Jean-Jacques Marigo
email: agnes.maurel@espci.fr

An Interface Model for Homogenization of Acoustic Metafilms

 [START_REF] Lemoult | Wave propagation control at the deep subwavelength scale in metamaterials[END_REF]] have been shown to be able to control the wave propagation with high flexibility. In comparison, phononic crystals have ap r i o r ihigher dimensions because of their wavelength-scale period. However, if the metamaterial has subwavelength period, many of the observed phenomena are attributable to Fabry-Perot type resonances, resonances in the hole or resonances of the wires. Therefore, these structures have a limitation in their thickness, which has to be at wavelength scale to produce such resonances (and thickness refers to the size in the direction perpendicular to the plane containing the periodic cells). In order to reduce the size of the devices, structures with subwavelength thickness have been developed. They are known as metasurfaces and metafilms. Despite the vanishing thickness in comparison to the incident wavelength, the capability of these ultrathin devices to control the wave propagation has been evidenced. This is because they are based on a resonance which is not related to their thickness. It can be a thin elastic membrane within the unit cell [Ma et ] have been largely used for massive structures leading to effective bulk modulus and effective mass density, being the acoustic counterparts of the effective permittivity and permeability in electromagnetism. An enlightening review can be found in Ref. [START_REF] Simovski | On electromagnetic characterization and homogenization of nanostructured metamaterials[END_REF]]. When metafilms or metasurfaces are concerned, first attempts have been proposed which use the same retrieval parameters as for bulk materials and affecting a thickness value more or less relevant (see the discussions in § 3o f[ S i m o v s k i (2011)] and in §2o f[ H o l l o w a yet al. (2009)]). It is now admitted that these approaches are not pertinent and rather, jump conditions or transmission conditions of the fields across a zero thickness surface are thought. In the context of electromagnetism, effective surface parameters are expressed in term of effective surface admittances [START_REF] Zhao | Homogenization of plasmonic metasurfaces modeled as transmission-line loads[END_REF]] or surface susceptibilities [START_REF] Holloway | A discussion on the interpretation and characterization of metafilms/metasurfaces: The two-dimensional equivalent of metamaterials[END_REF]; [START_REF] Dimitriadis | Surface susceptibility bianisotropic matrix model for periodic metasurfaces of uniaxially mono-anisotropic scatterers under oblique te-wave incidence, Antennas and Propagation[END_REF]; [START_REF] Kuester | Averaged transition conditions for electromagnetic fields at a metafilm, Antennas and Propagation[END_REF]; Holloway et al. (2005)]. In addition to be closely related to the notion of electric polarization density and surface magnetization, which are not easily transposable to the case of acoustic or elastic waves, these techniques require an inversion procedure to get the surface susceptibilities, from the measurement of the scattering coefficients. Thus, although the problem of the artificial thickness is avoided, the question of wether or not the obtained parameters depend on the scattering problem that has been considered remains.

In this paper, we present a homogenization method for vanishing thickness metamaterials which explicitly accounts for the microstructure of the unit cell. The problem ends with jump conditions for the acoustic pressure and for the normal acoustic velocity involving parameters being wave independent, by construction. This is because, as in the standard homogenization, the parameters are determined by solving (analytically or numerically) elementary cell problems in the static case (that is for zero frequency). The model relies on a separation of scales, a micro scale associated to the small scatterer size and a macro scale associated to the acoustic wavelength, and ε is the small parameter that measures the ratio of the two scales. Each scale is associated to a system of coordinates which is relevant or not to describe the variation of the acoustic fields wether we are close to the film or far from it. Thus, a separation of the space is used, into an outer region, typically the far field, where only the macro scale makes sense and an inner region, the near field, where both the micro and macro scales are needed. Expansions of the fields in power of ε are performed in both regions and finally, matching conditions are used between the two regions. This approach has been developed in the context of static elasticity, see [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF]; [START_REF] David | Homogenized interface model describing inhomogeneities located on a surface[END_REF]] for a complete description. The case of wave propagation has been less regarded. We mention the works of Capdeville and Marigo for seismic waves in the time domain [START_REF] Capdeville | Second order homogenization of the elastic wave equation for non-periodic layered media[END_REF] The paper is organized as follows. In Section 14.2, we apply the interface model for the problem of acoustic wave propagation through a thin film made of a periodic row of sound hard inclusions. The jump conditions are derived, Eqs. (14.27), with the interface parameters (B, C)beingassociatedtotheelemen taryproblems,Eqs. (14.17). Section 14.3 presents results in the case of rectangular inclusions in two dimensions, with thickness e.T h i sc h o i c ei sn o tc a s u a l . First, it allows for approximate, but explicit, solutions of the elementary problems. Thus, the interface parameters (B, C)a r eg i v e n ,a n d associated transmission and reflection coefficients of a plane wave at oblique incidence, Eqs. (14.33) (14.34). The validity of our interface model is inspected by comparison with full wave simulations, and discussed in the light of the comparison with the classical homogenization. The results show that the interface homogenization is accurate for kh, ke < 1, while classical homogenization can be used for kh < 1a n de>hand it largely underestimates the scattering properties of thin structured layers. The small parameter ε can be forced to values of order unity; typically for ε and ke of order unity, the discrepancy between the exact solution and the solution of the homogenized problem do not exceed few per percent. We end with an illustrative example of an ultra thin structured film obstructing a waveguide with rigid walls. Again, an explicit solution of the homogenized problem is possible and a good agreement with direct numerics is obtained. Technical calculations are collected in the Appendices.

X 1 X 2 X 3 h e X 1

The Homogenized Interface Model

The acoustic pressure P (X)s a t i s fie st heH e l m ho l t ze q ua t i o no ut si de the inclusions

∆+k 2 P (X)=0, (14.1)
with k the wavenumber (the time dependance is e -iωt ,w i t hω the frequency, and it is omitted in the following). Here, we restricted our study to sound hard inclusions, with Neumann boundary condition on the boundary of the inclusions ∂ n P =0 ,w h e r e∂ n denotes the normal derivative. Also, the wave propagation is associated to a wave source and to a proper radiation condition that we do not need to specify at this point. The inclusions form a periodic arrangement located near the (X 2 ,X 3 )p l a n e( w i t ht y p i c a lt h i c k n e s se along X 1 , Fig. 14.1(a)). For simplicity, we assume that h is the period of the cell along X 2 and X 3 (we could assume two different periods along X 2 and X 3 without additional complexity, except in the notations). We inspect the possibility to replace this problem by a simpler one, where the structured film is replaced by an interface associated to jump conditions for the pressure P and its normal derivative, or normal velocity, ∂ X 1 P (the nature of the equivalent interface, in terms of its thickness, will be discussed).

In our acoustic problem, the natural small parameter is ε = kh ≪ 1a n dt ob ec o n s i s t e n t ,w en e e dt ow r i t et h eH e l m h o l t ze q u ation in a dimensionless form, with x ≡ kX,a n dp(x) ≡ P (X). In x =( x 1 ,x 2 ,x 3 ), the period is now ε and the film thickness is ke (Fig. 14.2(a)); it is assumed that e and h are of the same order of magnitude. We get (∆ x +1)p(x)=0 ,w h i c hi sw r i t t e n ,f o rc o n v enience using u,

⎧ ⎨ ⎩ div x u(x)+p(x)=0, with u ≡ ∇ x p, u.n |∂D =0, (14.2)
and u is the acoustic velocity, in dimensionless form.

The inner/outer expansions and associated matching conditions

The idea is now to expand the solution with respect to the small parameter ε,n a m e l y

⎧ ⎨ ⎩ p = p 0 (x)+εp 1 (x)+ε 2 p 2 (x)+..., u = u 0 (x)+εu 1 (x)+ε 2 u 2 (x)+... (14.3)
In principle, this expansion can be used in the whole space (see e.g. [START_REF] Martin | Scattering of long waves by cylindrical obstacles and gratings using matched asymptotic expansions[END_REF]]). Nevertheless, if the spatial derivatives in Eq. (14.2) make ε to appear, the resolution may become tricky. Such complications are avoided if a displacement in x of order unity produces a variation in p of order unity, namely ∂ x i p ∼ p.This is ensured in the far field of the film, where

∂ X i P ∼ kP → ∂ x i p ∼ p.
The story is different in the near field: there, the film generates evanescent waves whose strongest variations are associated to the smallest scale of the microstructure, say ∂ X i P ∼ P/h → ∂ x i p ∼ p/ε. Then, the wavefield satisfies pseudo-periodic conditions, for instance along X 2 , P (X 1 ,X 2 + h, X 3 )=P (X 1 ,X 2 ,X 3 )e ik 2 h ,w i t hk 2 the component of the incident wavenumber along X 2 (the same applies along X 3 ). This behavior is associated to low variations of P (∂ x 2 p ∼ p)f o rn o nn o r m a li n c i d e n c ek 2 ̸ =0 .T h u s ,w eh a v et od e a lw i t h different scales in the far and near fields, and two scales in the near field.

The presented approach solves this annoying problem with two ingredients: first, a separation of the space into an inner and an outer regions, which correspond to the near and far fields, respectively. In the outer region, the natural coordinates x ≡ (x 1 ,x 2 ,x 3 )areadapted and the expansion, Eq. ( 14.3), applies. In the inner region, the rapid variations of p and u are accounted introducing a new system of coordinates y = x/ε such that ∂ y i p ∼ p for the rapid variations of p;n e x t ,t h es l o wv a r i a t i o n sa l o n gx 2 and x 3 are accounted for by keeping x ′ ≡ (x 2 ,x 3 )a sa d d i t i o n a lc o o r d i n a t e s .N o t et h a ti nt h ey coordinates, the cell has a thickness unity (Fig. 14.2(b)). Owing to this modification, the expansions read Outer expansion

⎧ ⎨ ⎩ p = p 0 (x)+εp 1 (x)+ε 2 p 2 (x)+..., u = u 0 (x)+εu 1 (x)+ε 2 u 2 (x)+... Inner expansion ⎧ ⎨ ⎩ p = q 0 (y, x ′ )+εq 1 (y, x ′ )+ε 2 q 2 (y, x ′ )+..., u = v 0 (y, x ′ )+εv 1 (y, x ′ )+ε 2 v 2 (y, x ′ )+... ( 14
.4) Finally, both regions are connected in some boundary region, where the evanescent field is vanishing at small x 1 values corresponding to y 1 = x 1 /ε →±∞.Thesematchingconditionsarewrittenusing Taylor expansions of p 0 (x 1 , x ′ )=p 0 (0, x ′ )+x 1 ∂ x 1 p 0 (0, x ′ )+••• = p 0 (0, x ′ )+εy 1 ∂ x 1 p 0 (0, x ′ )+...,sameforu 0 ,andidentifyingtheterms in ε n in the inner and outer expansions, Eqs. (14.4). We get

⎧ ⎪ ⎨ ⎪ ⎩ p 0 (0 ± , x ′ )= lim y 1 →±∞ q 0 (y, x ′ ), u 0 (0 ± , x ′ )= lim y 1 →±∞ v 0 (y, x ′ ), (14.5)
at leading order, and

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ p 1 (0 ± , x ′ )= lim y 1 →±∞ q 1 (y, x ′ ) -y 1 ∂p 0 ∂x 1 (0 ± , x ′ ) , u 1 (0 ± , x ′ )= lim y 1 →±∞ v 1 (y, x ′ ) -y 1 ∂u 0 ∂x 1 (0 ± , x ′ ) , (14.6)
at first order, and fortunately, we do not need to go at higher orders in the expansions. A last subtlety of the interface model is to assume that the functions q n and v n are periodic with respect to y 2 and y 3 .A g a i n ,t h i si sn o tm e a n i n g l e s si nt h ep r e s e n tc o n t e x t if we have in mind the condition of pseudo periodicity. This condition is handled by the variables x 2 and x 3 ,f o ri n s t a n c eq n (y,x 2 + ε, x 3 )=e ik 2 h q n (y,x 2 ,x 3 )( n o t et h a tt h ec o n d i t i o no fp s e u d ope r i odicity applies for the outer solution (p n , u n ), and from (14.5)-(14.6), it applies for (q n , v n )a l s o ) .I fo n et h i n k st ot h eq n in terms of separable functions q n (y, x ′ )=f (y)g(x ′ )(andthiswillbethecaseinthe quasi-static limit), we recover the form of a Floquet solution, with g(x ′ )=e ik ′ .x ′ /k and f periodic with respect to y ′ ≡ (y 2 ,y 3 ). Now, the matching conditions, Eqs. (14.5)-(14.6) can be written in an alternative form, that will be used later

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ p 0 = δq 0 , p 1 =l i m y m 1 →+∞ δq 1 (y m 1 ) -2y m 1 ∂p 0 ∂x 1 (0, x ′ ) , u 0 .e 1 = δv 0 1 , u 1 .e 1 =l i m y m 1 →+∞ δv 1 1 (y m 1 ) -2y m 1 ∂u 0 1 ∂x 1 (0, x ′ ) , (14.7)
(Fig. 14.2), where we have defined δf(y m 1 ) ≡ f (+y m 1 )f (-y m 1 ), and δf ≡ δf(+∞). (14.8)

In the above expressions, the dependence of f on y ′ and x ′ are omitted.

14.2.2. Equations governing the outer and inner terms at orders 0 and 1

The equations in the outer and inner problems read, from Eqs. (14.2)

Outer problem: div x u 0 + p 0 =0, u 0 = ∇ x p 0 , Inner problem:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ div y v 0 =0, ∇ y q 0 = 0, v 0 = ∇ y q 1 + ∇ x ′ q 0 , div x ′ v 0 +div y v 1 + q 0 =0, v 0 .n |∂D = v 1 .n |∂D =0.
(14.9)

For the inner problem, we must add, at orders 0 and 1, the p erio dic conditions of q n and v n , n =0, 1, with respect to y ′ . These are the equations that will yield the elementary problems, and, together with the matching conditions, Eqs. (14.5)-(14.6), the jump conditions in the outer problem.

The jump conditions for the outer problem of order 0

The first two equations of the inner problem in (14.9) show that q 0 does not depend on y and that dy

′ v 0 1 (y 1 = -∞, y ′ , x ′ )= dy ′ v 0 1 (y 1 =+ ∞, y ′ , x ′ )
. This latter relation is obtained by integrating div y v 0 =0o v e rY ∞ \D and using the boundary condition v 0 .n =0o n∂D and the periodicity of v 0 with respect to y ′ .I tf o l l o w sf r o mE q s .( 1 4 . 5 )t h a t( i )q 0 (x ′ )=p 0 (0 ± , x ′ )a n d (ii) dy ′ v 0 1 (±∞, y ′ , x ′ )=u 0 1 (0 ± , x ′ ), so that the jump conditions at leading order read p 0 = u 0 .e 1 =0.

(14.10)

The structured film is transparent at leading order, with the pressure and the normal velocity being continuous across the interface. In the context of static elasticity, for microcraks or microholes in an elastic body, this has been referred as the "the principle of the dressmaker" ([Marigo and Pideri (2011)]): "It is not necessary to sew entirely two pieces of fabrics in order to render invisible their relative opening, it is sufficient to sew them at a great number of points regularly spaced." This is not trivial ap r i o r i ,a n di tf a c t ,i ti sn o te x p e c t e di fw eh a v e the limiting case of a rigid wall ℓ = h in mind, and this will be discussed below. For now, the solution (p 0 , u 0 )a tt h i sl e a d i n go r d e r is the solution of the Helmholtz equation in the absence of interface, which means that (p 0 , u 0 )c o r r e s p o n dt ot h ep r e s s u r ea n dv e l o c i t y associated to a given incident wave. At this stage, the appropriate radiation condition has to be accounted for to determine the unique solution of the problem at leading order.

To capture the effect of the interface, we need to go at the next order to find the jump conditions.

The jump conditions for the outer problem of order 1 and the associated elementary problems

At order 1, we have to consider a problem on the inner terms (q 1 , v 0 ).

To do so, it is convenient to intro duce (q, v)d e fi n e db y

⎧ ⎪ ⎨ ⎪ ⎩ q(y, x ′ ) ≡ q 1 (y, x ′ ) -y 1 ∂p 0 ∂x 1 (0, x ′ ), v(y, x ′ ) ≡ v 0 (y, x ′ ) -u 0 (0, x ′ ). (14.11)
From the equations (14.9) in the inner problem, (q, v)s a t i s f y

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ div y v =0, v = ∇ y q, lim y 1 →±∞ v =0, v.n |∂D = -u 0 (0, x ′ ).n |∂D ,
q and v.n being y ′ -periodic, (14.12)

where we have used that q 0 (x ′ )=p 0 (0, x ′ )a n du 0 = ∇ x p 0 ,f r o m the outer problem. For the boundary conditions at y 1 →± ∞ ,w e also used the matching condition, Eq. (14.5), u 0 (0, x ′ )=v 0 (y 1 = ±∞, y ′ , x ′ ). If this problem is solved, the jump conditions can be determined.

Inspecting the form of p 1 in Eqs. (14.7), we see that the jump condition on p is simply determined using the definition of q in (14.11), yielding

δq 1 (y m 1 ) -2y m 1 ∂p 0 ∂x 1 (0, x ′ )=δq(y m 1 ), (14.13)
with the notations (14.8).

For the jump condition on u 1 ,w ei n t e g r a t et h et h i r de q u a t i o no f the inner problem in (14.9) over Y\D.T h i sl e a d st o

δv 1 1 (y m 1 ) - ∂u 0 1 ∂x 1 (0, x ′ )(2y m 1 -S D )+ ∂ ∂x α Y\D dy v α =0, (14.14)
where α =(2, 3), and with S D the volume of D in the y coordinates (we use the summation convention on repeated indices). We have used that (i) v 1 is periodic with respect to y ′ ,( i i )q 0 (x ′ )=p 0 (0, x ′ ) from the previous section and (iii) div x ′ u 0 + p 0 = -∂ x 1 u 0 1 (from the outer problem). Note that (2y m 1 -S D )i ss i m p l yt h ev o l u m eo fY\D. It is now sufficient to take the limit y m 1 → +∞ in Eqs. (14.13)-(14.14) and using (14.7) to get

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ p 1 = δq, u 1 .e 1 = - ∂u 0 1 ∂x 1 (0, x ′ ) S D - ∂ ∂x α Y∞\D dy v α (y, x ′ ), (14.15)
with

Y ∞ =(-∞, ∞) × (0, 1) 2 .
As written above, the jump conditions are not satisfactory in the sense that both q and v α depend on the value of u 0 on ∂D (in (14.12)). However, it is possible to split the problem (14.12) on (v,q)i n t os i m p l ep r o b l e m so nq (i) (y), i =1 , 2, 3, called elementary problems. These problems are "elementary" in the sense that they can be solved once and for all and this is because these problems are independent of x ′ and of the value of u 0 on ∂D.S p e c i fi c a l l y ,( q, v) can be decomposed into

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ q(y, x ′ )= ∂p 0 ∂x i (0, x ′ ) q (i) (y)+q(x ′ ), v(y, x ′ )= ∂p 0 ∂x i (0, x ′ ) ∇q (i) (y), (14.16)
with q (i) (y)s a t i s f y i n g the elementary problems:

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ∆q (i) =0, ∇ q (i) + y i .n =0, on ∂D, lim y 1 →±∞ ∇q (i) =0, q (i) being y ′ -periodic.
(14.17) It is easy to see that such decomposition ensures that (q, v)issolution of (14.12) (using u 0 = ∇ x p 0 ). Note that q(x ′ )h a sb e e ni n t r o d u c e d since q(y, x ′ )i n( 1 4 . 1 2 )i sd e fi n e du pt oax ′ -dependent field, but its determination is not needed.

We can now express δq and v α in (14.15) in terms of the elementary solutions q (i)

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ δq = ∂p 0 ∂x i (0, x ′ ) δq (i) , v α (y, x ′ )=u 0 i (0, x ′ ) ∂q (i) ∂y α (y), (14.18)
and the jump conditions (14.15) become

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ p 1 = ∂p 0 ∂x i (0, x ′ ) δq (i) , u 1 .e 1 = - ∂u 0 1 ∂x 1 (0, x ′ ) S D - ∂u 0 i ∂x α (0, x ′ ) Y∞\D dy ∂q (i) ∂y α (y), (14.19)
which can be written introducing the parameters b and c

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ p 1 = b i ∂p 0 ∂x i (0, x ′ ), u 1 .e 1 = c ij ∂u 0 i ∂x j (0, x ′ ), (14.20)
with b i = δq (i) and

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ c 11 = -S D , c 1β = - Y∞\D dy ∂q (1) ∂y β , c α1 =0, c αβ = - Y∞\D dy ∂q (α) ∂y β , (14.21)
where α, β take the values 2 and 3.

It is essential that the elementary problems, Eqs. (14.17), do not depend on the incident wave (as the former problem Eqs. (14.12), does, through u 0 in the boundary condition). This ensures that the parameters b and c are characteristic of the interface independently of the particular scattering problem that will be considered.

Construction of the up to order 1 effective problem

The final up to order 1 effective problem requires two additional steps. First, we have obtained jump conditions at order 0 and at order 1. Strictly, an iterative resolution of the problem is possible which would involve first the resolution of the order 0 outer problem (which does not see the structured layer) yielding (u 0 ,p 0 ), and then the resolution of the order 1 outer problem associated to the jump conditions (14.20) (the effective parameters entering in (14.20) being known after the elementary problems (14.17) have been solved). However, it is stressed in [START_REF] David | Homogenized interface model describing inhomogeneities located on a surface[END_REF]] that such procedure is not suitable in practice, notably if we have numerical resolutions in mind, and rather, a unique outer problem involving p 0 + εp 1 (and u 0 + εu 1 ) has to be considered. To do so, we use

f = f 0 + εf 1 + O(ε 2 )a n d f = ε f 1 +O(ε 2
)foranyfieldf with f 0 =0.F romEqs. (14.10) and (14.20), we get

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ p = ε b i 2 ∂p ∂x i (0 -, x ′ )+ ∂p ∂x i (0 + , x ′ ) + O(ε 2 ), u .e 1 = ε c ij 2 ∂u i ∂x j (0 -, x ′ )+ ∂u i ∂x j (0 + , x ′ ) + O(ε 2 ), (14.22)
where we have introduced for fields being discontinuous at x 1 =0 the mean value of their limits x 1 → 0 ± . Next, the above conditions will be written in an alternative form, equivalent at order O(ε 2 ). Loosely speaking, the new jump conditions have to restore the initial thickness of the structured layer, and this is required following the intuitive argument that the equivalent interface has to support the acoustic energy in the actual layer in order to ensure that it supports a positive energy, see [START_REF] Marigo | Homogenization models for thin rigid structured surfaces and films[END_REF]] for more details. In [START_REF] Abdelmoula | Comportement asymptotique d'une interphase élastique mince[END_REF]], this is done by considering that the problem is written in a domain of fixed size in which the film of size ε has been inserted, resulting in a shift of the boundaries. In [Delourme (2010); Delourme et al. (2012)], this is done to overcome troubles in the homogenized problem due to the signs of the (b i , c ij ), specifically some of these parameters have to be positive or negative to ensure existence and uniqueness of variational solutions in the homogenized problem. In this latter case, a family of equivalent jump conditions are proposed, which correspond to interfaces "enlarged" of an arbitrary thickness; here, we consider as h i f to fe.As h i f to fe in the real space corresponds to a shift of ke in the x-space and we assume that e/h = O(1), from which ke = εe/h = O(ε). To do so, we use the Taylor expansion of p n , n =0, 1,

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ p 0 ± εe 2h , x ′ = p 0 (0, x ′ ) ± εe 2h ∂p 0 ∂x 1 ± εe 2h , x ′ + O(ε 2 ), p 1 ± εe 2h , x ′ = p 1 (0 ± , x ′ )+O(ε), (14.23)
to get

p εe 2h , x ′ -p - εe 2h , x ′ = εe 2h ∂p 0 ∂x 1 - εe 2h , x ′ + ∂p 0 ∂x 1 εe 2h , x ′ + ε p 1 + O(ε 2 ). (14.24)
Finally, defining for any f ,t h em e a nv a l u eo ff across the interface of thickness ke

f = 1 2 f - εe 2h , x ′ + f εe 2h , x ′ , (14.25)
and coming back to (14.26), the new jump conditions read

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ p e = ε b i + e h δ i1 ∂p ∂x i + O(ε 2 ), u e .e 1 = ε c ij + e h δ i1 δ j1 ∂u i ∂x j + O(ε 2 ). (14.26)
Now, the homogenized model is written in the real space for the pressure field P (X)( F i g .1 4 . 3 ) :P satisfies the Helmholtz equation ∆P +k 2 P =0exceptin-e/2 <X 1 <e/2(thisregionisdisregarded) and it experiences jump conditions which link its values at X 1 = ±e/2, with P e ≡ P (e/2, X ′ ) -(-e/2, X ′ ),t ot h em e a nv a l u e s of its spatial derivatives

∂ X i P ≡ 1 2 [∂ X i P (-e/2, X ′ )+∂ X i P (e/2, X ′ )] X 1 X 2 X 3 ∆P + k 2 P =0
jump conditions e Texte Fig. 14.3. In the homogenized problem, the Helmholtz equation applies in the whole-considered-space except in the region -e/2 <X 1 <e / 2w h i c hi sd i s r egarded; the jump conditions (14.27) apply between X1 = -e/2a n de/2.

(the same for ∂ X 1 P ):

Jump conditions ⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ P e = hB i ∂P ∂X i , ∂P ∂X 1 e = h C ij ∂ 2 P ∂X i ∂X j , (14.27)
with (1) ,B α = δq (α) ,

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ B 1 = e h + δq
C 11 = e h -S D , C 1β = - Y∞\D dy ∂q (1) ∂y β , C α1 =0, C αβ = - Y∞\D dy ∂q (α) ∂y β , (14.28)
where α, β take the values 2 and 3.

Remarks

As previously said, it is essential that the elementary problems, Eqs. (14.17), do not depend on the incident wave. This ensures that the interface parameters are characteristic of the structured film only. For acoustic waves, this could be done for metafilms containing long hard scatterers wrapped in the unit cell, thus allowing for the resonance of the enrolled element. For elastic or electromagnetic waves, this could be done for metafilms containing inclusions with a contrast in its material properties such that the wavelength inside the inclusions is of the same order than the size of the inclusions.

The following remarks can be done on the elementary problems, Eqs. (14.17)

• The problem for i =1i sn o t h i n gb u tt h ep r o b l e mo fau n iform potential flow past a rigid obstacle in a duct. With velocity potential ϕ(y)=q (1) + y 1 ,w eg e t∆ ϕ =0 ,∂ n ϕ |∂D =0a n d lim y 1 →±∞ ϕ = y 1 + b ± .T h ec o e ffi c i e n tb 1 = b +b -is called the blockage coefficient and it has been calculated for several shapes of obstacles (see [START_REF] Martin | Scattering of long waves by cylindrical obstacles and gratings using matched asymptotic expansions[END_REF]] and references therein). This coefficient can be non zero even for zero surface obstacle S D =0,reducingtoaplateoflengthℓ. • For inclusion shap e b eing symmetric with resp ect to y 2 , q (1) is symmetric also. It results that Y\D dy ∂ yα q (1) =0. • The cell problem for i =2 , 3i sn e e d e do n l yf o ro b l i q u ei n c idence of the wave. Besides, for inclusion shape being symmetric with respect to y 2 , q (α) , α =2 , 3a r ea n t i s y m m e t r i c ;i nt h i sc a s e , δq (α) =0.

Finally, we have said that the present derivation of the jump conditions follows from previous works [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF]; [START_REF] David | Homogenized interface model describing inhomogeneities located on a surface[END_REF]] in the static case (k =0).F orthesak eofcompleteness, we report in the Appendix 14.5 a short discussion on the links with this work.

The Case of Sound Hard Rectangular Inclusions in 2D

In the following, we consider a 2 dimensional array made of rectangular inclusions infinite along X 3 (otherwise of size ℓ along X 2 and thickness e along X 1 ). This inclusion shape being symmetric with respect to y 2 ,t h ep r e c e d i n gr e m a r k so nt h ee l e m e n t a r yp r o b l e m s apply and the jump conditions (14.27) simplify to Jump conditions

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ P e = hB ∂P ∂X 1 , ∂P ∂X 1 e = hS ∂ 2 P ∂X 2 1 + hC ∂ 2 P ∂X 2 2 , (14.29)
with S ≡ (e/h -S D ), B ≡ e/h + δq (1) and C ≡-Y∞\D dy ∂ y 2 q (2) .

As previously, f means the mean value of f at X 1 = ±e/2.

The interface parameters

An obvious interface parameter is S = e/h(1ℓ/h). Next, (B, C) can be calculated by solving numerically the elementary problems, Eqs. (14.17). A simple procedure to calculate (B, C)( b a s e do n mode matching) and the associated scripts are provided in the Appendice 14.6. Alternatively, we can use approximate but explicit expressions. An estimate of C is, as a rule of thumb,

⎧ ⎨ ⎩ C ≃ e 1 - ℓ h - π 8 1 - ℓ h 2
, if this leads to a positive value, C ≃ 0, otherwise, (14.30) which means that C (which is always positive) can be neglected for small e/h values. A more precise estimate of B can be found in [START_REF] Flagg | Sway added-mass coefficients for rectangular profiles in shallow water[END_REF]] (see also [START_REF] Suh | Closed-form solutions for wave reflection and transmission by vertical slotted barrier[END_REF]]). It is written as the sum of two terms, one of which does not vanish for e =0( b l oc k a g e coefficient of a flat plate)

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ B = e h -ℓ + B 0 , B 0 = 2 π log d 2 + 1 2d , with d ≡ tan π(h -ℓ) 4h .
(14.31)

In fact, Ref. [START_REF] Flagg | Sway added-mass coefficients for rectangular profiles in shallow water[END_REF]] provides an approximation of

B 0 ≃ 2 π 1 -log (4(1 -ℓ/h)) + 1 3 (1 -ℓ/h) 2 + 281 180 (1 -ℓ/h
) valid for ℓ/h close to 1. The expression of B 0 in Eq. (14.31) above can be found in [START_REF] Morse | Theoretical Acoustics[END_REF]] (the calculation is done for flat plates using techniques of complex variables). The parameters (B, C) are reported in Fig. 14.4.

Validity of the interface model -The scattering of an incident plane wave

We consider the problem of the scattering of an incident plane wave at oblique incidence θ on the structured film (Fig. 14.5). When replaced by the equivalent interface associated to the jump conditions Eqs. (14.29), the problem reduces to a one dimensional problem whose solution reads P (X 1 < -e/2,X 2 )=e ik sin θX 2 [e ik cos θ(X 1 -e/2) + Re -ik cos θ(X 1 -e/2) ], P (X 1 >e/2,X 2 )=e ik sin θX 2 Te ik cos θ(X 1 +e/2) , (14.32) with

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ R =i a -b (1 -ia)(1 -ib) ,T = 1+ab (1 -ia)(1 -ib)
,

with a ≡ kh 2 cos θ S + C tan 2 θ ,b≡ kh 2 cos θB. (14.33) Note that for ℓ → h,w eh a v eB → +∞ (thus b → +∞)a n d C → 0. We obtain R →-1/(1 -ia)a n dT → ia/(1 -ia), with a ∝ S ̸ =0exceptforaflatarra ye =0.Ob viously ,w ew ouldexpect R =1,T =0inthiscasesincethearra ybecomesasoundhardwall.
It is a classical problem in homogenization theories when a new small parameter is introduced, here ε ′ =1ℓ/h.I nt h ep r e s e n tc a s e ,t h e problem ε ′ → 0a p p e a r si nt h ej u m pc o n d i t i o n s ,E q .( 1 4 . 2 6 ) ;i th a s been assumed that εB ≪ 1(asithasbeenassumedthatεC ≪ 1and εS ≪ 1). Inspecting B in Eq. (14.31), we see that this assumption can fail when (hℓ) → 0, depending on how fast ε and ε ′ go to zero: (i) if e ̸ =0 ,i tf a i l si fε/ε ′ →∞and (ii) if e =0 ,i tf a i l si f ε log ε ′ →∞ .T h i sl a t t e rl i m i tc o r r e s p o n d st ot h eo n ee s t a b l i s h e d in [Sanchez-Hubert and Sanchez-Palencia (1982)] for perforated thin plate.

In the actual problem, the incident wave generates an evanescent field in the vicinity of the structured film, which may have as t r o n gi n fl u e n c eo nt h er e fl e c t i o na n dt r a n s m i s s i o no ft h ep l a n e wave. Below, we inspect the validity of our homogenized solution, namely its ability to incorporate the effect of the evanescent field in the interface parameters.

To b egin with, we rep ort in Figs. 14.6 and 14.7 the pressure fields for a structured interface of thickness e,w i t hke =0.5a n d1a n d ,i n the unit cell, different aspect ratio ℓ/h.F ortheconsideredgeometries, the interface parameters are given in the table below. The notation X[ Y ]i su s e d ,w i t hXt h ev a l u eo fB or C given by the estimates To go further, we insp ect the variations of the reflection co efficient as a function of k and e (say for a fixed h value). This is done by comparing R in Eq. (14.33) (blue dotted lines in Figs. 14.8) with R ex computed numerically (plain orange lines). We also report the behavior of R (grey dotted lines) given by the classical homogenization of layered media (see Appendix 14.7.1). In this approach, the propagation inside the structured layer is explicitly accounted for (while the boundary layer effects are disregarded) and the structured film is replaced by an equivalent film of thickness e filled with an homogeneous anisotropic medium. Classical homogenization has been shown efficient to predict the extraordinary optical transmission at the Brewster angle [Maurel et

coefficient R R = (c 2 -1)e ike cos θ/2 [c -itan(ke/2)] [c +i/ tan(ke/2)] , (14.34)
where c = h cos θ/(hℓ)( t h ed e r i v a t i o ni sg i v e ni nt h eA p p e n d i x 14.7.1). From Figs. 14.8, the ranges of validity of b oth homogenizations can be defined: the interface homogenization is valid for ke, kh < 1, while the classical homogenization is valid for e/h > 1, specifically ke > 1 >kh(inspecting higher values of kh would reveal that both homogenizations fail for kh > 1, as expected). The error in R for small e/h is attributable to the behavior of B when ke → 0, namely

lim e→0 R ≃-i kh 2 cos θB 0 , (14.35)
and B 0 ̸ =0w h i l e R → 0( ad i s c u s s i o no nt h ej u m pc o n d i t i o n s obtained from the classical homogenization in the limit ke → 0i s presented in the Appendix 14.7.2). Thus, classical homogenization underestimates the scattering properties of thin films.

Application to the wave propagation in a guide obstructed by a hard perforated screen

In addition to yield explicit solutions of several scattering problems, homogenization approaches are of particular interest if we have numerical resolutions in mind. This is particularly true if the problem under consideration involves very different scales between the largest one and the smallest one, and this is what we exemplify in this section. We consider the high frequency propagation of an acoustic wave in a rigid guide of height H,w i t hkH =1 0 0( F i g .1 4 . 9 ) .T h eg u i d e is obstructed, from wall to wall, by a rigid perforated screen with h = H/100 and e = H/2000 (thus, kh =1andke =0.05). Thus, the smallest scale is e and the largest scale is H,w i t hH =2 0 0 0 e and this is demanding in terms of any computational method.

In general, the solution of guided wave problems can be expanded in modes, associated to transverse functions ϕ n (X 2 )w i t hv a n i s h i n g first derivative at X 2 =0,H,n a m e l y

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ P (X 1 ,X 2 )= N n=0 P n (X 1 ) ϕ n (X 2 ), ϕ n (X 2 )= 2 -δ n0 cos nπX 2 H , (14.36)
(the √ 2standsforpropernormalizationofthetransv ersefunctions). At this stage, P stands for the solution of the actual problem or for the solution of the homogenized one. For N → +∞,( 1 4 . 3 6 )i st h e modal expansion of the solution, the P n being the projections of P on the basis ϕ n ;i nt h en u m e r i c s ,afi n i t ea n ds u ffi c i e n t l yl a r g eN -value is used within a multimodal approach which accounts for the mode coupling due the presence of the screen, see [START_REF] Maurel | Propagation of guided waves through weak penetrable scatterers[END_REF]; [START_REF] Maurel | Wave propagation through penetrable scatterers in a waveguide and through a penetrable grating[END_REF]].

However, in the homogenized problem, no coupling is expected. This is because the jump conditions, Eqs. (14.29), apply for each mode owing to the orthogonality of the transverse functions, and they take the form

⎧ ⎪ ⎨ ⎪ ⎩ P n = hBP ′ n , P ′ n = hSP ′′ n -hC nπ H 2 P n , (14.37)
where we used that ϕ ′′ n = -(nπ/H) 2 ϕ n .N e x t ,i nt h ep a r t so ft h e waveguide free of scatterer, each mode satisfies P ′′ n + k 2 n P n =0,with

k n ≡ k 2 - nπ H 2 (14.38)
the component of the wavenumber along X 1 .Exceptatthein terface -e/2 <X 1 <e/2, this is the case and we can write for each P n

P n (X 1 )= ⎧ ⎨ ⎩ P inc n
e ikn(X 1 +e/2) + r n e -ikn(X 1 +e/2) , for X 1 < -e/2,

P inc n t n e ikn(X
⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ r n =i a n -b n (1 -ia n )(1 -ib n ) ,t n = 1+a n b n (1 -ia n )(1 -ib n ) , with a n ≡ k n h 2 S + C nπ k n H 2 ,b n ≡ k n h 2 B.
(14.40)

The interface parameters for this structuration (e/h =0 .05, ℓ/h =0 .95) are :

B =2 .62[2.64], C =1 .51 0 -3 [1.71 0 -3 ]a n d S =2.510 -3 .
Figures 14.10 and 14.11 report the wavefields P ex (X)c a l c u l a t e d numerically and P (X) given by the explicit expressions, Eqs. (14.36) to (14.40). The wavefield in the absence of the film for the same incident wave P inc (X 1 ,X 2 )=P inc n e iknX 1 ϕ n (X 2 )i sre po rt e df o rc o mparison (in both cases, P inc is a combination of the 100 first modes). The fields P (X)a n dP ex (X)a r ef o u n di ng o o da g r e e m e n t( w i t ha relative discrepancy of about 1% in both cases), which confirms that the jump conditions correctly account for the effect of the structured screen.

Let us now inspect the reflection and transmission coefficients mode by mode. In the numerics, the scattering properties are encapsulated in matrices T and R being N × N matrices. We calculated R and T at the interfaces of the film, namely P ex m (-e/2) = [δ mn + R mn ]P inc n (-e/2), and P ex m (e/2) = T mn P inc n (-e/2). We have checked that the scattering matrices are essentially diagonal (the relative weight of the off diagonal terms is about 10 -4 ), which confirms that the mode coupling is negligible. Figures 14.12 show the transmission and reflection coefficients (r n ,t n )f r o mE q s .( 1 4 . 4 0 )a n dt h e diagonal terms (R nn , T nn )( t h es c a t t e r i n gp r o p e r t i e sd on o td e p e n d on the source, so they are the same in the cases of the Figs. 14.10 and 14.11). The agreement is relatively good, with 1% and 5% respectively for the reflection and transmission coefficients, averaged for n =1,...100. As can be expected, the full wave calculations have been demanding. Indeed, it requires 3 different scales to be resolved, the largest one H,t h ei n t e r m e d i a t eo n e ,g i v e nb yt h ei n c i d e n tw a v e n u m b e r s with k n , n =0t o1 0 0 ,f r o mw h i c ht h et y p i c a lw a v e l e n g t hi sa b o u t H/100, and the smallest one, given by e = H/2000. This latter small scale imposes the truncation N in the numerics, which means that |k N |e ∼ 1. Thus Nπe/H ∼ 1, leading to N ∼ H/e =2000.Ingeneral, the smallest scale corresponds to the highest evanescent mode excited in the near field of the structured film, and indeed, N =2000-3000 has been needed to reach a reasonably converged solution P ex (for instance, the variation in P ex ,E q .( 1 4 . 3 6 ) ,f r o mN =2 0 0 0t o3 0 0 0 is about 0.8 %). This need in a high truncation is expected and not expected. Indeed, it has been said, and it has been checked, that no mode coupling occurs our configuration (see Appendix 14.9). Thus, in the present cases, the modes at n>100 are not expected to contribute significantly to the total field since we have considered an incident wave being a combination of the first 100 modes only. The reason why high truncation is needed is in fact related to the convergence of R nn and T nn with N ,ac o n v e r g e n c ew h i c hi s often disregarded; the Appendix 14.9 presents a discussion on this aspect.

Concluding Remarks

We have presented an interface mo del to identify the effective b ehavior of a thin structured film composed of sound hard inclusions. Parameters characteristic of the interface enter in jump conditions for the acoustic pressure field and for the acoustic normal velocity. They are given by the resolution of elementary problems written in the static limit, and they are therefore wave independent by construction. We have validated this model in the case of a layer made of rectangular inclusions and for a plane wave at oblique incidence on the structured layer. Explicit expressions of the scattering coefficients deduced from the interface parameters have been shown to be accurate with a range of validity being kh < 1a n dke < 1. Results have been compared to the scattering coefficients given by classical homogenization of bulk material; these latter have a range of validity kh < 1ande/h > 1, and largely underestimates the scattering properties of thin structured layers. Finally, the validity and the interest of such jump condition has been exemplified in the more involved case of an ultrathin structured layer obstructing a guide from wall to wall (there, due to the high frequency propagation in the guide, the numerical cost is heavy).

The present model is applicable to more involved geometries of the inclusions and to inclusions being penetrable for the wave as well as for three dimensional cases. More generally, it can be extended to a large class of wave problems, in acoustics, in elasticity and in electromagnetism.

Appendix A The Present Derivation as a

Simplified Case of a Previous One

In [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF]], the problem of an elastic body containing microcracks or micro holes periodically located on a surface is considered. The elastic body is submitted to external forces (or boundary conditions) on its boundaries, which impose variations of the external stresses on the scale which is typically the size of the body; this latter is assumed to be much larger than the spacing between the holes. The size of the body in the elastic problem is the equivalent of the incident wavelength in our acoustic problem. The increased complexity in this work is that it requires to deal with vectors instead of scalars p, q → U, V (the elastic displacement vector) and a tensor instead of a vector u, v → σ, τ (the stress tensor). Otherwise, the derivation of the interface parameters is the same.

One noticeable difference is that the calculation of the jump condition are performed after τ 0 and τ 1 have been extended by 0 in D. This is possible in the static case since the equilibrium translates in div y τ 0 =0a n dd i v y τ 1 +div x ′ τ 0 =0w i t hb o u n d a r yc o n d i t i o n s τ 0 .n |∂D = τ 1 .n |∂D =0.Th us,extendingτ n by 0 in D is compatible with the equilibrium and makes the boundary conditions on ∂D to be automatically satisfied. This is not possible in our case, and this is why the integrations are performed in Y\D instead of in Y in [START_REF] Marigo | The effective behavior of elastic bodies containing microcracks or microholes localized on a surface[END_REF]]. The two results are of course identical. Namely, the Eq. ( 40) in this reference reads

σ 1 i1 = - ∂ ∂x α Y dy τ iα (x ′ , y), (14.41)
and τ ≡ τ 0σ 0 (0, x ′ )h a sb e e ne x t e n d e di m p l i c i t l yb y-σ 0 (0, x ′ ) in D.I tf o l l o w st h a t

σ i1 = D dy ∂σ 0 iα ∂x α (0, x ′ ) - ∂ ∂x α Y\D dy τ iα (x ′ , y), (14.42)
and owing to

∂σ 0 iα ∂xα = - ∂σ 0 i1
∂x 1 given by the outer problem, we get

σ 1 i1 = - ∂σ 0 i1 ∂x 1 (0, x ′ )S - ∂ ∂x α Y\D dy τ iα (x ′ , y), (14.43)
which is equivalent to our jump condition Eq. (14.15). Finally, the elementary problems satisfied in the static elastic case are given by Eqs. ( 28) are the equivalent of our Eqs. (14.17).

Appendix C -Multimodal Approach to Compute (B, C)f o rR e c t a n g u l a rR i g i d Structuration

For piecewise constant geometry in the elementary problems, mo de matching is a simple way to get B and C.I nbo t he l e m e n t a r yp r o blems (14.17), the idea is to project the solution q (i) onto bases of transverse functions of y ′ being adapted in the different regions along y 1 .B e l o w ,w eg i v eas i m p l ep r o c e d u r et oc o m p u t e( B, C)f o r rectangular rigid inclusions in two dimensions (in this case, only two bases of transverse functions (f - n (y ′ ),f + n (y ′ )a r en e e d e d ) .T h es i mplicity lies in the fact that the procedure does not require more than the inversion of a matrix. The size N of the matrix fixes the spatial resolution : with a cell of size unity along y 2 ,thespatialresolutionis typically 1/N .A sw ea s s u m e dt h a ta l lt h ed i m e n s i o n so ft h ei n c l usions is of order unity, large N is not required (otherwise, the calculation may become tricky since mode matching has low convergence, typically 1/N ).

In the following, we consider h =1forsimplicit y . Because q (1) is defined up to a constant and because the symmetry of the inclusion w.r.t. y 1 ,w ec a nc o n s i d e raf u n c t i o nq (1) odd w.r.t. y 1 ,w he nc ea ppl y i ngD i ri c hl e tbo unda ryc o ndi t i o na ty 1 =0(namely q (1) (0,y 2 )=0 ) .I tf o l l o w st h a tq (1) → b 1 /2f o ry 1 → +∞ (and b 1 = δq (1) ). We use Q(y)=y 1 +q (1) (y)-b 1 /2whichsatisfies∆Q =0, ∇Q.n |∂D =0,and(Qy 1 ) → 0f o ry 1 → +∞.F i n a l l y ,w eu s et h a t

ϕ/2 -ϕ/2 dy 2 ∂ y 1 Q(0,y 2 )= 1/2
-1/2 dy 2 ∂ y 1 Q(+∞,y 2 )=1( b yi n t e g r a t i o n of ∆Q =0).Th us,Q can be expanded as

Q(y)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Q -(y)= y 1 ϕ + q - 0 f - 0 0 ≤ y 1 ≤ e/ 2 + N - n=1 q - n sinh a n y 1 sinh a n e/2 f - n (y 2 ), Q + (y)=y 1 e/2 ≤ y 1 , + N + n=-N + ,n̸ =0
q + n e -|bn|(y 1 -e/2) f + n (y 2 ), (14.44) with a n = nπ/ϕ, b n =2nπ,a n dw h e r e

f + n (y 2 )=e ibny 2 ,f - n (y 2 )= 2 -δ n0 ϕ cos a n y 2 + nπ 2 , (14.45)
are the transverse functions (forming a basis) adapted for solutions being respectively periodic and for solutions with zero derivatives at y 2 = ±ϕ/2.

In the above projection, b 1 being unknown has ben encapsulated in the first coefficient q - 0 of the expansion, and it is easy to see that

Q -(0,y 2 )=q - 0 f - 0 = -b 1 /2( t h u sb 1 = -2q - 0 / √ ϕ), by construction.
Now, we will ask to Q ± to match (on average) their values and their first derivatives at y 1 = e/2, and this latter matching on the derivatives will be done accounting for the Neumann boundary conditions at y 1 = e/2a n d|y 2 | >ϕ / 2( n o t et h a tQ -satisfies by construction the Neumann boundary condition on at y 2 = ±ϕ/2, because of the choice of the f - m ). To that aim, we use the following relations

⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ϕ/2 -ϕ/2 dy 2 Q -(e/2,y 2 )f - m (y 2 )= ϕ/2 -ϕ/2 dy 2 Q + (e/2,y 2 )f - m (y 2 ), ϕ/2 -ϕ/2
Q + = ∂ y 1 Q -for |y 2 | <ϕ / 2.
We get a matrix system for the two vectors q -=( q - n ) n=0,...,N -and q + =(q + n ) n=0,...,N + of the form (14.50)

I -t F * FA 1 B q - q + = S s S d , ( 14
F mn = ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ √ ϕ sinc b m ϕ 2 ,n =0, = ϕ 2 sinc (a n -b m ) ϕ 2 e inπ/2 n ̸ =0, +sinc (a n + b m ) ϕ 2 e -inπ/2 , ( 14
The system is of the form Mq = s with the matrix M being square (this is not always the case in systems written using mode matching). Next, M is invertible if one has taken care to consider only the antisymmetric modes. Thus, the system can be solved to find q by inverting M or it can be solved in the least squares sense (as done by the operation M\s in Matlab). Once the system has been inverted, b 1 is obtained and thus, from (14.28), B = e + b 1 B = e -2 ϕ q(1). (14.51) 14.6.2. Calculation of C in the elementary problem i =2

The procedure to get C in the elementary problem i =2i ss i milar. We consider the solution Q = Q (2) + y 2 satisfying ∆Q =0 , ∇Q.n |∂D =0a n dQ → y 2 for y 1 →∞ .B e c a u s eq (2) is even in y 1 , we can project Q onto the bases (f ± n )a sf o l l o w

Q(y)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ Q -(y)= N - n=1 q - n cosh a n y 1 cosh a n e/2 f - n (y 2 ), -e/2 ≤ y 1 ≤ 0 Q + (y)=y 2 + N + n=-N + ,n̸ =0 q + n e -|bn|y 1 f + n (y 2 ), 0 ≤ y 1 .
(14.52) As previously, we ask to Q ± to match (on average) their values and their first derivatives at y 1 = e/2, and we get a matrix system of the form

I -t F * FA 2 B q - q + = S 0 , (14.53)
with the same definitions as previously and now, A 2 = diag(a n tanh(a n e/2)) and S n = ϕ/2 -ϕ/2 dy 2 y 2 f - n (y 2 ), specifically

S n = -2 2 ϕ 1 a 2 n . ( 14
.54)

Once the system has been inverted, we can determine with the tensor a eff and b eff given by the resolution of two elementary problems. The case of layered media, with inclusions infinitely long along X 1 and of width ℓ along X 2 ,w eh a v ea(X 2 )a n db(X 2 )a n d explicit solution is possible. It ends with diagonal effective mass density tensor a eff =diag(a 1 eff ,a 2 eff )a n de ff e c t i v eb u l km o d u l u sg i v e nb y (14.59) where the limits correspond to the Neumann case (a = b =0).Inthis limit, the propagation inside the layered medium is described by the wave equation a 1 eff ∂ X 2 1 P +a 2 eff ∂ X 2 2 P +k 2 b eff P =(1-ℓ)[∂ X 2 1 P +k 2 P ]= 0, from which we deduce that the component of wavenumber along X 1 is k.

C = -dy ∂Q (2) ∂y 2 = 0 -e dy 1 ϕ/2 -ϕ/2
⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ a 1 eff = ⟨a(X 2 )⟩ =[ℓa +(1-ℓ)] → 1 -ℓ, a 2 eff = ⟨a(X 2 ) -1 ⟩ -1 =[ℓ/a +(1-ℓ)] -1 → 0, b eff = ⟨b(X 2 )⟩ =[ℓb +(1-ℓ)] → 1 -ℓ,
Imagine now that we cut the layered medium to create a slab, which occupies the space X 1 ∈ [-e/2,e/2], and outside the slab, the medium has a(X)=b(X)=1 .F o rt h er e a ls t r u c t u r e ,t h er e s u l t i n g configuration corresponds to an array of rectangular hard inclusions as studied in Section 14.2. For the equivalent homogenized medium, it corresponds to a slab of thickness e filled with the homogeneous and anisotropic material given by (a eff ,b eff ). In this latter configuration, the problem of the scattering for an incident wave coming from X 1 = -∞ with incidence θ is trivial and it has a solution of the form

P (X 1 ,X 2 ) = e ik sin θX 2 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ [e ik cos θX 1 + Re -ik cos θ(X 1 +e/2) ], for X 1 < -e/2, Ae ikX 1 + Be -ikX 1 , for |X 1 |≤e/2,
Te ik cos θ(X 1 -e/2) , for X 1 >e/2, (14.60)

where the component of the wavenumber along X 2 , k sin θ,isimposed by the continuity of P (X 1 ,X 2 )a tt h ei n t e r f a c e sX 1 = ±e/2. Next, applying the relations of continuity for P and a 1 (X)∂ X 1 P (with a 1 (X)=a 1 eff =1-ℓ in the slab and a 1 (X)=1outside)atX 1 = ±e/2, leads to usual forms of the scattering coefficients

R = 2i(1 -c 2 )sinke e -ike cos θ/2 (1 + c) 2 e -ike -(1 -c) 2 e ike , T = 4ce -ike cos θ/2 (1 + c) 2 e -ike -(1 -c) 2 e ike , ( 14
.61) where c ≡ cos θ/(1ℓ). The form of R in the above equation is written in an alternative form in Eq. (14.34). in the film and ∆P + k 2 P =0o u t s i d e ,a n dt h er e l a t i o n so fc o n t inuity of P and a 1 ∂ X 1 P apply at X 1 = ±e/2( w i t ha 1 = a 1 eff inside the layer, a 1 =1o u t s i d e ) .E x p a n d i n gt h es o l u t i o ni n t ot h eb a s i so f transverse functions, P (X 1 ,X 2 )= n≥0 P n (X 1 ) ϕ n (X 2 ), the system is decoupled, with In the problem considered in Section 14.3.3, we have said that the high truncation N =2 0 0 0-3000 was expected and not expected.

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ a 1 eff P ′′ n + k 2 b eff -
Expected because the smallest scale to be resolved is e = H/2000 which imposes N ∼ H/e (and this corresponds to the highest evanescent mode with wavenumber k N ∼ Nπ/H). Unexpected because we have checked that the mode coupling is negligible; the incident wave being a combination of the first 100 modes, the absence of mode coupling ensures that the modes at n>100 are not excited. To be clear, we have compared P ex (X)c o m p u t e dw i t h30 0 0m o d e st ot h ef o llowing field

P diag (X)= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 100 n=0 P inc
n [e ikn(X 1 +e/2) + R nn e -ikn(X 1 +e/2) ],X 1 < -e/2, 100 n=0 P inc n T nn e ikn(X 1 -e/2) ,X 1 >e/2, (14.67) which is precisely the field composed of the 100 first modes generated directly by the incident wave. The difference |P ex -P diag | is less than 0.4% in both cases, thus the evanescent field generated by higher modes is indeed negligible. The reason why high truncation has been needed is in fact related to the convergence of R nn and T nn for n ≤ 100 with N ,aconvergencewhichisoftendisregarded.InFig. 14.16, we have reported the variation of |R nn |, n ≤ 100, for several truncations N =1 0 0t o3 0 0 0 .A si tc a nb es e e n ,i n c r e a s i n gN produces a significant variation in |R nn |,7 0 %o na v e r a g ef o ri n c r e a s i n g the truncation N from N =100toN =3000.Thislo wcon v ergence has already been observed for inclusions presenting a contrast in the mass density [START_REF] Maurel | Wave propagation through penetrable scatterers in a waveguide and through a penetrable grating[END_REF]]; it is a convergence law in 1/N , due to the low regularity of the field (with discontinuous pressure gradient). Amusingly, for insufficient truncation N =1 0 0 ,t h es c a ttering coefficients coincide with the solution (r n , tn )givenbyclassical homogenization, which underestimates the weight of the evanescent field (Eqs. (14.66), see Appendix 14.8).

14. 1 .

 1 Introduction Acoustic metamaterials consisting in massive materials perforated by periodic subwavelength holes [Pendry et al. (2004); Garcia-Vidal et al. (2005); Zhu et al. (2011)] or more sparse structures involving periodic arrangement of wires

  al. (2014); Zhao et al. (2014)], or resonances of labyrinthine or curled elements squeezed in the unit cell [Peng et al. (2014); Li et al. (2014); Xie et al. (2014)]. Because of their subwavelength unit cell, homogenization techniques are natural tools to describe the effective properties of metamaterials. Standard homogenization methods [Zhu et al. (2013); Maurel et al. (2013)] or more empirical methods, as the retrieval techniques [Arslanagic et al. (2013); Castanié et al. (2014)] (often referred as Nicolson-Ross-Weir technique) and approximated modal approaches [Pendry et al. (2004); Garcia-Vidal et al. (2005); Kelders et al. (1998)

  ; Capdeville et al. (2010a); Guillot et al. (2010); Capdeville et al. (2010b); Capdeville and Marigo (2013)], and the Refs. [Bonnet-Bendhia et al. (2004); Delourme (2010); Delourme et al. (2012)] in the frequency domain. Note also works using alternative forms of the matched asymptotic expansions [Sanchez-Hubert and Sanchez-Palencia (1982); Martin and Dalrymple (1988); Kakuno et al. (1992)].

  , are obtained. Next, a comparison with the classical homogenization of layered media is possible since again, explicit solutions are available in this case [Garcia-Vidal et al. (2005); Oleinik et al. (2009)], Eq.

Fig. 14 . 1 .

 141 Fig. 14.1. Scattering problem for (a) the microstructured film (metafilm) and (b) the equivalent interface. X =( X1,X2,X3)i st h ec o o r d i n a t es y s t e mi nt h e real space.

Fig. 14

 14 Fig. 14.2. (a) Geometry of the structured film in non dimensional coordinates x =( x1,x2,x3); (b) inner problem in coordinates y =( y1,y2,y3). D is the region occupied by the rigid inclusion. We define Y =( -y m 1 ,y m 1 ) × (0, 1) 2 , Y∞ =lim y m 1 →+∞ Y,a n dx ′ =(x2,x3), y ′ =(y2,y3).

  However, one can think to introduce a dependance on the frequency in the elementary problems, as it has been done for classical homogenization (high frequency homogenization or resonant homogenization) [Bouchittéa n dF e l b a c q( 2 0 0 4 ) ;F e l b a c qa n dB o u c h i t t é( 2 0 0 5 ) ; Craster et al. (2011); Antonakakis et al. (2013)].

Fig. 14 Fig. 14 . 5 .

 14145 Fig. 14.4. Interface parameters B and C computed numerically by solving the elementary problems (14.17) (symbols), see Appendix 14.6. (a) (Be/h)a sa function of ℓ/h for e/h =0.5,1and2;plainlinessho wthev aluesgivenby(14.31). Same representation for C,w i t hp l a i nl i n e sg i v e nb y( 1 4 . 3 0 ) .

Fig. 14 . 6 . 1 Fig. 14 . 7 .

 1461147 Fig. 14.6. Wavefields in the (X1,X2)p l a n ef o rke =0 .5, kh =1 ,a n dkℓ =0 .5 or 0.9 (a) P ex (X)a n d( b )P (X)c o m i n gf r o mt h ei n t e r f a c em o d e l ,E q s .( 1 4 . 3 2 ) -(14.33). The white arrow indicates the wavenumber of the incident plane wave.

(

Fig. 14 . 8 .

 148 Fig. 14.8. Variations of the reflection coefficients R ex calculated numerically (orange plain lines), R given by the interface model, Eq. (14.33) (blue dotted lines), and R given by classical homogenization, Eq. (14.34) (grey dotted lines). (a) as a function of kh for e/h =1 0 ,1 ,1 0 -1 (from the top to the bottom), and (b) as a function of e/h for kh =10 -2 , 10 -1 ,1( f r o mt h et o pt ot h eb o t t o m ) .I n all plots, ℓ =0 .9h and θ = π/4.

Fig. 14 . 9 .

 149 Fig. 14.9. Configuration of the rigid waveguide obstructed by a perforated screen at -e/2 <X1 <e/2. The screen has a periodic spacing h = H/100 and thickness e = H/2000. The height of the rigid material in the screen is ℓ/h =0 .95. The guide is studied in the high frequency regime, with kH =100.

1 Fig. 14 .

 114 Fig. 14.10. Scattering by the perforated screen in the guide (geometry of Fig. 14.9). (a) Wavefield P inc in the absence of the film (b) and (c) Wavefields P ex calculated numerically and P in the homogenized problem, Eqs. (14.36) to (14.40). (d) shows the profile P inc (X1,H/2) (in arbitrary unit) and( e )s h o wt h ep r o fi l e s P ex (X1,H/2) (blue line) and P (X1,H/2) (black symbols); the discontinuity of the field across the structured film is visible in the zoom of the profile for -H/20 < X1 <H/20.

1 Fig. 14 .

 114 Fig. 14.11. Same representation as in Fig. 14.10 for an incident wave of beam type.

Fig. 14 .

 14 Fig. 14.12. Comparison of the reflection and transmission coefficients of the modes, (Rnn, Tnn)f o =1 ,...,100 from direct numerics (computed with a truncation N =3000),and(rn,tn)g i v e nb yt h ei n t e r f a c em o d e l ,E q s .( 1 4 . 4 0 ) .

Fig. 14 .

 14 Fig. 14.13. Mode matching configuration. The solutions Q -and Q + are written for 0 ≤ y1 ≤ e/2h and y1 ≥ e/2h respectively (with proper conditions at y1 =0), and the resolution involves only matching conditions at y1 = e/2h.

1 δ

 1 n0 ,n =0,...,N - S d n = -sinc(nπϕ),n = -N + ,...,-1, 1,...N + .

Fig. 14 .

 14 Fig. 14.14. Static fields computed for the elementary problems i =1a n d2 . (a) (c) q (1) (y)a n d( b )( d )q (2) (y). (a-b) for e/h =0 .1a n dℓ/h =0 .5,(c-d) for e/h =0.01 and ℓ/h =0.9.

a

  Fig. 14.15. In the classical homogenization, the actual structured film (a) is replaced by a homogeneous anisotropic film of thickness e. The derivation of ( R, T )i sd o n ea c c o u n t i n gf o rbo u n d a r yc o n d itions being the continuity of p and of a eff ∇p.n,w h i c hi st r u ea tt h i s leading order (see [Marigo and Maurel (2017)]).

  one (Eqs.(14.60)), owing to k cos θ → k n andrn = 2i(1c 2 n )sinke e -ikne/2 (1 + c n ) 2 e -ike -(1c n ) 2 e ike , tn = 4c n e -ikne/2 (1 + c n ) 2 e -ike -(1c n ) 2 e ike , (14.66) with c n ≡ k n /k(1ℓ). 14.9. Appendix F -Discussion on the Convergence of the Scattering Coefficients in the Modal Method

Fig. 14 .

 14 Fig. 14.16. Reflection coefficients of the modes in the guide obstructed by a perforated screen (same calculations as in Fig. 14.12). |Rnn| for n =0t o1 0 0 is shown for several truncation order N in the expansion Eq. (14.36) (orange curves). For low truncation, N =100,|Rnn| are found to be close to the rn given by the classical homogenization in (14.66) (blue symbols); increasing N makes |Rnn| to converge toward rn given by the interface homogenization, |rn| in Eq. (14.40) (black symbols).

  14.30)-(14.31), and Y the value obtained by solving the associated elementary problem (see Fig.14.4 and Appendix 14.6). Eq. (14.32), with Eqs.(14.33). We have considered θ = π/4, kh =1 .T h i sε = kh =1v a l u ei so b v i o u s l yal i m i t i n gc a s ef o rt h e validity of any homogenization model but the fields P ex and P are found in good agreement, with ||P -P ex ||/||P ex || ∼ 1% for ke =0.5 (in both cases reported in Figs.[START_REF] Delourme | Modèles et asymptotiques des interfaces fines et périodiques en électromagnétisme[END_REF].6) and ||P -P ex ||/||P ex || ∼ 5% for ke =1(inbothcasesreportedinFigs.14.7);||.|| is the norm L 2 , considering the field in |X 1 | >e/2.

	e/h =0.5 ℓ/h =0.5 ℓ/h =0.9
	B	1.22 [1.25] 6.18 [6.22]
	C	0.15 [0.17] 0.05 [0.05]
	e/h =1 ℓ/h =0.1 ℓ/h =0.5
	B	1.12 [1.12] 2.22 [2.25]
	C	0.58 [0.60] 0.40 [0.41]
	The field P ex (X)referstothesolutionoftheactualproblemcal-
	culated numerically (a multimodal method is used, see [Maurel et al.
	(2014)]) and P (X)referstothepredictioncomingfromthein terface
	model,

  ,t n )t h er e fl e c t i o na n dt r a n s m i s s i o nc o e ffi c i e n t so ft h e mode n.N o w ,i ti ss u ffi c i e n tt oa p p l yt h ej u m pc o n d i t i o n s , Eqs.(14.37), to get (r n ,t n )

1

-e/2) , for X 1 >e/2, (14.39) with (r n

  The first relation is the matching of the values in the region y 2 ∈ [-ϕ/2,ϕ/2] where Q -is defined. The second relation has more information: we have used that the ∂ y 1 Q + =0aty 1 = e/2f o r|y 2 | >ϕ/2, from which

	dy 2	∂Q -∂y 1	(e/2,y 2 )f + m	* (y 2 )=	1/2 -1/2	dy 2	∂Q + ∂y 1	(e/2,y 2 )f + m	* (y 2 ),
	with f + m m is real). 1/2 * the conjugate of f + m (f --1/2 dy 2 ∂Q + ∂y 1 (e/2,y 2 )f + n * (y 2 )= ϕ/2 -ϕ/2 dy 2	∂Q + ∂y 1	(14.46) (e/2,y 2 )f + m * (y 2 ),
										(14.47)
	afterwards we ask, on average, ∂ y 1

  .48) with I the N -× N -identity matrix, A 1 =d i a g ( a n cotanh(a n e/2)) with A 1 (1, 1) = 0, B =d i a g ( |b n |), and F mn =

	ϕ/2 -ϕ/2 dy 2 f + n	* (y 2 )f -n
	(y 2 ). The source terms (S s , S d )c o r r e s p o n dt ot h ep r o j e c t i o no ft h e loading (y 1 /ϕ and y 1 )o n t of -n and f + n * .TheexpressionsofF mn and S s,d n are given below

  2 appearing for large enough thickness. Both corrections are due to boundary film effect, see[START_REF] Marigo | Homogenization models for thin rigid structured surfaces and films[END_REF]].14.8. Appendix E -Scattering Coefficients for the Modes in a WaveguideIf the film of the equivalent anisotropic medium occupies the spaceX 1 ∈ [-e/2,e/2] and X 2 ∈ [0,H]a sc o ns i de r e di nt heSe c t i o n1 4 . 3 . 3 , it is possible to calculate the reflection and transmission coefficients (r n , tn ) mode by mode (and a mode is defined in Eq. (14.36)). This is because no mode coupling occurs in this case, again owing to the orthogonality of the transverse functions. The problem first reads

	a 1 eff	∂ 2 P ∂X 2 1	+ a 2 eff	∂ 2 P 2 ∂X 2	+ k 2 b eff p =0,	(14.64)

  continuous at X 1 = ±e/2, and P ′ n (-e/2 -)=a1 eff P ′ n (-e/2 + ), P ′ n (e/2 + )=a 1 eff P ′ n (e/2 -). In the case of our layered sound hard medium, a 1 eff = b eff =1-ℓ and a 2 eff =0,whic hleadstoP ′′ n +k 2 P n =0 in the layer, |X 1 |≤e/2. Thus, the problem is identical to the previous

			a 2 eff	nπ H	2	P n =0, for |X 1 |≤e/2
	P ′′ n + k 2 -	nπ H	2	,		outside
						(14.65)
	with P n