann map T ω per , the associated scattered field does not propagate high-frequency components in the exterior domain R 3 \D. One can then verify that the scattered field admits the same approximation (3.25) than in the unmodulated case. In particular, it is fully independent on the modulation ρ(t): everything happens as if ρ(t) would be replaced with any positive constant independent of ω and δ (see also Remark 3.2).

5.1.4.

Subwavelength resonances in the exceptional coupling situation Λ = {(0, 0), (m, l)}

We now consider the more interesting case where the modulation ρ(t) is tuned in such a way there exists another pair of coinciding eigenvalues µ m /v 2 r = λ l ̸ = 0 with (m, l) ̸ = (0, 0): Λ = {(0, 0), (m, l)}.

From the point (iii) of Proposition 5.4, the constant γ 0 is positive, but the constant γ m defined in (5.10) does not have a clear sign. In fact, we conjecture from numerical experiments that the modulation ρ(t) can be tuned in such a way both cases γ m > 0 and γ m < 0 are possible. The following proposition shows that complex resonances with positive imaginary parts generally arise.

Proposition 5.5. Assume that D is a disk and that the modulation is invariant by time-reversal: ρ(t) = ρ(-t).

Let us denote by P (ξ) the second order polynomial P (ξ) := γ 0 γ m ξ 2 + (M 00,00 γ m + M ml,ml γ 0 ) ξ + (M 00,00 M ml,ml -M 2 ml,00 ), (5.18) and by ∆ its discriminant: ∆ = (M 00,00 γ m -M ml,ml γ 0 ) 2 + 4γ 0 γ m M 2 ml,00 .

(5.19)

Then, the three following situations are possible depending on the sign of γ m and ∆:

• if γ m > 0, then the four subwavelength resonances of (4.2) are real at the leading order and satisfy

ω ± i (δ) ∼ ±ξ 1 2 i v r δ 1 2 , 1 ≤ i ≤ 2,
where the eigenvalues (ξ i ) i=1,2 of (5.16) are the real positive numbers given by

ξ 1 = -M 00,00 γ m -M ml,ml γ 0 - √ ∆ 2γ 0 γ m , ξ 2 = -M 00,00 γ m -M ml,ml γ 0 + √ ∆ 2γ 0 γ m ;
(5.20)

• if γ m < 0 and ∆ > 0, then the reduced wave problem (4.2) admits exactly two resonances ω ± 1 (δ) which are purely imaginary at first order, and two resonances ω ± 2 (δ) which are purely real at first order:

ω ± 1 (δ) ∼ ±i -ξ 1 v r δ 1 2 , ω ± 2 (δ) ∼ ±ξ 1 2 2 v r δ 1 2 ,
where the eigenvalues ξ 1 and ξ 2 of (5.20) are respectively negative and positive; • if γ m < 0 and ∆ < 0, then the four resonances of (4.2) are complex at the order in O(δ 1 2 ), and their leading asymptotics are given by

ω ± 1 (δ) = ±v r δ 1 2 √ a + ib and ω ± 2 (δ) = ±v r δ 1 2 √ a -ib, (5.21)
where a and b are the positive numbers a = -M 00,00 γ m -M ml,ml γ 0

2γ 0 γ m , b = √ -∆ 2γ 0 γ m ,
and where √ • is a determination of the square root in C.

Proof. In this context, the matrix of the eigenvalue problem (5.16) reads, for a given ξ ∈ C:

T + ξG =   ξγ 0 + M 00,00
M 00,ml M ml,00 ξγ m + M ml,ml .

  .

This matrix is not invertible when P (ξ) = 0. Recalling that T is a symmetric negative definite matrix due to the symmetry assumption on D and ρ, which implies that M 00,00 M ml,ml -M 2 ml,00 ≥ 0. The result follows by discussing the complex nature of the root of the polynomial (5.18) depending on the sign of γ m and ∆. □

This proposition shows that at least if γ m < 0 and in the most symmetric case where D is a ball and ρ is symmetric upon time-reversal, there exist outgoing nonzero solutions to the time-modulated scattering problem (1.1), which grow exponentially in time and with a growth rate of order O(δ 1 2 ).

Modal decomposition and point scatterer approximation of the scattered field

We now follow the lines of Section 3.3 to obtain a modal decomposition and a far field approximation for the scattered solution û -ûin based on the formula (5.7). In the remainder of this section, we consider the "exceptional" coupling situation where there are two pairs of coinciding Sturm-Liouville and Neumann eigenvalues: Λ = {(0, 0), (m, l)} for some (m, l) ∈ N × N\{(0, 0)}.

(5.22)

Modal decomposition of the scattered field in the exceptional coupling

In order to obtain a modal decomposition of u f (ω, δ) based on (5.7), we first need an approximation of the function u f (ω, δ) solution to (5.1) where f ∈ V ′ is the linear form (5.3).

Proposition 5.6. The function u f (ω, δ) solution to (5.1) with f given by (5.3) reads at first order

u f (ω, δ) = -δ ûin (0)p -1 0 |D| 1 2   (m,l)∈Λ M 00,ml p m ϕ l + w 00   + O(ωδ), (5.23)
where w 00 is the function of the Proposition 5.2 satisfying ⟨ w 00 , p m ϕ l ⟩ H = 0 for all (m, l) ∈ Λ.

Proof. The problem (5.1) reads in strong form

               1 v 2 r ∂ tt u f - 1 ρ(t) ∆u f + (m,l)∈Λ ⟨u f , p m ϕ l ⟩ H 1 T 1 ρ(t) p m (t)ϕ l (x) - 1 v 2 r (2iω∂ t + ω 2 )u f = 0, (t, x) ∈ R × D, 1 ρ(t) ∂u f ∂n = δ ∂ u in ∂n -T ω per [ u in ] , (t, x) ∈ R × ∂D, u f is T -periodic.
Following the proof of Proposition 3.3, we find that

∂ ûin ∂n -T ω per [û in ] = -û in (0)T 0 per [1 D ] + O(ω) = -û in (0)p -1 0 |D| 1 2 T 0 per [p 0 ϕ 0 ] + O(ω).
Hence, u f (ω, δ) = δ(u 0,1 + u 0,1 ) + O(ωδ), where

u 0,1 (t, x) = -û in (0)p -1 0 |D| 1 2 (m,l)∈Λ T 0 ∂D T 0 per [p 0 ϕ 0 ]p m ϕ l dσ(x) dt p m (t)ϕ l (x) = -û in (0)p -1 0 |D| 1 2
(m,l)∈Λ M 00,ml p m (t)ϕ l (x), and u 0,1 is the unique solution to

                   1 v 2 r ∂ tt u 0,1 - 1 ρ(t) ∆ u 0,1 = u in (0)p -1 0 |D| 1 2 (m ′ ,l ′ )∈Λ M 00,m ′ l ′ 1 T 1 ρ(t) p m ′ (t)ϕ l ′ (x), 1 ρ(t) ∂ u 0,1 ∂n = -û in (0)p -1 0 |D| 1 2 T 0 per [p 0 ϕ 0 ], ⟨ u 0,1 , p m ϕ l ⟩ H = 0 for all (m, l) ∈ Λ, u 0,1 is T -periodic.
Comparing with (5.13), we obtain the result with u 0,1 = -û in (0)p -1 0 |D| 1 2 w 00 . □

The next proposition establishes a "modal decomposition"in the exceptional coupling situation (5.22). For simplicity, we consider only the "least favourable" case where both resonant frequencies are complex at first order, that is the situation of (5.21). This case is the "least favourable" in the sense that the resonances are damped and there is no amplification of the input wave field (a resonance which is purely real at first order would result in an amplification of the incident field by a factor O(δ -1 2 )). However, we find that the scattered field has an amplitude of the same order of the incident wave field and contains high frequency components generated by the mode p m (t)ϕ l (x). In the other cases, one observes an amplification of the scattered field near the real part of the resonant frequency. Proposition 5.7. Assume that the discriminant ∆ of (5.19) is negative and Λ = {(0, 0), (m, l)}. Then, for a real subwavelength frequency ω = O(δ

2 ), we have the following modal decomposition for the wave field û(t, x) solution to (4.2) for x ∈ D:

û(t, x) = ûin (0)p -1 0 |D| 1 2 P ω 2 v 2 r δ ω 2 v 2 r δ
γ m M 00,00 + M ml,ml M 00,00 -M 2 00,ml p 0 (t)ϕ 0 (x) +

ω 2 v 2 r δ γ 0 M 00,ml p m (t)ϕ l (x) +O(δ 1
2 ),

(5.24) where P (ξ) is the second order polynomial of (5.18), which is negative and bounded from below by a constant independent of ω and δ.

Proof. We use the exact formula (5.7). In order to have an approximation of the coefficients x ml (ω, δ), we need to compute the right-hand side F = (F 00 , F ml ). By integrating (5.23) against p m ′ ϕ l ′ /ρ, we obtain

F m ′ l ′ = -δ ûin (0)p -1 0 |D| 1 2 M 00,m ′ l ′ + O(ωδ) for (m ′ , l ′ ) ∈ {(0, 0), (m, l)}.
Then, computing explicitly the inverse of the matrix (I -C(ω, δ)) yields

(I -C(ω, δ)) -1 = - 1 det((I -C(ω, δ)))   ω 2 v 2 r γ m + δM ml,ml
-δM 00,ml -δM ml,00

ω 2 v 2 r γ 0 + δM 00,00   + O ω 3 + δω + δ 2 det((I -C(ω, δ)))
.

For a real frequency ω = O(δ

2 ) in the regime where the polynomial P of (5.18) has no real roots (∆ < 0), the determinant can be approximated by det

((I -C(ω, δ))) = δ 2 P ω 2 v 2 r δ + O(ω 5 + δω 3 + δ 2 ω) = δ 2 P ω 2 v 2 r δ 1 + O(δ 1
2 ) .

Therefore, the coefficients x ml (ω, δ) of (5.7) read

x 00 = ûin (0)p -1 0 |D| 1 2 P ω 2 v 2 r δ ω 2 v 2 r δ γ m + M ml,ml M 00,00 -M 2 00,ml + O(δ 1
2 ),

x ml = ûin (0)p -1 0 |D| 1 2 P ω 2 v 2 r δ ω 2 v 2 r δ
γ 0 + M 00,00 M 00,ml -M 00,ml M 00,00 + O(δ

2 ), which yields the result. □ Remark 5.2. The expansion (5.24) still holds the other situations in which ∆ > 0 and where P (λ) has a real root up to a correction of the approximation error O(δ

2 ): in that case P ω 2 v 2 r δ becomes a resonant denominator which vanishes as ω becomes closer to the resonant frequency. A more precise approximation would be obtained by using a higher order expansion of det ((I -C(ω, δ))), which would capture the leading order of the imaginary parts of the resonances, expected to be of order O(δ) (see e.g. [START_REF] Feppon | Modal decompositions and point scatterer approximations near the Minnaert resonance frequencies[END_REF]Section 4.3]).

Remark 5.3. In this situation where the resonances are complex at the leading order in O(δ 1 2 ), (5.24) shows that these "resonances" are damped to the point where there is no significant amplification of the incident field in the regime ω = O(δ 1 2 ). However, we still find an order one high-frequency response which periodically oscillates in time at the "fast" frequency Ω.

Far field expansion of the scattered field in the exceptional coupling situation

We now determine a far field approximation for the scattered field. The scattered field û-ûin can be rewritten thanks to the periodic single layer potential S ω D,per . Using the inversion property of Proposition 4.13, we can indeed write the following representation formula for û in R 3 \D, assuming (H3):

û -ûin = S ω D,per [(S ω D,per ) -1 [û |∂D -ûin ]] in R 3 \D. (5.25)
In what follows, we denote by ĉl,n : S 2 → R the functions defined for any θ in the unit sphere S 2 by ĉl,n (θ) :

= ∂D e -i nΩ v 0 y•θ (S nΩ v 0 D ) -1 [ϕ l ](y) dσ(y), (5.26)
where ϕ l is the Neumann eigenmode of (1.5). From the functions (ĉ l,n ) n∈Z , we construct for any (m, l) ∈ N × N a T -periodic distribution G ml : R × S 2 by the formula

G ml (t, θ) := n∈Z pm,n ĉl,n (θ)e -inΩt , (5.27)
where we recall that (p m,n ) n∈N denote the Fourier coefficients of the Sturm-Liouville eigenmode p m (t). The functions G ml allow to obtain the far field patterns generated by the modes p m (t)ϕ l (x) as shown in the following lemma.

Lemma 5.1. The following far field expansions hold in the regimes ω → 0 and |x| → +∞ and for (m, l) ∈ N×N:

S ω D,per [(S ω D,per ) -1 [p m ϕ l ]](t, x) = G ml t - |x| v 0 , x |x| (1 + O(|x| -1 ) + O(ω))Γ ω v 0 (x).
Proof. For a given ϕ ∈ H -1 2 (∂D), the far-field expansion of the single layer potential

S ω+nΩ v 0 D [ϕ](x) reads S ω+nΩ v 0 D [ϕ](x) = ∂D e -i nΩ v 0 y• x |x| ϕ(y) dσ(y) (1 + O(|x| -1 ) + O(ω))Γ ω+nΩ v 0 (x).
By using (H3), we have the asymptotic expansion

(S ω+nΩ v 0 ) -1 = (S nΩ v 0 D ) -1 + O(ω). Consequently, we obtain that for any (m, l) ∈ N × N, S ω D,per [(S ω D,per ) -1 [p m (t)ϕ l (x)]](t, x) = n∈Z pm,n S ω+nΩ v 0 D [(S ω+nΩ v 0 D ) -1 [ϕ l ]](x)e -inΩt = n∈Z pm,n ∂D e -i nΩ v 0 y• x |x| (S nΩ v 0 D ) -1 [ϕ l ](y) dσ(y) (1 + O(ω) + O(|x| -1 ))Γ ω+nΩ v 0 (x)e -inΩt . (5.28) Using Γ ω+nΩ v 0 (x)e -inΩt = Γ ω v 0 (x)e -inΩ t- |x| v 0
, we obtain the result. □ Remark 5.4. It is easy to verify that G 00 is the constant function given by Using this result and the modal decomposition (5.24), we obtain the far field expansion of the scattered field, assuming the exceptional coupling situation (5.22).

G 00 (t, θ) ≡ G 00 = -p 0 |D| -1 2 cap(D), ∀(t, θ) ∈ R × S 2 . ( 5

Rescalings of the point-scatterer approximation formula

In what follows, we reproduce the Foldy-Lax approximation argument of Section 3.4 in order to formally derive a homogenized equation for the heterogeneous system (6.2). To do so, we rewrite the far field approximation formula (5.30) in this dilute setting.

First, the Neumann eigenfunctions and eigenvalues of (1.7), and the Sturm-Liouville eigenfunctions and eigenvalues of (1.6) must be replaced according to the dilute regime (6.1) as follows:

ϕ l → s -3 2 ϕ l (•/s), λ l → λ l s 2 , p m → p m (•/s) , µ m → µ m s 2 .
We verify that the obtained eigenfunctions are normalized since

1 sT sT 0 1 ρ(t/s) p m (t/s) 2 dt = 1 T T 0 1 ρ(t) p m (t) 2 dt = 1 and sD s -3 |ϕ l (x/s)| 2 dx = D |ϕ l | 2 dx = 1.
Now, we observe that the set Λ of (1.7) remains invariant by the rescaling (6.1) since

Λ = (m, l) ∈ N × N | λ l s 2 = µ m s 2 v 2 r = (m, l) ∈ N × N | λ l = µ m v 2 r .
The next lemma summarizes how the different quantities occuring in the point-scatterer approximation formula (5.30) are affected by the rescaling.

Lemma 6.1. The coefficients (γ m ) (m,l)∈Λ , (M ml,m ′ l ′ ) (m,l),(m ′ ,l ′ )∈Λ×Λ , the polynomial P of (5.24), the coefficients ĉl,n (θ) of (5.26) and the functions (G ml ) (m,l)∈Λ of (5.27) are changed as follows after the rescaling (6.1):

(i) γ m → sγ m , (ii) M ml,m ′ l ′ → s -1 M ml,m ′ l ′ , (iii) P (ξ) → s -2 P (s 2 ξ), (iv) ĉl,n (θ) → s -1 2 ĉl,n (θ), (v) G ml (t, θ) → s -1 2 G ml (t, θ), (vi) A ω 2 v 2 r δ → sA s 2 ω 2 v 2 r δ and B ω 2 v 2 r δ → s 3 2 B s 2 ω 2 v 2 r δ
.

Proof. Changing ρ(t) in ρ(t/s), we find that the function p 1 m of (5.9) needs to be changed according to the transformation p 1 m → sp 1 m (•/s), from where the point (i) follows. Then one can verify that the sT -periodic Dirichlet-to-Neumann map T 0 sD,per associated to the dilute domain D N,s verifies T 0 sD,per [p m (•/s)ϕ l (•/s)] = s -1 T 0,D [p m ϕ l ](•/s, •/s), which implies that the transformation (ii) for the matrix (M ml,m ′ l ′ ) (m,l),(m ′ ,l ′ )∈Λ×Λ . The point (iii) follows by inserting the rescalings (i) and (ii) in (5.18). Then, the rescaling of the coefficients ĉl,n (θ) of (5.26) can be computed from the integral s∂D e -i nΩ sv 0

y•θ S nΩ sv 0 sD -1 [ϕ l (•/s)](y) dσ(y) = s 2 ∂D e -i nΩ v 0 y•θ s -1 S nΩ v 0 D -1 [s -3 2 ϕ l ](y) dσ(y),
which yields the updates (iv)-(v). The result (vi) is a consequence of (i)-(iii). □

The rescaling rules of Lemma 6.1 yields the following point-scatterer approximation formula for a single rescaled resonator sD centered at zero and modulated by the sT -periodic function ρ(•/s):

û(t, x) -ûin (x) ≃ sû in (0) A ω 2 s 2 v 2 r δ + B ω 2 s 2 v 2 r δ G ml t s - |x| v 0 s , x |x| Γ ω v 0 (x) as |x| s → +∞. (6.3)
Assuming now the setting of Figure 2 where we consider N time-modulated resonators located at the centers (y i ) 1≤i≤N , the resonator located at y i receives the incident source field and N -1 scattered waves coming from the other centers (y j ) 1≤j̸ =i≤N . If s is sufficiently small in the sense that the scatterers located at (y j ) 1≤j̸ =i≤N behave as point sources for the center y i , we can expect that the total field ûN,s of (6.2) can be approximated by the contribution of ûin and these scattered fields:

ûN,s (t, y i ) -ûin (y i ) ≃ s 1≤j̸ =i≤N A ω 2 s 2 v 2 r δ + B ω 2 v 2 r δ G ml t s - |y i -y j | sv 0 , y i -y j |y i -y j | Γ ω v 0
(y i -y j )û in (y j ).

(6.4) Based on this formula, we can discuss the formulation of an effective medium theory following the Foldy-Lax argument of Section 3.4. [START_REF] Kuchment | Floquet Theory for Partial Differential Equations[END_REF][START_REF] Allaire | The Bloch transform and applications[END_REF] taken to be L 2 (R 3 ), because wave packets which propagate in such structures decay at infinity. We are only aware of [START_REF] Barth | The Bloch Transform on Lp-Spaces[END_REF] regarding the definition of the Bloch transform of L p functions with a general p ≥ 1.

In the context of scattering due to a time-modulated media, where the periodicity affects the time variable rather than the space variable, it is not physically relevant to assume that the wave fields decay at infinite times, due to the energy conservation. Hence, there is a need for introducing a Bloch transform on a larger space of functions, which do not necessary decay at infinity.

In this appendix, we propose a definition of the Bloch transform of tempered distributions. Our construction is quite classical: we transpose the ideas that are considered for defining the Fourier transform of tempered distribution. Namely, we start by defining the Bloch transform B in the Schwartz space of smooth rapidly decaying functions in Appendix A.1. We also define the Floquet transform U, which is its counterpart in the Fourier space. Then, in Appendix A.2, we extend B to tempered distributions by duality. We show that the main properties of this extension of the Bloch transform remain true, namely the fact that the Bloch transform of a distribution is equal, up to a multiplicative constant, to the Floquet transform of its Fourier transform. In Appendix A.3, we show an extension of the invariance property of the Bloch transform, namely B[ρf ] = ρBf for a periodic smooth function ρ and a tempered distribution f . In fact, this invariance property holds for periodic distributions ρ, whenever the multiplication ρf makes sense (see Definition A.3). Finally, we compute the Bloch transform of a time-harmonic function t → e -iωt for ω ∈ R in Appendix A.4, and we highlight how we can retrieve from this formalism the time harmonic ansatz (4.1) for the solution to the time-modulated wave equation (1.1).

In what follows, we denote by S(R) the Schwartz space S(R) of smooth rapidly decaying functions (see e.g. [START_REF] Rudin | Functional analysis[END_REF]):

S(R) := f ∈ C ∞ (R) | sup x∈R x p d q f
dx q < +∞, for any p, q ∈ N .

We denote by S ′ (R) the space of tempered distributions T , which are linear forms on S(R) satisfying the continuity property ⟨T, f ⟩ S ′ ,S ≤ C sup Since f ∈ S(R), this series and all its derivatives converge normally, which implies that U f ∈ V(R × R) where V(R × R) is the space of C ∞ functions ϕ(t, α) which are T -periodic in the first variable and Ω quasi-periodic in the second variable:

V(R × R) := {ϕ ∈ C ∞ (R × R) | ∀t ∈ R, ∀α ∈ R, ϕ(t + T, α) = ϕ(t, α
) and ϕ(t, α + Ω) = ϕ(t, α)e iΩt }.

V(R × R) is a Fréchet space when equipped with the topology generated by the family of semi norms N p,q (ϕ) = sup t∈R,α ′ ∈R |∂ q α ∂ p t ϕ(t, α ′ )|, p, q ∈ N.

Lemma A.1. The transform U is an isomorphism from S(R) to V(R × R), whose inverse is the mapping U Proof. It is straightforward to see that f = U -1 (U f ) with U -1 given by (A.3), which proves that U is injective.

In order to prove that U is also surjective, it is enough to show that U -1 ϕ ∈ S(R) for any ϕ ∈ V(R × R), because the Fourier inversion formula for trigonometric series yields U(U -1 ϕ) = ϕ. For an arbitrary n ∈ N, we have:

α p d q dα q U -1 ϕ(α) = α p T T 0 ∂ q α ϕ(t, α) dt = α p T T 0 ∂ q α ϕ(t, α -nΩ + nΩ) dt = α p T T 0 ∂ q α ϕ(t, α -nΩ)e inΩt dt,
where we use the fact that for any q ∈ N, ∂ q α ϕ is Ω-quasi-periodic in the second variable. Using p + 1 integration by parts with n := ⌊α/Ω⌋, we obtain α p d q dα q [U -1 ϕ(α)] = α p (-1) p+1 T (inΩ) p+1 □ By using the properties of the Floquet transform U, we obtain the following result.

Corollary A.1. The Bloch transform is an isomorphism from S(R) to V(R × R), and the following inversion formula holds true:

f (t) = 1 Ω Ω 2
-Ω 2 Bf (t, α)e -iαt dα, for any f ∈ S(R).

(A.6)

We have also the following inversion formula for the Fourier transform of f ∈ S(R):

f (α) = T 0
Bf (t, α) dt.

We now state the well-known Plancherel identities which hold for the Floquet and Bloch transforms.

Proposition A.1. The following Plancherel identities hold for any f, g ∈ S(R): We now use the Plancherel identities of Proposition A.1 to extend B to tempered distributions by duality. Let us introduce the space of distributions V ′ (R × R) on V(R × R).

Definition A.1. We call Φ a distribution on V(R × R) a linear form on V(R × R) for which there exist an integer p ∈ N and an independent constant C > 0 such that |⟨Φ, ϕ⟩ V ′ ,V | ≤ C sup t,α∈R 0≤i,j≤p |∂ i t ∂ j α ϕ(α, t)| for all ϕ ∈ V(R × R).

We denote by V ′ (R × R) the corresponding space of such distributions.

Any function (t, α) → ϕ(t, α) integrable on (0, T )×(-Ω 2 , Ω 2 ) can be identified to a distribution Φ ϕ ∈ V ′ (R×R) by defining

⟨Φ ϕ , ψ⟩ V ′ ,V := 1 2π T 0 Ω 2
-Ω/2 ϕ(t, α)ψ(t, α) dt dα, for all ψ ∈ V(R × R),

where we choose 2π = ΩT as the normalisation constant. We can now define the Bloch transform of a tempered distribution by mimicking the duality identity (A.5).

Definition A.2. For a given tempered distribution f ∈ S ′ (R), the Bloch transform of f is the distribution Bf ∈ V ′ (R × R) defined by the formula ⟨Bf, ϕ⟩ V ′ ,V := 1 T ⟨f, B -1 ϕ⟩ S ′ ,S , for any ϕ ∈ V(R × R).

(A.9)

For a given tempered distribution f ∈ S(R), the Floquet transform of f is the distribution U f ∈ V ′ (R × R) defined by the formula In what follows, we still write (A.2) and (A.5) for Bf and U f , remembering that these mean (A.9) and (A.10) in the distributional sense. Clearly, the property of Lemma A.2 remains true for f ∈ S ′ (R).

⟨U f , ϕ⟩ V ′ ,V := 1 Ω ⟨ f , U -1 ϕ⟩ S ′ ,
Proposition A.2. The following properties hold:

(i) the Bloch transform B and Floquet transform U of (A.9) and (A.10) are bijective mappings from S ′ (R) onto V ′ (R × R) whose reciprocal map are the transform denoted by B -1 : V ′ (R × R) → S ′ (R) and U -1 : V ′ (R × R) → S ′ (R) defined for any ϕ ∈ V ′ (R × R) by ⟨B -1 ϕ, f ⟩ S ′ ,S = T ⟨ϕ, Bf ⟩ V ′ ,V , ∀f ∈ S(R), ⟨U -1 ϕ, f ⟩ S ′ ,S = Ω⟨ϕ, U f ⟩ V ′ ,V , ∀ f ∈ S(R);

(ii) for any f ∈ S ′ (R), it holds Bf = 1 T U f , where f is the Fourier transform of the tempered distribution f . Proof. (i) is a straightforward consequence of the definition and the inverse mapping properties of B and U defined on the Schwartz class. The point (ii) is obtained as follows: using the Definition A.2, we find, for any ϕ ∈ V(R × R):

⟨Bf, ϕ⟩ = 1 T ⟨f, B -1 ϕ⟩ = 1 2π ⟨ f , U -1 ϕ⟩ = Ω 2π ⟨U f , ϕ⟩ = ⟨ 1 T U f , ϕ⟩. □ A.

Invariance property by multiplication with a periodic distribution

We now extend one the main properties of the Bloch transform, which is its invariance by multiplication by a smooth periodic function (or by a L ∞ periodic function for the Bloch transform in L 2 ).

An important observation is to remark that, although the multiplication of two distributions is in general not well determined, the product of a tempered distribution by a T -periodic distribution ρ can be defined upon a certain condition. We recall that a T -periodic distribution can be written in the form of an arbitrary trigonometric series p∈Z ρ p e -ipΩ• with coefficients (ρ p ) p∈Z growing at most polynomially as p → ∞, see e.g. [START_REF] Rudin | Functional analysis[END_REF][START_REF] Donoghue | Distributions and Fourier transforms[END_REF].

  k D and k → K k * D are analytic operator valued mappings: we still denote by S k D and K k * D the continuous extension of these operators to complex numbers k ∈ C. We denote by S D and K * D the single layer potential and Neumann-Poincaré operator associated to the Laplace operator on D, which correspond to S D := S 0 D and K * D := K 0 * D . Finally, we denote by Φ the solution to the exterior problem     

Definition 3 . 1 . 1 2

 311 The Dirichlet-to-Neumann map with wave number k ∈ R is the operator denoted by T k : H (∂D) → H -1 2 (∂D) and defined by

ω v 0 0 D

 00 follows because the operators S ω v 0 D and K ω v 0 * D are analytic in ω and S ω v is invertible for small values of ω ∈ C. We can then compute the expansion of the Dirichlet-to-Neumann operator thanks to a Neumann series:

Corollary 3 . 1 .

 31 For real values of the frequency ω satisfying ω = O(δ 1 2

Proposition 4 . 1 .

 41 For any sufficiently large R > 0 and f ∈ H 1 2 (∂D), the solution w f,k to the problem

4. 1 . 3 .

 13 Assumptions on the modulation ρ: coincidence of a finite number of Sturm-Liouville and Neumann eigenvalues

Proposition 4 . 3 .

 43 There exist a sequence of eigenfunctions (p m (•; ω)) m∈N and eigenvalues (µ m (ω)) m∈N parameterized by ω ∈ (-Ω/2, Ω/2) solving the eigenvalue problem

  By convention, we assume that p m (t) ≡ p m (t; 0) and µ m ≡ µ m (0) where we recall the definition (1.6) for the Sturm-Liouville eigenfunctions (p m (t)) m∈N and eigenvalues (µ m ) m∈N . Let us also recall the definition (1.5) of the Neumann eigenfunctions (ϕ l ) l∈N and eigenvalues (λ l (x)) l∈N . The following proposition motivates the consideration of the set Λ introduced in (1.7). Proposition 4.4. The set of solutions

  0, ∂u m ∂n = 0 on ∂D, which shows that u m is a Neumann eigenfunction of -∆ with eigenvalue µ m /v 2 r . □ Throughout our analysis of the reduced T -periodic wave equation (4.2), we assume the following assumptions on the modulation ρ.

2 . 1 2

 21 Time-periodic Dirichlet-to-Neumann map and its main properties Definition 4.1 (Time-periodic Dirichlet-to-Neumann map). Assume (H1). The Dirichlet-to-Neumann map associated to the time-dependent Helmholtz equation (4.10) is the operator T ω per : L 2 per ((0, T ), H (∂D)) → L 2

□ 4 . 3 .

 43 Well-posedness of the time-periodic reduced wave equation

  ∂D, is ϕ = 0 (this can be seen using the layer potential representation ϕ = D

Lemma 4 . 3 .ω+nΩ v 0 D

 430 The periodic single layer potential S ω D,per can be extended to periodic distributions by settingS ω D,per [ϕ](t, x) := n∈Z S [ φn ](x)e -inΩt ,(4.36)

ω+nΩ v 0 D) - 1 .-

 01 From the jump relation, we infer that denotes the interior Dirichlet-to-Neumann map of the Helmholtz operator ∆ + (ω + nΩ) 2 /v 2 0 , and T ω+nΩ v 0is the exterior one (Definition 3.1). By standard variational estimates, the mapping T

ω 2 v 2 r

 2 Gx + δM x = 0, which yields the expansion(5.15). □ Remark 5.1. The asymptotic expansion (5.14) is given for ω, δ → 0 without assuming any dependency between these two parameters. If further ω = O(δ 1 2

Proposition 5 . 4 .

 54 Let us denote by (p m,n ) n∈Z the Fourier coefficients of the Sturm-Liouville eigenfunction p m : p m (t) = n∈Z pm,n e -inΩt .

. 29 )

 29 Remark 5.5. Far field expansions of retarded potentials featuring such a function G ml depending on the propagation direction x/|x| are classical, see e.g.[START_REF] Cakoni | On the factorization method for a far field inverse scattering problem in the time domain[END_REF] Lemma 4.3].

x∈R 0≤i,j≤p x i d j f dx j , (A. 1 )A. 1 .

 11 for some integer p ∈ N and independent constant C > 0. The Bloch and Floquet transforms on the Schwartz space We start by introducing a variant of the Floquet transform for functions defined in the Fourier space. For any f ∈ S(R), we consider the transformed function U f defined by the formulaU f (t, α) := p∈Z f (α + pΩ)e -ipΩt , ∀(t, α) ∈ R × R. (A.2)

- 1 :

 1 V(R × R) → S(R) given by U -1 ϕ(α)

ff

  C > 0 independent of α. This shows that U -1 ϕ ∈ S(R). □ For a given function f ∈ S(R) defined in the time domain, the Bloch transform of f is the C ∞ (R×R) function Bf defined byBf (t, α) = n∈Z f (t -nT )e iα(t-nT ) , (t, α) ∈ R × R. (A.5)The following lemma relates the Bloch transform B in the time-domain and the Floquet transform U in the Fourier space.Lemma A.2. We haveBf (t, α) = 1 T U f (t, α) = 1 T p∈Z f (α + pΩ)e -ipΩt ,where f is the Fourier transform of f :f (α) := ∞ -∞ f (t)e iαt dt.Proof. Let us compute the trigonometric coefficients of B(f, α): for any p ∈ N, (t -nT )e iα(t-nT ) e ipΩt dt = (t)e i(α+pΩ)t dt = 1 T f (α + pΩ).

2 U 2 f 0 Ω 2 -Ω 2 U 2 BfT 0 Ω 2 -Ω 2 BfA. 2 .

 2202220222 f (t, α)U ĝ(t, α) dα dt.(A.8) Proof. This classical property is an easy consequence of the result of Lemma A.2. Using the usual Plancherel identity for the Fourier transforms f and ĝ of f and g, we writeR (α + pΩ)ĝ(α + pΩ) dα = 1 2π T f (t, α)U ĝ(t, α) dα dt (t, α)Bg(t, α) dt dα = 1 Ω (t, α)Bg(t, α) dα dt.□ Bloch transform of tempered distributions

  S , for any ϕ ∈ V(R × R).(A.10)Remark A.1. The fact that U maps S ′ (R) to V ′ (R × R) is a direct consequence of the inequality (A.4), and similarly for B.

Definition A. 3 .

 3 Let ρ be a T -periodic distribution and denote by (ρ p ) p∈N the Fourier coefficients of ρ: ρ(t) = p∈Z ρ p e -ipΩt . (A.11)

The work of F. Feppon was supported by ETH Zürich through the Hermann-Weyl fellowship of the Institute for Mathematical Research (FIM).

We warmly thank Euan Spence for his insights on high-frequency bounds for the Helmholtz equation.

Corollary 5.1. Assume that the discriminant ∆ of (5. [START_REF] Ammari | Full low-frequency asymptotics for the reduced wave equation[END_REF]) is negative, and that the exceptional coupling situation (5.22) is satisfied. The scattered field generated by the time modulated resonator admits the following far field expansion as |x| → +∞ and ω = O(δ

2 ) + O(|x| -1 ))Γ ω v 0 (x), (5.30) where the constant coefficients A (ξ) and B (ξ) are given by A (ξ) := cap(D) 1 -ξγ m M 00,00 + M ml,ml M 00,00 -M 2 00,ml

P (ξ) γ 0 M 00,ml ξ, (5.31) where P is the second order polynomial of (5.18).

Proof. This result is obtained by inserting (5.24) into the layer potential representation formula (5.25). Using Lemma 5.1 and remembering that ûin = u in (0)p -1 0 |D| 1 2 + O(ω) on ∂D yields: û(t, x) -ûin (x) = ûin (0)p -1 0 |D|

γ m M 00,00 + M ml,ml M 00,00 -M 2 00,ml G 00

The result follows by substituting G 00 with (5.30). □ Remark 5.6. Using the Fourier expansion of G ml , we can also write û(t, x)-ûin (x) as a superposition of spherical waves with wave numbers (ω + nΩ)/v 0 for all n ∈ Z:

Remark 5.7. The point scatterer approximation (5.30) appeals to two remarks: first, the scattered field propagates outgoing waves with high frequency components, more precisely with wave numbers ω+nΩ v0 for any n ∈ Z. Second, these high frequency waves propagate with an amplitude which depends on the space direction x |x| . This suggests that a suitable tuning of the geometry of the resonator D could be used for confining the energy of the scattered high frequency pulse to a desired spatial direction.

Forewords: effective medium theory for fast time-modulated high-contrast metamaterials

In this opening section, we consider a metamaterial obtained by filling a bounded domain U with N timemodulated high-contrast resonators D around centers (y n ) 1≤n≤N and rescaled by a factor s: we assume the homogenization setting of Section 3.4 which is also illustrated on Figure 2. In this dilute regime where s → 0 and N → +∞, the incident frequency is of order one (ω = O(1)) while the modulation frequency is rescaled by a factor 1/s; we assume that the modulation is sT -periodic and is of the form ρ(•/s) for a T -periodic modulation ρ. The rescaling laws can be summarized as follows:

Therefore, the modulation frequency is still much larger than the incident frequency: Ω ≫ ω. Then, we assume that the size factor is of order s = O(δ

). Finally, we still assume that the modulation ρ is tuned to the "exceptional" coupling situation (5.22) featuring two pairs of coinciding Sturm-Liouville and Neumann eigenvalues.

In this part, we denote by ûN,s the solution to the scattering problem in the time-modulated, heterogeneous medium D N,s :

6.2. Effective medium theory for the time-modulated medium in a subcritical regime

Using the law of large numbers, we deduce that ûN,s can be approximated by ûN,s (t, x) ≃ ûin (x) + sN

where K(t, θ) is the sT -periodic integral kernel defined for any t ∈ R and θ ∈ S 2 by

and where A and B are the coefficients defined by (5.31). When sN → 0, this equation can be seen as the first "Born" approximation of the solution ûeff to an effective integral equation.

Corollary 6.1. Assume that the Foldy-Lax approximation (6.4) is valid and the "subcritical" regime sN → 0.

Then we can expect the convergence of the total wave field ûN,s to an effective wave field ûeff solution to the integral equation

where V dx is the probability distribution of the centers (y i ) 1≤i≤N .

Therefore, in the regime where sN → 0, the effective medium is governed by the integral equation (6.5), which further emphasizes that the output signal carries a high frequency component in the far field. This "subcritical" regime corresponds to the one in which higher-order reflections, which correspond to those coming from subsequent interactions between the high-frequency components of (6.4) and the resonators themselves, can be neglected: indeed, (6.4) shows that these reflections are of order O((sN ) 2 ). Nevertheless, the scattered field remains small in this situation (of order O(sN )), and it seems that it would be more relevant to consider the case where sN converges to a constant. Unfortunately, it does not seem clear that one can write a homogenization theory for (6.2) in the "critical" regime sN → α for some α > 0. Indeed, the higher order reflections induce scattered waves with arbitrarily small wavelengths (equal to 2πv0 ω+nΩ/s for any n ∈ Z), which can resolve the lower order reflections entirely on a "small" resonator y i + sD. Hence, these rescaled resonators y i + sD can neither be considered "small" by the higher order reflecting waves nor be approximated by point sources. Hence, a different analysis (for instance, a high frequency homogenization approach in the spirit of [START_REF] Craster | High-frequency homogenization for periodic media[END_REF]) is needed to derive a homogenized medium in the regime sN = O(1), left for a future work. However, it is still possible to write a homogenized equation for the low frequency component of the scattered field in the regime sN = O(1), following the ideas of [START_REF] Ammari | Transient anomaly imaging by the acoustic radiation force[END_REF]. Let us introduce the time-averaging operator P defined by

Then, (6.3) and (5.27) imply that the far field of P û generated by a single rescaled resonator sD is

Repeating the Foldy-Lax approximation argument, one can expect that P û(t, x) converges to the solution û * eff to the following Lippmann-Schwinger equation with varying refractive index:

where C(ξ) := A (ξ) -B (ξ) pm,0 |D| -1 2 cap(D) , with A(ξ) and B(ξ) defined in (5.31). This equation has properties similar to those of the effective equation (3.28) obtained in the unmodulated case, namely the possibility to achieve negative index wave propagation and amplification close to the resonance frequency. Therefore, the true interest of the time-modulated medium lies in the generation of high frequency scattered waves.

Appendix A. Bloch transform of tempered distributions

The Bloch transform (or its variant, the Floquet transform) is a classical mathematical tool in nanophotonics and condensed matter theory; it allows to study the properties of periodic crystals, which are lattice structures featuring a periodically repeated pattern in the three directions of space R 3 . Its most importance property is the invariance with respect to the multiplication by a periodic function (see Appendix A.3), which allows to reduce a spectral problem posed in H 1 (R 3 ) into a parameterized family of elliptic problems posed on the unit-cell of the periodic lattice with periodic boundary conditions. The Bloch transform is in most textbooks Let f ∈ S ′ (R) be a tempered distribution. Whenever the series p∈Z ρ p (f e -ipΩ• ) is convergent in S ′ , we say that the "product" of ρ and f is well-defined and we write ρf := p∈Z ρ p f e -ipΩ• ∈ S ′ (R).

(A.12)

Remark A.2. The definition (A.12) makes sense thanks to two remarks. First, the product of the tempered distribution f ∈ S ′ (R) with the smooth bounded function t → e -ipΩt is a well-defined tempered distribution. This series converges in S ′ (R) if and only if the series p∈Z ρ p ⟨f e -ipΩ• , ϕ⟩ S ′ ,S converges for any ϕ ∈ S(R).

When this is the case, the Banach-Steinhaus theorem implies that the linear form ρf defined by ⟨ρf, ϕ⟩ := lim

belongs automatically to S ′ (R).

Similarly, we can define the product of a T -periodic distribution with a distribution Φ ∈ V ′ (R × R).

Definition A.4. Let ρ be a T -periodic distribution whose Fourier series can be written as (A.11) and Φ ∈ V ′ (R × R). Whenever the series p∈Z ρ p (Φe -iΩp• ) converges in V ′ (R × R), we say that the product ρΦ is well-defined and we write ρΦ

We can now extend the classical invariance result B(ρf ) = ρBf , which is obvious for f ∈ S and ρ ∈ C ∞ per ((0, T ), R) a smooth T -periodic function.

Proposition A.3. Let f ∈ S ′ (R) and ρ be a T -periodic distribution. If ρf is well-defined (in the sense of Definition A.3), then the product ρBf is also well-defined (in the sense of Definition A.4) and it holds

Proof. For any ϕ ∈ V(R × R), we have by using the Definition A.2:

It is clear from Corollary A.1 that for ϕ ∈ V(R × R), e -iΩp• B -1 ϕ = B -1 (e -iΩp• ϕ). Therefore, we arrive at

From the Definition A.4, we obtain that the product ρBf is well-defined, and that ⟨B(ρf ), ϕ⟩ V ′ ,V = ⟨ρBf, ϕ⟩ V ′ ,V .

□

Remark A.3. One can verify that if ρ ∈ L ∞ per ((0, T ), R) and f ∈ L 2 (R), then the tempered distribution ρf of (A.12) is well-defined and coincides with the "usual" product ρf . Therefore, we retrieve the well-known invariance property B(ρf ) = ρBf which holds on these spaces.

A.4. The Bloch transform of time-harmonic functions

We conclude this appendix by computing the Bloch transform of time-harmonic functions t → e -iωt , ω ∈ R. These functions do not belong to L 2 (R) but they are tempered distributions.

which can also formally be written as

Proof. Using Proposition A.2 and the fact that the Fourier transform of e -iω• is α → 2πδ 0 (ω -α), we find

which is the result (A.14). The distributional sense (A.13) is a consequence of (A.10), which states that

□

As an application, we can use Lemma A.3 to formally deduce that the solution u(t, x) to (1.1) must be in the time-harmonic form (4.1). Due to Lemma A.3, the Bloch transform of the incident field u in (t, x) of (1.2) is equal to Bu in (t, x; α) = Ω p∈Z ûin (x)δ(ω -α -pΩ)e -ipΩt , (A. [START_REF] Ammari | A High-Frequency Homogenization Approach Near the Dirac Points in Bubbly Honeycomb Crystals[END_REF] where we observe some separability between the variable α and ω. Then, if 1/ρ(t)u is well-defined, the Bloch transform Bu solves the following set of partial differential equations: