Study of propagation of acoustic waves in junction of thin slots

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thin slots

Adrien Semin, Patrick Joly

To cite this version:

Adrien Semin, Patrick Joly. Study of propagation of acoustic waves in junction of thin slots. [Research

Report] RR-7265, INRIA. 2010, pp.56. �inria-00476322�

a p p o r t

d e r e c h e r c h e

Study of propagation of acoustic waves in junction of

thin slots

Patrick Joly — Adrien Semin

N° 7265

Centre de recherche INRIA Paris – Rocquencourt

Domaine de Voluceau, Rocquencourt, BP 105, 78153 Le Chesnay Cedex

Patrick Joly

_{, Adrien Semin}

Thème : Modélisation, simulation et analyse numérique Équipe-Projet Poems

Rapport de recherche n° 7265 — April 2010 —56pages

Abstract: In this document, we analyze via the theory of matched asymptotics the propagation of a time domain acoustic wave in a junction of thin slots. This allows us to propose Improved Kirchhoff conditions for the 1D limit problem. These conditions are analyzed and validated numerically.

Key-words: Matched asymptotics, wave equation, time domain, thin slots, Kirchhoff conditions

∗_{mail : patrick.joly@inria.fr} †_{mail : adrien.semin@inria.fr}

Résumé : Dans cet article, nous utilisons la théorie des développements asymp-totiques raccordés pour analyser la propagation d’ondes acoustiques à travers une jonction de fentes minces. ceci nous permet de proposer des conditions de Kirchhoff améliorées pour le problème limite 1D. Ces conditions sont analysées et validées numériquement.

Mots-clés : Méthodes asymptotiques, équation des ondes, temporel, fentes minces, conditions de Kirchhoff

Introduction

One can consider time harmonic wave propagation and time domain propaga-tion in thin domains that are juncpropaga-tions of thin slots whose thickness ε is small with respect to the wave length λ and converge, when ε tends to 0, to a 1-dimensional graph. A contrario, one can imagine that we start from a 1D graph of straight lines that we thicken symmetrically so that we obtain a ε-thickness 2D graph (an example is given by the figure 1).

2D graph with thickness ε 1D graph

Figure 1: Configuration of the 1D graph (left) and the ε-thickness 2D graph (right)

On this 2D graph (that will be call Ωε_{in the following), one wants to solve the}

following problem: find uε

∈ C0(R∗ +, H1(Ωε))∩ C1(R∗+, L2(Ωε)) satisfying                        ∂2_uε ∂t2 (t, x)− ∆u ε_{(t, x) = 0 for (t, x)} ∈ R∗ +× Ωε ∂uε ∂~n(t, x) = 0 for (t, x)∈ R ∗ +× ∂Ωε uε_{(0, x) = f}ε _{for x ∈ Ω}ε ∂uε ∂t (0, x) = g ε _{for x ∈ Ω}ε (1)

with (fε_{, g}ε_{)∈ H}1_(Ωε_{)× L}2_(Ωε_{) suth that we assume, for ε small enough, that}

these functions vanish on the different junctions, and do not depend on the crosswise variable on the slots.

The limit model is known for very long time but its justification seems to be very recent, with the works of Jacob Rubinstein, Michelle Schatzman [7,8] and Peter Kuchment [6]. In this limit model, the solution only depends of the topology of the graph and satisfy an one-dimensional time domain acoustic equation. Moreover, this solution is continuous at each node of the graph, and satisfy the Kirchhoff laws. To be more precise, the sum of the outgoing normal derivatives (with appropriate constants) is equal to zero.

For studying the behaviour of uε_{for small ε, we will use the method of matched}

at the beginning of the 70’s, initially to analyse boundary layer phenomena. This approach has been developed quasi-independently (one does not find a lot of cross citations in the publications) by two mathematical schools, from two rather different points of view (see also [2,10] for more references). In this article, the use of the matched asymptotics expansions is based on a recent work of Patrick Joly and Sébastien Tordeux [5]. As it is classical with this type of problem, the work can be naturally divided into three steps. The first one is related to obtaining the formal asymptotic expansion and constitutes the most algebraic and calculator part. The next two cover two different aspects of the mathematical justification and make use of various techniques for the analysis of PDE’s.

• Step 1: Derive formal expansion. One starts from an ansatz, i.e. a form of a priori behaviour for the asymptotic expansion that is injected in the equations of the initial (ε dependant) problem, to deduce a series of (ε independent) elementary problem that are supposed to characterize the terms of the asymptotic expansion.

• Step 2: Show that the various terms of the asymptotic expansions are well defined, i.e. that the above elementary problems are well posed. This is not necessary straightforward since these problems are quite often not standard.

• Step 3: Justify the asymptotic expansion, namely establish error esti-mates between the true solution and truncated asymptotic expansions. A very particular case has been treated in [3]: in this article, we consider the case of two slots of same width being connected by one junction. The outline of this paper is the following:

• in section1, we explain the model problem we consider, and we claim the main results of this article,

• in section2, we use the technic of matched asymptotics to give the devel-opment of the exact solution of the whole domain. One can remark that, for giving our approximated model, we only need the first terms of our development, but one can also see that the approximated model we need is the first order of a generalized family of approximated problems, • in section3, we justify the expansions of section2, and we prove that we

can build some approximate function which differs from the exact solution with an error as small as we want,

• finally in section 4, we explain how we build one 1D problem (in space) whose solution differs from the exact solution with an arror as small as we want.

1

Model problem and main results

1.1

The model problem

In this section we introduce the geometry and the equations of our problem. We consider a domain made of the junction of N straight slots (see figure2). More precisely, we consider the union of N thin rectangles and a junction zone. The ith

rectangle has length Li and thickness ciε, where ε is the small parameter in

the analysis. A geometrical characteristic of this domain is the relative width of the rectangles, given by the numbers ci. For the analysis, we consider in fact

a family of such thin domains denoted Ωε _{with varying ε. We make the choice}

(this has an influence on the asymptotic analysis) that each rectangle of Ωε

expands symmetrically from a fixed segment Si whose one of the two vertices

does not depend on i (see again figure 2).

Jε

O

t_i

n_i

Figure 2: Configuration of the domain for the general case

Analytically, we have

Ωε= Ωε1∪ Ωε2∪ . . . ∪ ΩεN ∪ Jε (2)

with Jε_{= εJ where J does not depend on ε, and}

( Ωε i =  x_{∈ R}2 _{/ 0 < x· ti< Li, −}ciε 2 < x· ni< ciε 2

t_i and n_i are given by figure2 (3) The problem we consider is: find uε

∈ C0(R∗+, H 1

(Ωε)∩ C1(R∗+, L 2

(Ωε)) such that (we recall here the wave system (1) we want to solve)

                       ∂2_uε ∂t2 (t, x)− ∆u ε_{(t, x) = 0 for (t, x)} ∈ R∗+× Ωε ∂uε ∂~n(t, x) = 0 for (t, x)∈ R ∗ +× ∂Ωε uε(0, x) = fε for x ∈ Ωε ∂uε ∂t (0, x) = g ε for x ∈ Ωε

For this system, we can define the associate energy Eε_{(t, u}ε_{) =} 1 2ε Z Ωε|∇u ε_{(t, x)} |2₊     ∂uε ∂t (t, x)     2 dx (4)

and we have (thanks to the Neumann boundary condition)

Eε_{(t, u}ε_{) =} Eε_{(0, u}ε_{) =} 1 2ε Z Ωε|∇f ε_(x) |2₊ |gε_(x) |2_{dx (5)}

In this energy, we add the ε−1 _{term to get the initial energy independant of ε}

when fε _{and g}ε_{satisfy somme good properties (namely hypothesis1.2detailed}

below).

Remark 1.1. In the following, we assume that Li = +∞. In fact, for t small

enough, solution obtained on a finite domain with homogeneous boundary con-ditions is the same as solution obtained on a semi-finite domain. But effects of the junction does not depend on the choice of Li, neither on conditions

eventu-ally put at si= Li.

1.2

The 1D limit problem

When ε tends to 0, the domain Ωε _{"degenerates" into a "1D domain", namely}

the union of the n segments Si(left part of the figure2). Intuitively, one expects,

assuming fε_{and g}ε_{have good properties (with respect to ε), that the solution}

uε _{"converges" to a "1D function" (in space), namely a function of time and}

the arclength si along Si, solution of a "1D problem". It remains to give a

more precise mathematical meaning to such a statement. To describe the "limit problem" inside the slots, we will use local normalized tangential and normal coordinates (si, bνi), that express that Ωεi is isomorphic to the rectangle

b Ωi=]0, +∞[× i −c₂i,ci 2 h

through the map

x_{7→ (si, bνi) = (x· ti, x· ni/ε)} from Ωε_i into b_Ωi

The reader can note that domains bΩi overlap (see for instance figure3).

Before writing our 1D limit problem, we consider additional hypothesis on fε

and gε_{, namely:}

Hypothesis 1.2. There exists ε0 such that, for 0 < ε < ε0, for vε = fε or

vε_{= g}ε_,

• supp vε

∩ Jε₌

∅, • on each Ωε

i, vε(s, νi) = vε(s) (called in the following vεi),

• vε _{converges (in an appropriate space) to a function v}0 _{(which is a 1D}

c

c

c

Figure 3: Configuration of the limit domain (red dotdashed) and the 2D rect-anges bΩi (black boxes)

Under the hypothesis, let us define the following problem: find (u0

i)16i6N such that                ∂2_u0 i ∂t2 (t, si)− ∂2_u0 i ∂s2 i (t, si) = 0 for (t, si)∈ R∗+×]0, +∞[ u0i(0, si) = fi0 for si∈]0, +∞[ ∂u0 i ∂t (0, si) = g 0 i for si∈]0, +∞[ (6) and u0 j(t, 0) = u0k(t, 0) for j 6= k and X j cj ∂u0 j ∂sj (t, 0) = 0 (7)

The system (6) can be seen as the formal limit of the system (1) when ε tends to 0. The coupling conditions (7), known also in the literature as the Kirchhoff conditions, link the functions at the node of the graph.

Intuition says that, on any subdomain in the slots, error between the exact solution and the approximated solution is in O(ε). More precisely, one has the following proposition:

Proposition 1.1. Given (δi)16i6N, one can define:

• the sets Ωε

i,δ given as (see for instance figure4):

Ωε

i,δ={x ∈ Ωεi such that x· ti >δi}

• the 2D function fu0

i defined as

f u0

εσ

εσ

εσ

Figure 4: Configuration of the sets Ωε i,δ

Then there exist some 1D function (fu1

i)16i6N in space such that N X i=1 1 ε Z Ωε i,δ        ∂(uε− fu0 i) ∂t      2 +   ∇(uε− fu0 i)   2   =   N X i=1 Z +∞ δi       ε ∂fu1 i ∂t      2 +_ε∇fu1 i    2     + O(ε4₎ (8)

Proof. The proof of this proposition is a direct consequence of corollary 3.8

taking k = 1.

Remark 1.3. We can find Cauchy data such that fu1

i = 0 (for example, we choose

Cauchy data such that the solution is a single wave contained in the first slot and going to infinity - this solution does not see the junction and does not have any interest for us). But one can see that when the junction is "excited", one has fu1

i 6= 0 for any later time.

1.3

An improved 1D approximate model

To describe our improved problem, we need to introduce some additional nota-tions for our problem.

1.3.1 Additional information about the geometry

Definition 1.4 (Definition of σi). We introduce σi, for 1 6 i 6 N, as the

smallest value for which (see figure5for an example)

{x ∈ Ωεi such that x · ti>εσi} ∩  Jε ∪[ j6=i Ωεj   = ∅

Γ

Γ

Γ

σ

σ

σ

Figure 5: Configuration of an example of junction for three slots (in scaled coordinates).

From the definition of σi, we add additional sets: we define Ωεi,ext as (see for

instance figure6)

Ωεi,ext={x ∈ Ωεi / x· ti> εσi}

and we define our new junction eJε_as

e Jε= Jε_∪ N [ i=1 {x ∈ Ωε / x_{· t}i< εσi} 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111000000000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 000000 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 111111 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 000000000000 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111 111111111111

e

J

Ω

Ω

Ω

Figure 6: Configuration of the new sets Ωε

i,ext and eJε

Note that we also have the decomposition Ωε_{= Ω}ε

with domains that do not intersect each others. We keep eJε_{= ε bJ, where bJ is a}

fixed domain (i.e. not depending of ε) of R2_{. On domain bJ, we also define}

• Γi, for 1 6 i 6 N (see again figure 5) as

Γi= n x_{∈ ∂ bJ / x· ti= σi and |x · ni| <} ci 2 o , • Γneu as the remaining boundary (if exists).

Remark 1.5. One can easily imagine situations where boundary ΓN does not

exist (for instance, a junction of three slots of same width with an angle of 2π/3 between two slots: the junction is an equilateral triangle whose edges can be identified to some Γi).

1.3.2 Solution of auxiliary problems and DtN maps

To allow writing of Improved Kirchhoff conditions, one uses the solution of N −1 auxiliary problems. Let us introduce, once Σihas been identified to the segment

]0, ci[, Ti as the non-local DtN operator defined as

         Ti: H 1 2(]0, c i[) → H− 1 2(]0, c i[) ϕ 7→ Tiϕ = ∞ X p=1 pπ ci Z ci 0 ϕ(b ν′)wi,p(bν′)dbν′  wi,p (9)

where wi,pis an orthonormal family of L2(]0, ci[) given by

wi,0(bν) = r 1 ci , wi,p(bν) = r 2 ci cos  pπbν ci  (10) Proposition 1.2. We have the following properties

• Ti∈ L  H12(]0, c i[), H− 1 2(]0, c i[)  • for any ϕ∈ H12(]0, c i[), one has < Tiϕ, ϕ >−1 2, 1 2 >0

Proof. The proof is detailed in the appendixA.1.

Once we defined our DtN operators, we can define the following problem: find Wi∈ H1( bJ), for 1 6 i 6 N− 1, such that

                   ∆Wi = 0 in bJ ci(∇Wi· ~n + TiWi) = 1 on Γi

ci+1(∇Wi· ~n + Ti+1Wi) = −1 on Γi+1

ck(∇Wi· ~n + TkWi) = 0 on Γk, k6= {i, i + 1}

∇Wi· ~n = 0 on Γneu

(11)

with the additional condition _Z

b J

Wi= 0 (12)

The additional condition (12) has been added to claim the following proposition (whose proof is detailed in appendixA.2):

Proposition 1.3. There exists a unique Wi in H1( bJ) satisfying (11) and (12).

We can then compute the matrix K ∈ MN −1(R) given by

Ki,j= 1 cj Z Γj Wi− 1 cj+1 Z Γj+1 Wi (13)

Another useful property is given by the following proposition (whose proof is detailed in appendixA.3)

Proposition 1.4. K is a symmetric definite positive matrix.

1.3.3 Build the approximate problem

Thanks to sections1.3.1and1.3.2, we can now give our improved Kirchhoff con-ditions. First, we introduce, for a family of functions vi∈ C1(R∗+, H1(Si))
_16i6N,

with Si is the segment defined in section 1, the following vectors:

Vε_{(t) = (v}

j(t, εσj))16j6N ∈ RN (14-i)

∂SVε(t) = (∂sjvj(t, εσj))16j6N ∈ R

N _(14-ii)

They represent the value of the i-th function and its normal derivate at the point on the boundary εΓi (the blue circled point on figure7).

εσ

εσ

εσ

Figure 7: Points (blue circles) used for the definition of the vectors (14-i) and (14-ii).

One can see that this definition will remain valid even if we consider instead a family of functions vε

i ∈ C1(R∗+, H1(εσi, +∞))

16i6N indexed by ε.

Let us now define the "jump matrix" J , the "average matrix" A and the "weight matrix" C by the following formulas:

• A is the N × N matrix given by (denoting | bJ_{| the area of b}J):

A = _N| bJ2|    1 . . . 1 .. . ... 1 . . . 1    (15)

• J is the N × N matrix given by

J = PTK−1P (16) with K is the previously defined matrix and P is defined by

P =        1 ₋₁ 0 . . . 0 0 1 ₋₁ 0 0 .. . . .. ... ... ... ... 0 0 1 −1 0 0 . . . 0 1 −1        (17)

• C is a diagonal matrix of MN(R), given by

C =          c1 0 . . . 0 0 c2 . .. ... .. . ... ... ... ... .. . . .. cN −1 0 0 . . . 0 cN         

The Improved Kirchhoff model that we propose is the following: find (˜uε i)16i6N such that                ∂2_u˜ε i ∂t2 (t, si)− ∂2_u˜ε i ∂s2 i (t, si) = 0 for (t, si)∈ R∗+×]εσi, +∞[ ˜ uεi(0, si) = fiε for si∈]εσi, +∞[ ∂ ˜uε i ∂t (0, si) = g ε i for si∈]εσi, +∞[ (18) and C∂SU˜ε(t) =  1 εJ + εA ∂2 ∂t2  ˜ Uε(t), for t ∈ R∗ + (19)

Remark 1.6. It is clear that, when ε tends to 0, (18) gives (6). The not so clear point is that, when ε tends to 0, (19) gives (7). However, at least formally, ˜Uε_(t)

tends to ˜U0_{(t), and the formal limit of (19) multiplied by ε leads to the fact}

that ˜U0_{(t) is in the kernel ofJ , which is equal to}

Vect{(1, . . . , 1)t

}

This gives the left part of (7). Then, going back to (19) and doing the scalar product with (1, . . . , 1)t_{leads to}

X j cj ∂uεj ∂sj (t, εσj) = ε| b J_| N X j ∂2_u j ∂t2 (t, εσj)

For the Improved Kirchhoff problem (18,19), one may define the approximate energy e Eε_{(t, v) =} N X i=1 ci 2 Z +∞ εσi    ∂v ∂si(t, s ′₎     2 +     ∂v ∂t(t, s ′₎     2! ds′ + 1 2ε(J V ε_(t)) · Vε_{(t) +}ε 2  A∂V ε ∂t (t)  ·∂V ε ∂t (t) (20)

Proposition 1.5. We have the following energy conversation relation

e Eε_{(t, ˜u}ε_{) = e} Eε_{(0, ˜u}ε_{) =} N X i=1 ci 2 Z +∞ εσi   ∂f ε ∂si (s′)     2 +|gε_(s′₎ |2 ! ds′ (21)

This proposition will be proved in section4.2.

One idea, of course, is to compare the difference between the solution of the exact problem (1) and the solution of the 1D Improved problem (18, 19). We compare this in four steps:

• we estimate the difference between the solution of the exact problem and an expansion in powers of ε,

• we estimate the difference between the solution of the 1D Improved prob-lem and another expansion in powers of ε,

• we remark in section4.2that the two expansions are the same one for at least the two first terms,

• we use some classical triangular inequality to conclude.

Finally, one gets the following proposition (which is an improvement of propo-sition1.1):

Proposition 1.6. Given (δi)16i6N, one can define:

• the set Ωε

i,δ given as (see for instance figure4):

Ωεi,δ={x ∈ Ωεi such that x· ti >δi}

• the 2D function fuε

i defined as

f uε

i(t, x) = ˜uεi(t, x· ti) for t∈ R+ and x∈ Ωεi,δ

Then if fε i ∈ H

5

(Si) and gεi ∈ H 4

(Si), for 1 6 i 6 N (see back hypothesis 1.2),

there exist some 1D functions (ui)16i6N in space such that: N X i=1 1 ε Z Ωε i,δ        ∂(uε i − fuεi) ∂t (t, x)      2 +_∇(uε i − fuεi)(t, x)    2   dx = "_N X i=1 Z +∞ δi   ε2∂u_∂ti(t, si)     2 +ε2_∇ui(t, si)  2 ! dsi # + O(ε6) (22)

Proof. For this proof, we denote by Eδ the following quantity: Eε_{δ(v) =} N X i=1 1 ε Z Ωε i,δ   ∂v_∂t(t, x)     2 +_{|∇v(t, x)|}2 ! dx

Thanks to the hypothesis on the Cauchy data, we can bring the third order approximation for both uε

i and fuεi. We denote then by uεi,2the approximation

of uε

i to second order, and guεi,2the approximation of fuεi to second order. Then

one has that   Eεδ(uεi − fuεi)− Eεδ(uεi,2− guεi,2)    6 Eεδ(uεi − uεi,2) + Eδε(fuεi ∗ guεi,2) (23)

Now, for the left member size, since u0

i = fu0i and u1i = fu1i (this point will be

proved later in section4.2), there exist a function ui such that

uεi,2− guεi,2= ε2ui

One can easily see that uiis 1D in space and does not depend on ε. To conclude,

we use both corollaries3.8and4.1for n = 2 to treat the right member of (23), and we obtain the desired result.

2

The formal expansions

As we said in the introduction, as the problem is multi-scale (in space), it is not possible to write a uniform expansion for the solution everywhere in the domain Ωε_{. The method of matched asymptotics will lead us, we have to consider N + 1}

distinct zones, respectively N slot zones and a junction zone, in which different expansions will be obtained. However, contrarily to the naive intuition, this domain decomposition does not correspond to the partition (2) of Ωε_{: in the}

method of matched asymptotics, the different domains must overlap, the idea being that the different expansions must "coincide" in the overlapping zones.

2.1

An overlapping domain decomposition

In the following, we will denote by C the class of positive continuous functions of ε > 0 that tend to 0 when ε→ 0, less rapidly than ε| log(ε)| (a typical example is εβ_{, with β strictly less than 1).}

C₌  ϕ : R∗+→ R∗+ / lim ε→0ϕ(ε) = 0 and limε→0 ϕ(ε) ε| log(ε)| = +∞  (24) σ1ε σ2ε σ3ε 2ϕ(ε) ϕ(ε) σ1ε σ2ε σ3ε 2ϕ(ε) ϕ(ε)

Figure 8: Slots zones Ωε

i(ε) (left figure) and junction zone Jε(ε) (right figure).

Given ϕ in C, we define, for 1 6 i 6 N, the i-th slot zone by (see figure8): Ωε_{i(ε) ={x ∈ Ω}ε_{i / ϕ(ε) 6 x· ti} (⊂ Ω}ε_i)

The junction zone is defined by (see again figure8):

Jε_{(ε) = ε bJ∪} N [ i=1 {x ∈ Ωε i / 0 6 x· ti< 2ϕ(ε)} (contains Jε)

in such a way that we have Ωε_{= Ω}ε

1(ε)∪ Ωε2(ε)∪ · · · ∪ ΩεN(ε)∪ Jε(ε) with N

overlapping regions (see figure 9) Oi(ε) = Ωεi(ε)∩ Jε(ε) = n(x, y)∈ Ωε _{/ ϕ(ε) < x} · ti< 2ϕ(ε) and − ciε 2 < x· ni< ciε 2 o The mapping x_{7→ (x · ti, x· ni/ε)} (25)

σ

ε

σ

ε

σ

ε

2ϕ(ε)

ϕ(ε)

Figure 9: Configuration of the overlapping domains Oi(ε) (grey zones)

maps the domain Ωε

i(ε) into the rectangle bΩi(ε) with

b Ω_{i(ε) =]ϕ(ε), +∞[×}i₋ci 2, ci 2 h

Note that bΩ_{i(ε) increases when ε decreases and converges to bΩi} when ε tends to 0. In the same way,

x_{7→ bx = x/ε} (26) maps the domain Jε_{(ε) onto bJ(ε), a domain which increases when ε decreases}

and converges to the unbounded domain:

bJ = bJ_∪ N [ i=1 b Bi, Bbi={bx / bx · ti>0, |bx · ni| < ci/2}

2.2

Local expansions and basic equations

We formulate our Ansatz for the asymptotics expansions which consists, in each zone after scaling ((25) or (26)), in looking for power series expansions with respect to ε1

. In other words, we look for functions

uki : bΩi→ C and Uk: bJ→ C, k ∈ N,

such that, at least formally

uε(t, si, εbνi) = ∞ X k=0 εkuki(t, si, bνi) + o(ε∞), in R∗+× bΩi(ε) (27) uε(t, εbx_{) =} ∞ X k=0 εkUk(t,xb_{) + o(ε}∞_), in R∗_{+× bJ(ε)} (28)

It remains to obtain the equations that will determine the functions uk i and

Uk_{. For the u}k

i’s, we substitute formally the expansions (27) in the 2D acoustic

1_{This is a particular case. In [5], one can see that the authors look for power series}

equation written in R∗

+× bΩi(ε), using the scaled coordinates (s, bν), and we

identify the terms with the same power of ε. Straightforward manipulations lead to: ∂2_u0 i ∂bν2 i = 0, ∂ 2_u1 i ∂bν2 i = 0, ∂ 2_uk i ∂t2 − ∂2_uk i ∂s2 i − ∂2_uk+2 i ∂bν2 i = 0, k > 0 (29)

with the Neumann boundary condition along the "lateral" sides ∂uk i ∂bνi  t, si,− ci 2  = ∂u k i ∂bνi  t, si, ci 2  = 0, t_{∈ R}∗+, si∈]0, +∞[, k > 0 (30)

From (29) and (30), we deduce that

Proposition 2.1. For 1 6 i 6 N , for k_{∈ N, one has}

uki(t, si, bνi) = uki(t, si), k > 0 (31-i) ∂2uki ∂t2 − ∂2uki ∂s2 i = 0, k > 0 (31-ii)

Proof. We prove (31-i) and (31-ii) for a given i by induction on k. For k = 0, 1, the first two equations of (29) combined with (30) show that u0

i and u1i are

independent of bν. Then, integrating the third equation of (29) written for k = 0 (respectively k = 1) with respect to bν and using the boundary conditions (30) for k = 2 (respectively for k = 3), we see that u0

i (respectively u1i) satisfies

(31-ii).

Assume that (31-i) and (31-ii) holds up to k = p. Then, the third equation of (29) written for k = p − 1 combined with (30) show that up+1

i is independent of

b

ν. Next, integrating the third equation of (29) written for k = p+1 with respect to bν and using the boundary conditions (30) leads to (31-ii) for k = p + 1.

Remark 2.1. Now we may explain at least formally why the second line of

hypothesis1.2is almost necessary. The proposition2.1tell us that each term of the asymptotic expansion (27) does not depend on bνi. Now assume that there

exists a family of function (fk

i)k∈N defined on bΩi such that

fε_(s i, εbνi) = ∞ X k=0 εk_fk i(si, bνi) + o(ε∞), in bΩi(ε)

We can see that uk

i(0,·) = fik. Since uki does not depend on bν and is continuous

with respect to t, one can deduce that fk

i should neither depends on bν. In the

same way, we get the same result for gk i.

Moreover, from Cauchy data in (1), we deduce uk i(0,·) = fik, 1 6 i 6 N, k > 0 (32-i) ∂uk i ∂t (0,·) = g k i, 1 6 i 6 N, k > 0 (32-ii)

To obtain the equations for the Uk_{’s, we substitute formally the expansion (28)}

we identify the terms with the same power of ε. This permits us to see that the Uk_{’s satisfy embedded Laplace’s equations}

∆U0_{= 0, ∆U}1_{= 0, ∆U}k+2₌∂2Uk

∂t2 , k∈ N, in R ∗

+× bJ (33)

with Neumann boundary conditions ∂Uk

∂~n = 0, on ∂bJ, k∈ N (34)

Remark 2.2. In the sequel, we shall adopt the convention that all quantities

super-indexed by k (such as Uk_{, u}k

i, . . . ) are 0 for negative values of k. This

will be useful to simplify some formulas. For instance, with this convention the last equation of (33) is also valid for k = 0, 1.

2.3

Matching conditions

Equations (31) to (34) are not sufficient to characterize the functions (uk i, Uk)’s:

we miss boundary conditions at s = 0 for the uk

i’s and additional conditions at

infinity for the Uk_{’s. These conditions, namely the matching conditions, will}

couple the uk

i’s and the Uk’s. To derive them, one writes that the two expansions

(27) and (28) must coincide in each overlapping zone Oi(ε). Introducing the

canonical semi-strip Bi defined by

Bi=

n b

x_{∈ bBi / b}x_{· ti> σi} o

(see the definition of σi given by1.4)

and calling Uk

i the restriction of Uk on the i-th canonical semi-strip let us allow

to express the following matching conditions:

+∞ X k=0 εkUik(t, si/ε, νi/ε) + o(ε∞) = +∞ X k=0 εkuki(t, si, νi/ε) + o(ε∞) in R∗+× Oi(ε) (35) To express more precisely these matching conditions, it is useful to describe the form of the functions Uk

i’s in the semi-strips Bi: this is the object of the next

section.

2.3.1 Modal expansion of solutions of embedded Laplace equations Let consider, for a given i, Uk

i :Bi→ R satisfying

∆Ui0= 0, ∆Ui1= 0, ∆Uik+2=

∂2_Uk i ∂t2 , in Bi, ∂Uk i ∂bνi = 0 for _|bνi| = ci/2 (36) Later, in section2.3.2, the results of the present section will be applied to the re-striction of the Uk_{’s on B}

i, where Uk’s are the coefficients of the expansion (28).

In Bi, we shall use the local coordinates (bsi, bνi) = (xb· ti,bx· ni) such that

b

The behaviour of the fields Uk

i in the semi-strips Bi is easily described by using

separation of variables in (bsi, bνi) coordinates, which introduce naturally the

orthonormal family ewi,p defined by

e wi,0(bνi) = r 1 ci, wei,p(bνi) = r 2 cicos  pπbνi ci + pπ 2  (37)

Remark 2.3. One can easily see that

e

wi,p(bνi) = wi,p(bνi+ ci/2)

where wi,p is the family defined by (10).

The "basis" ewi,pis adapted to the Neumann conditions at bνi=±ci/2, and there

exists 1D functions (in space) Uk

i,p(t, bsi) such that

Uik(t, bsi, bνi) =

X

p∈N

Ui,pk (t, bsi) ewi,p(bνi) (38)

with (thanks to the fact that ewi,p is orthonormal):

Ui,pk (t, bsi) =

Z ci/2

−ci/2

Uik(t, bsi, bνi′) ewi,p(bνi′)dbνi′

If we substitute formally the expression (38) into the equations (36), we obtain

∀k 6 1, ∀p ∈ N, ∂ 2_Uk i,p ∂bs2 i (t, bs i)−  pπ ci 2 Ui,pk (t, bsi) = 0 (39-i) ∀k > 2, ∀p ∈ N, ∂ 2_Uk i,p ∂bs2 i (t, bsi)−  pπ ci 2 Ui,pk (t, bsi) = ∂2_Uk−2 i,p ∂t2 (t, bsi) (39-ii)

The resolution of (39) is a tedious but quite simple exercise on ordinary differ-ential equations (the idea is to solve this system with respect to bsi on a given

time t fixed). In what follow, we reproduce some results of [5], that we present in a slightly different form, more adapted to the purpose of this report. After having remarked that the change of unknowns

Ui,pk (t, bsi) = exp(±pπ

ci bs

i)Vi,p,±k (t, bsi)

leads to the equations (with the convention of the remark 2.2) ∂2_Vk i,p,± ∂bs2 i (t, bsi)± 2pπ ci ∂Vk i,p,± ∂bsi (t, bs i) = ∂2_Vk−2 i,p,± ∂t2 (t, bsi) (40)

One can see, for k = 0, 1, that the equation (40) does not depend on time. We can write Vk _{as sum of products of functions depending on t and functions}

depending on si. The idea is to work with separation of variables and to search

Vk

i,p,±, for k ∈ N, under the form Vi,p,±k (t, bsi) = φki,p,±(bsi)ψki,p,±(t), and one can

see that the family φ has to satisfy the following equation (for this part, we don’t care about dependance in time for functions Vk

i,p,±): ∂2_φk i,p,± ∂bs2 i (bs i)± 2pπ ci ∂φk i,p,± ∂bsi (bsi) = C k i,p,±φk−2i,p,±(bsi) (41)

Remark 2.4. One can see later (in the proof of lemma 2.3) that is it judicious to take Ck

i,p,±= 1, for each quartet (i, p, k,±). In [3] we took Ci,p,±k =−1, this

coefficient being dictated by the coefficient of the right member of embedded Laplace equations.

We introduce, for each p ∈ N, two sequences of polynomial solutions (in space) of (41)

(nk

i,p, dki,p), k∈ N,

which are defined inductively on k, for each p ∈ N and are identically 0 for odd values of k.

• The value p = 0 plays a particular role, since equation (41) degenerates (the equation with "± = +" is the same as the equation with "± = −"). For k = 0, 1 one has

n0_{i,0(bsi) = bsi, n}1_{i,0(bsi) = 0,} d0_{i,0(bsi) = 1, d}1_{i,0(bsi) = 0} (42) continuing for k > 2 with

∂2nk i,0 ∂bs2 i = nk−2_i,0 , nk_{i,0(0) =} ∂n k i,0 ∂bsi (0) = 0 (43-i) ∂2_dk i,0 ∂bs2 i = dk−2_i,0 , dk_{i,0(0) =} ∂d k i,0 ∂bsi (0) = 0 (43-ii) It is easy to see that, for even k, one recovers the monomials of the series expansion of exp(bsi): n2m_{i,0(bsi) =} bs 2m+1 i (2m + 1)!, d 2m i,0(bsi) = bs 2m i (2m)!, (44) • For p > 1, one starts from

n0_{i,p(bsi) = 1, n}1_{i,p(bsi) = 0,} d0_{i,p(bsi) = 1, d}1_{i,p(bsi) = 0, for p > 1 (45)} Then, (nk

i,p, dki,p) are defined as the polynomial solutions of

∂2_nk i,p ∂bs2 i +2pπ ci ∂nk i,p ∂bsi = nk−2

i,p , nki,p(0) = 0 (46-i)

∂2_dk i,p ∂bs2 i − 2pπ ci ∂dk i,p ∂bsi = dk−2

i,p , dki,p(0) = 0 (46-ii)

Note the difference of sign between (46-i) and (46-ii) for the first derivative term.

The following proposition gives some properties about the functions (nk i,p, dki,p):

Proposition 2.2. Let (nk

i,p, dki,p)k∈N,p>1 be a polynomial family of

func-tions satisfying (45) and (46), then

– this family is well-defined and is unique,

– (n2m

i,p, d2mi,p) have degree m,

– we have a relation between n2m

i,p and n2mi,p given by:

n2m_{i,p(bsi) = d}2m_i,p(−bsi) – we have a relation between n2m

i,p and n2mi,1 given by:

n2m_{i,p(bsi) = p}−kn2m_{i,1(p bsi)}

Proof. The proof of this proposition is very similar to the proof of the

proposition 2.3 of [4].

Remark 2.5. The fact that the functions n and d are identically 0 for odd values

of k is due to the fact that the equations satisfied by U2m+1

i,p , for m ∈ N, are

identical to the equations satisfied by U2m

i,p (this is a choice of notation that

permit us the writing of future lemmas, regardless whether k is odd or even). Next, we construct two families of functions nk

i,p and dki,p from Bi into R, for

(k, p)∈ N2_{, by}

nk_i,p(x_{b) = exp(pπbsi/ci)n}k_{i,p(bsi) ewi,p(bνi) (47-i)} dk

i,p(xb) = exp(−pπbsi/ci)di,pk (bsi) ewi,p(bνi) (47-ii)

that constitute particular families of embedded Laplace’s equations: ∀p ∈ N, ∀k ∈ N, ∆nk+2

i,p = nki,p, ∆dk+2i,p = dki,p, in Bi (48)

with homogeneous Neumann boundary conditions at bνi=±ci/2.

The families (nk

i,p, dki,p) allow us to express the following lemma

Lemma 2.3 (Fundamental lemma for the expansion of Uk

i). Let (Uik)k∈N be a

family of functions satisfying (36), then there exists two sequences (ηk

i,p, δki,p)(k,p)∈N2

of complex functions depending only on t such that:

Uik(t,bx) = X p∈N k X l=0 " ∂l_ηk−l i,p ∂tl (t)n l i,p(xb) + ∂l_δk−l i,p ∂tl (t)d l i,p(bx) # in R∗+× Bi (49)

Proof. We will prove the relation (49) by induction on k. • Initialization: for k = 0, 1 and for a given t ∈ R∗

+, one can see that

∆Uk

0(t,·) vanishes. Hence, there exists two sequences of complex numbers,

that we call ηk

i,p(t) and δi,pk (t), such that

Uik(t,bx) =

X

p∈N



ηki,p(t)n0i,p(xb) + δi,pk (t)d0i,p(bx)



in R∗

+× Bi (50)

For k = 0, the expressions (50) and (49) are the same ones. For k = 1, since n1

i,p = d1i,p = 0 for all p∈ N, one can see also that the expressions

(50) and (49) remain the same ones.

• Heredity: let us admit that the sequences (ηk

i,p, δki,p) have been

con-structed up to l = k − 1. Then let us introduce (remember that n1 i,p =

e Uik(t,bx) = X p∈N k X l=1 " ∂l_ηk−l i,p ∂tl (t)n l i,p(bx) + ∂l_δk−l i,p ∂tl (t)d l i,p(bx) # = X p∈N k X l=2 " ∂l_ηk−l i,p ∂tl (t)n l i,p(bx) + ∂l_δk−l i,p ∂tl (t)d l i,p(bx) #

We have (successively, we use (48), apply the change of index l → l − 2, and use (49) for k − 2): ∆ eUik(t,xb) = X p∈N k X l=2 " ∂l_ηk−l i,p ∂tl (t)∆n l i,p(bx) + ∂l_δk−l i,p ∂tl (t)∆d l i,p(xb) # = X p∈N k X l=2 " ∂l_ηk−l i,p ∂tl (t)n l−2 i,p (bx) + ∂l_δk−l i,p ∂tl (t)d l−2 i,p (bx) # = X p∈N k−2 X l=0 " ∂l+2_ηk−2−l i,p ∂tl+2 (t)n l i,p(xb) + ∂l+2_δk−2−l i,p ∂tl+2 (t)d l i,p(bx) # = ∂ 2_Uk−2 i ∂t2 (t,xb) The function Uk

i(t,·) − eUik(t,·) being harmonic in Bi, we know there exist two

sequences complex numbers that we choose to denote respectively by (ηk

i,p(t))p∈N

and (δk

i,p(t))p∈N, such that

Uk i(t,xb)− eUik(t,bx) = X p∈N  ηk

i,p(t)n0i,p(bx) + δki,p(t)d0i,p(bx)



and the proof is complete.

From the previous lemma, it is natural to introduce the (increasing) spaces Vik = span  (nli,p, dli,p), p∈ N, l 6 k , _Vi= [ k∈N Vik

and let us introduce, for the functions U ∈ V0

i, the two natural families of linear

forms Ni,p and Di,p defined such that:

∀U ∈ V0

i, U (xb) = ∞

X

p=0

Ni,p(U )n0i,p(x) +Di,p(U )d0i,p(x)

Remark 2.6. One can see these linear forms as "traces" of the function on bsi= 0

(although bsi= 0 does not belong toBi).

Another way to write lemma2.3is Corollary 2.4. Let (Uk

i)k∈N be a family of functions satisfying (36), then:

• Uk i(t,·) ∈ Vik • It can be written as Uik(t,bx) = X p∈N k X l=0 " ∂l Ni,p(Uik−l(t,·)) ∂tl (t)n l i,p(bx) + ∂l Di,p(Uik−l(t,·)) ∂tl (t)d l i,p(bx) #

2.3.2 Derivation of the matching conditions

To express the matching conditions, we simply write that the two expansions (27) and (28) must coincide in the overlapping zones Oi(ε) (see back figure9

page16). In such zones, we have the following relations (we recall here (35) by denoting Uk i the restriction of Uk to Bi) +∞ X k=0 εkUik(t, si/ε, νi/ε) + o(ε∞) = uε(t, si, νi) = +∞ X k=0 εkuki(t, si, νi/ε) + o(ε∞)

for t ∈ R, ϕ(ε) < si < 2ϕ(ε) and|νi| < εci/2. We denoteR the right equality

of this relation and L the left one.

Treatment of R This is the simplest one. One has to consider that each function uk

i does not depend on the third variable νiand the fact that φ belongs

to C defined in (24), which implies that si tends to 0 as ε tends to 0. By using

a Taylor development on si= 0 for each function uki, one has

uε(t, si, νi) = X k∈N X m∈N  1 m! ∂m_uk i ∂sm i (t, 0)  εksmi + o(ε∞) (51)

Treatment of L Here one has more work to do. First, we use the corollary2.4

of the fundamental lemma to express each Uk

i on the family (nmi,p, dmi,p)(p,m)∈N2

defined in section2.3.1, and we get

uε(t, si, νi) = X k∈N εkX p∈N k X m=0 h ∂m Ni,p(Uk−m(t,·)) ∂tm (t)n m i,p  si ε, νi ε  + ∂ m Di,p(Uk−m(t,·)) ∂tm (t)d m i,p  si ε, νi ε  i + o(ε∞)

that can be rewritten as (summing for m 6 k is like summing for k > m, for convergent series) uε(t, si, νi) = X m∈N X k>m X p∈N εkh ∂ m_N i,p(Uk−m(t,·)) ∂tm (t)n m i,p  si ε, νi ε  + ∂ m_D i,p(Uk−m(t,·)) ∂tm (t)d m i,p  si ε, νi ε  i + o(ε∞) Since the function ϕ belongs to C, and since the functions dm

i,pare exponentially

decreasing at bsi = +∞, the corresponding terms in the previous sum can be

"put into" the o(ε∞_{) part. For the rest of the sum, we distinguish the terms for}

to p > 1 (which are exponentially increasing at infinity): uε(t, si, νi) = r 1 ci X m∈N X k>2m εk∂ 2m Ni,0(Uk−2m(t,·)) ∂t2m (t) 1 (2m + 1)!  si ε 2m+1 + r 1 ci X m∈N X k>2m εk∂ 2m_D i,0(Uk−2m(t,·)) ∂t2m (t) 1 (2m)!  si ε 2m +X p>1 X m∈N X k>m ∂m_N i,p(Uk−m(t,·)) ∂tm (t)n m i,p  si ε  exp  pπsi εci  e wi,p  νi ε 

Using the change of index k → k+2m+1 in the first line (respectively k → k+2m in the second line) of the previous equation, one gets

uε(t, si, νi) = r 1 ci X m∈N X k>−1 ∂2m Ni,0(Uk+1(t,·)) ∂t2m (t) 1 (2m + 1)!ε k_s2m+1 i + r 1 ci X m∈N X k∈N ∂2m Di,0(Uk(t,·)) ∂t2m (t) 1 (2m)!ε k_sm i +X p>1 X m∈N X k>m Sp,m,n(ε, si, νi) (52) with Sp,m,n(ε, si, νi) = ∂m Ni,p(Uk−m(t,·)) ∂tm (t)n m i,p  si ε  exp  pπsi εci  e wi,p  νi ε 

Conclusion Finally, the formal identification of the expressions (51) and (52) in the overlapping zone Oi(ε), as functions of siand ε, will lead us to our

match-ing conditions.

First, for p > 1, after multiplication of (51) and (52) by ewi,p ν_εi
and integration

over νi, we get X m∈N nm_i,p si ε  X k>m ∂m_N i,p(Uk−m(t,·)) ∂tm (t)   = 0

The proposition2.2implies that the functions nm

i,p are linearly independent and

one deduces from that Ni,p(Uk−2m(t,·)) = 0, ∀m > 0, ∀k > 2m, that is to say:

Ni,p(Uk(t,·)) = 0, p > 1, k > 0, 1 6 i 6 N (53)

which express the absence of exponentially growing terms. We can see that (53) leads to the following condition

Next, it remains to identify power series expansions.The identification of the terms in εk_sm

i , distinguishing even and odd values of m, leads to

∂2m_D i,0(Uk(t,·)) ∂t2m (t) = √_c i∂ 2m_uk i ∂s2m i (t, 0) ∂2m Ni,0(Uk(t,·)) ∂t2m (t) = √_c i ∂2m+1_uk−1 i ∂s2m+1_i (t, 0) Using the fact that each uk

i solves the 1D time wave equation, we have

∂2m_uk i ∂s2m i (t, 0) = ∂ 2m_uk i ∂t2m (t, 0)

and we get the "Dirichlet" and "Neumann" matching conditions, namely: Di,0(Uk(t,·)) = √ciuki(t, 0), ∀k ∈ N, ∀t ∈ R∗+ (55-i) Ni,0(Uk(t,·)) = √ci∂u k−1 i ∂si (t, 0), ∀k ∈ N, ∀t ∈ R∗ + (55-ii)

2.4

Summary

Finally, one gets the coupled system of equations and matching conditions, that are (we recall here the equations (31), (32), (33) and (34), and the matching conditions (55)):

• the equations written on the slots zone:

uki(t, si, bνi) = uki(t, si), k > 0 ∂2_uk i ∂t2 − ∂2_uk i ∂s2 i = 0, k > 0 uki(0,·) = fik, 1 6 i 6 N, k > 0 ∂uki ∂t (0,·) = g k i, 1 6 i 6 N, k > 0

• the equations written on the junction zone:

∆U0_{= 0, ∆U}1_{= 0, ∆U}k+2₌ ∂2Uk

∂t2 , k∈ N, in R ∗ +× bJ

∂Uk

∂~n = 0, on ∂bJ, k∈ N

• the matching conditions between the slots zones and the junction zone: Di,0(Uk(t,·)) = √ciuki(t, 0), ∀k ∈ N, ∀t ∈ R∗+ Ni,0(Uk(t,·)) = √ci ∂uk−1_i ∂si (t, 0), _{∀k ∈ N, ∀t ∈ R}∗ +

3

Justification and error estimates

In this paragraph, our goal is to prove that the functions (uk

i)k∈N,16i6N and

(Uk₎

k∈N are uniquely defined, and that there exists an approximate function

built from these functions which differs from the solution of the exact problem with some power of ε that is increasing with the order of the approximation we consider.

3.1

Existence and uniqueness of the formal expansion

In this section, our goal is to prove that the equations (31), (32), (33) and (34), together with the matching conditions (55), define unique families (uk

i)k∈N,16i6N

and (Uk₎

k∈N. To reach our goal, we first formulate an equivalent problem where

the unknowns (Uk₎

k∈N are restricted to the junction bJ.

3.1.1 Restriction to a bounded domain of the problem for the Uk

Our goal in this section is to characterize the restrictions of the functions Uk_’s

to the junction bJ by giving exact Dirichlet to Neumann boundary conditions at the interfaces Γi (see again figure 5).

Let us start with the corollary 2.4 of the fundamental lemma, applied to the restriction of Uk_{to the semi-slot B}

i, and then apply (53), then we get, separating

l = 0 from l > 1:          Uik(t,xb) = Ni,0(Uk(t,·))n0i,0(bx) + X p∈N Di,p(Uk(t,·))d0i,p(bx) + k X l=1 ∂l_N i,0(Uk−l(t,·)) ∂tl n l i,0(bx) + X p∈N k X l=1 ∂l_D i,p(Uk−l(t,·)) ∂tl d l i,p(bx)

Remember that, for our junction zone bJ, the boundary Γi is parametrized by

Γi= n x_{∈ ∂ bJ /} x_{· ti= σi} and |x · n_{i| <} ci 2 o ,

then, in the previous sum, we make the index change l → 2l (remember also that nl_i,p and dl_i,p vanish for odd values of l), and the use of analytically expressions of nl

                             Uk i(t, σi, bνi) = Ni,0(Uk(t,·)) σi √_c i +_Di,0(Uk(t,·)) 1 √_c i + X p>1 Di,p(Uk(t,·)) exp  −pπσ_c i i  e wp,i(bνi) + _√1 ci k/2 X l=1 ∂2l_N i,0(Uk−2l(t,·)) ∂t2l σi2l+1 (2l + 1)! + _√1 ci k/2 X l=1 ∂2l Di,0(Uk−2l(t,·)) ∂t2l σ2l i (2l)! + k/2 X l=1 X p>1 ∂2l Di,p(Uk−2l(t,·)) ∂t2l exp  −pπσ_c i i  d2l_{i,p(σi) ewp,i(bνi)} (56) From (56), since ( ewp,i)p∈N is an orthonormal family of L2(Γi), one can see that

Z

Γi

Uk

i(t, σi, bνi) ewp,i(bνi)dbνi=Di,p(Uk(t,·))e “ −pπσi_ci ” + k/2 X l=1 ∂2l_D i,p(Uk−2l(t,·)) ∂t2l e “ −pπσi_ci ” d2l_i,p(σi) (57)

Moreover, on Γi, the normal derivative of Ukis the same as the derivative among

si. After some computations, one has

                                    ∂Uk i ∂si (t, σi, bνi) = Ni,0(Uk(t,·)) 1 √_c i − X p>1 pπ ciD i,p(Uk(t,·))e “ −pπσi_ci ” e wp,i(bνi) + _√1 ci k/2 X l=1 ∂2l Ni,0(Uk−2l(t,·)) ∂t2l σ2l i (2l)! + _√1 ci k/2 X l=1 ∂2l Di,0(Uk−2l(t,·)) ∂t2l σ_i2l−1 (2l_{− 1)!} − k/2 X l=1 X p>1 pπ ci ∂2l_D i,p(Uk−2l(t,·)) ∂t2l e “ −pπσi_ci ” d2l_{i,p(σi) ewp,i(bνi)} + k/2 X l=1 X p>1 ∂2l Di,p(Uk−2l(t,·)) ∂t2l e “ −pπσi_ci ”∂d2li,p ∂si (σi) ewp,i(bνi) (58) The most important point is that, using (57), the sum of the second and fifth lines of (58) is equal to −X p>1 pπ ci Z Γi Uik(t, σi, bνi) ewp,i(bνi)dbνi

We can now, by using (55-ii) and by using the definition of non-local DtN oper-ator Ti given by (9), sum up the previous information by writing the following

relation: on Γi, one has

                       ∂Uk ∂~n (t,·) + TiU k_(t, ·) = ∂u k−1 i ∂si (t, 0) + k/2 X l=1 ∂2l+1_uk−2l−1 i ∂t2l_∂s i (t, 0) σ 2l i (2l)! + _√1 ci k/2 X l=1 ∂2lDi,0(Uk−2l(t,·)) ∂t2l σ2l−1i (2l− 1)! + k/2 X l=1 Si,k,l(σi) (59) with Si,k,l(σi) = X p>1 ∂2l Di,p(Uk−2l(t,·)) ∂t2l exp  −pπσ_c i i _∂d2l i,p ∂si (σi) ewp,i

For the right hand side of (59), one can see that we have two parts: • one part linked to the knowledge of Uk′

, for k′_{< k,}

• one more interesting part linked to the derivative of uk−1

i (t, si) at si = 0.

Finally, we obtain a problem "equivalent" to ((31), (32), (33), (34), (55)) by re-placing (55-ii) to the DtN condition (59). The precise statement is the following (denoting C1,2(V ) = C0(R∗+, H1(V ))∩ C1(R∗+, L2(V )):

Theorem 3.1. Let (ukj)∈ C1,2(]0, Lj[)
_16j6nand



Uk ∈ C1,2,loc(bJ)



be a so-lution of ((31), (32), (33), (34)) with the matching conditions (55), then

n

(uk1, . . . , ukn, Uk)∈ C1,2(]0, L1[)× · · · × C1,2(]0, LN[)× C1,2( bJ), k > 0

o

is solution of ((31), (32)), ((33), (34))Jb, (55-i) and (59), where ((33), (34))Jbholds

the restriction of ((33), (34)) respectively to bJ and ∂bJ_{∩ ∂ bJ.}

Reciprocally, if (ukj)∈ C1,2(]0, Lj[)
16j6nand  Uk _{∈ C}1,2,loc(bJ)  are solution of ((31), (32)), ((33), (34))_Jb, (55-i) and (59), then by extending Uk _{to each}

Bi via           Uik(t,xb) = √ci∂u k−1 i ∂si (t, 0)n0i,0(bx) + X p∈N Di,p(Uk(t,·))d0i,p(xb) +√ci k X l=1 ∂l+1_uk−l−1 ∂tl_∂s i (t, 0)nli,0(bx) + X p∈N k X l=1 ∂l_D i,p(Uk−l(t,·)) ∂tl d l i,p(xb) Thenn(uk1, . . . , ukn, Uk)∈ C1,2(]0, L1[)× · · · × C1,2(]0, LN[)× C1,2,loc(bJ), k > 0 o

Proof. The direct statement has been proved. For the reciprocal, let us consider

n

(uk1, . . . , ukn, Uk)∈ C1,2(]0, L1[)× · · · × C1,2(]0, LN[)× C1,2,loc(bJ), k > 0

o so-lution of ((31), (32)), ((33), (34))_Jb, (55-i) and (59), and let us extend Uk _{to each}

Bi. It is quite easy to prove that Uk satisfies embedded Laplace equations on

Bi (the proof of this point is the same as the proof of the fundamental lemma 2.3). By looking the modal expansion of the corollary and the modal expansion written above, (55-i) is immediate. A priori, the most difficult point is to show that Dirichlet and Neumann traces are continuous on Γj; however these

condi-tions are satisfied thanks to (59) (this condition has been built to satisfy such things).

3.1.2 Auxiliary lemmas

Existence and uniqueness is done by induction on k. According to theorem3.1, it suffices to consider the problem ((31), (32)), ((33), (34))_Jb, (55-i) and (59). To clearly identify the recursion process, it is useful to reformulate this problem in a more decoupled way (we mean between uk

i and Uk, at each step k), which is

also useful from the computational point of view.

To achieve such a decoupling the idea is first to consider (59) as a boundary condition for Uk_{, next to formulate a 1D transmission problem for u}k

i. In this

sense, we have to prove first the following two technical lemmas. Lemma 3.2. Given Φ _{∈ L}2( bJ) and (gi ∈ H−

1 2(Γ

i))16i6N, there exists U ∈

H1_{( bJ), which is unique up to an additive constant, such that}

       ∆U = Φ, in bJ, ∂U ∂~n = 0, on ∂ bJ\ [ i Γi ! ∂U ∂~n + TiU = gi, on Γi (60)

if and only if one satisfies the compatibility condition (in a weak sense)

X i Z Γi gi= Z b J Φ (61)

Moreover, Wi being defined by (11) and (12), any solution of (60) satisfies

1 ci Z Γi U₋ 1 ci+1 Z Γi+1 U = Z b J WiΦ + N X j=1 Z Γj Wigj (62)

Proof. The existence-uniqueness proof is a classical exercise about Lax-Milgram

lemma and Poincaré-Virtinger’s inequality (the important point is that Ti :

H12(Γ

i)→ H−

1 2(Γ

i) is a positive symmetric operator whose kernel contains the

space of constant functions - see appendix A.1). The compatibility condition (61) is obtained by integrating the first equation of (60), using Green’s formula and the symmetry of Ti (see again appendixA.1).

To obtain (62), we multiply the equation for U by Wi and integrate over bJ.

Using Green’s formula twice, and the fact that Wi is harmonic, we get

X j Z Γj ∂Wi ∂~n U− ∂U ∂~nWi= Z b J WiΦ

Using the boundary conditions on Γj for the Neumann traces of U and Wi

together with the symmetry of Ti, we obtain (62).

Lemma 3.3. Given (δi)16i6N −1 ∈ H1loc(R∗+) and α ∈ L2loc(R∗+) such that

δi(0) = 0 for any i, and given (fi)16i6N −1 ∈ H1(]0, +∞[) and (gi)16i6N −1 ∈

L2_{(]0, +}

∞[) , there exists a unique family (ui)16i6N ∈ C0(R∗+, H1(]0, +∞[) ∩

C1_(R∗ +, L2(]0, +∞[)) such that                    ∂2_u i ∂t2 − ∂2_u i ∂s2 i = 0, in R∗+×]0, +∞[ ui(0,·) = fi, ∂ui ∂t (0,·) = gi, N X i=1 ci ∂ui ∂si (_{·, 0) = α(·)} ui+1(·, 0) − ui(·, 0) = δi(·), ∀1 6 i 6 N − 1 (63)

Moreover, the norm of the solution is bounded by the norm of the Cauchy data, the norms of the Kirchhoff data and√t.

Proof. The idea is to say that solution of problem (63) can be decomposed into three functions:

• the solution (u0

i) of the following problem:

                     ∂2_u0 i ∂t2 − ∂2_u0 i ∂s2 i = 0, in R∗ +×]0, ∞[ u0i(0,·) = fi, ∂u0 i ∂t (0,·) = gi, N X i=1 ci∂u 0 i ∂si (·, 0) = 0 u0i+1(·, 0) − u0i(·, 0) = 0, ∀1 6 i 6 N − 1

This problem admits a unique solution (u0

i)16i6N ∈ C0(R∗+, H 1 (]0, +∞[)∩ C1_(R∗ +, L 2_{(]0, +}

∞[)) (and one has a priori estimates about the solution). • the solution (u1

i) of the following problem:

             ∂2_u1 i ∂t2 − ∂2_u1 i ∂s2 i = 0, in R∗ +×]0, +∞[ u1i(0,·) = 0, ∂u1 i ∂t (0,·) = 0, u1i(·, 0) = X j<i δj(·), ∀1 6 i 6 N

The solution of this system is simply written as (extending each δj(τ ) to

0 when τ < 0):

u1i(t, si) =

X

j<i

and it satisfies

u1i+1(·, 0) − u1i(·, 0) = δi(·)

Hypothesis on each δj ensure that (u1i)16i6N ∈ C0(R∗+, H1(]0, +∞[) ∩

C1_(R∗ +, L2(]0, +∞[)). However, we have N X i=1 ci ∂u1 i ∂si (_{·, 0) = −} N X i=1 X j<i ciδj′(·) • the solution (u2

i) of the following problem:

                     ∂2_u2 i ∂t2 − ∂2_u2 i ∂s2 i = 0, in R∗ +×]0, +∞[ u2i(0,·) = 0, ∂u2 i ∂t (0,·) = 0, N X i=1 ci ∂u2 i ∂si (_{·, 0) = α(·) +} N X i=1 X j<i ciδj′(·) u2i+1(·, 0) − u2i(·, 0) = 0, ∀1 6 i 6 N − 1

Solution of this equation can be written as follow:

u2i(t, si) =− 1 P cj  Z t−si 0 α(τ )dτ + N X i=1 X j<i ciδj(t− si)  

Under the hypothesis about α and each δj, one has (u2i)16i6N ∈ C0(R∗+, H1(]0, +∞[)∩

C1_(R∗

+, L2(]0, +∞[)).

Finally, one can check that ui = u0i+ u1i+ u2i satisfies problem (63), and

unique-ness is obtained thanks to uniqueunique-ness of the problem with classical Kirchoff conditions (if there exists two solutions of (63), the difference satisfies the wave equation with classical Kirchhoff conditions and null initial conditions). Now, we compute for a given time t ∈ R∗

+ the differents norms of the solution

of problem (63). One can easily check that kui(t,·)kL2_(]0,+∞[)6 u0 i(t,·) _L2_(]0,+∞[)+ u1 i(t,·) _L2_(]0,+∞[)+ u2 i(t,·) _L2_(]0,+∞[) ∂u_∂ti(t,·) _L2_(]0,+∞[) 6 ∂u 0 i ∂t (t,·) _L2_(]0,+∞[) + ∂u 1 i ∂t (t,·) _L2_(]0,+∞[) + ∂u 2 i ∂t (t,·) _L2_(]0,+∞[) ∂u_∂sii(t,·) _L2_(]0,+∞[) 6 ∂u 0 i ∂si (t,_{·) L}2_(]0,+∞[) + ∂u 1 i ∂si (t,_{·) L}2_(]0,+∞[) + ∂u 2 i ∂si (t,_{·) L}2_(]0,+∞[)

By using some classical estimates, we can say that the norm u0 i(t,·)

_L2_(]0,+∞[)

is bounded by square root of order 1 polynom (in t) depending on the Cauchy datas, and the norm of its derivatives is bounded by a constant depending only on Cauchy data. For the functions u1

i and u2i, by using explicit computation,

• for terms u1 i: u1 i(t,·) _L2_(]0,+∞[)6 X j<i cikδjkL2_(]0,t[) ∂u 1 i ∂t (t,·) _L2_(]0,+∞[) 6X j<i ci δ′ j _L2_(]0,t[) ∂u 1 i ∂si (t,_{·) L}2_(]0,+∞[) 6X j<i ci δ′ j L2_(]0,t[) • for terms u2 i: u2 i(t,·) _L2_(]0,+∞[)6 1 P cj  √tkαkL2_(]0,t[)+ N X i=1 X j<i cikδjkL2_(]0,t[)   ∂u 2 i ∂t (t,·) _L2_(]0,+∞[) 6 _P1 cj  kαkL2_(]0,t[)+ N X i=1 X j<i ci δ′ j _L2_(]0,t[)   ∂u2 i ∂si (t,_{·) L}2_(]0,+∞[) 6 _P1 cj  kαkL2_(]0,t[)+ N X i=1 X j<i ci δ′ j _L2_(]0,t[)  

and the proof is complete.

Remark 3.1. In the case where α_{∈ L}2(]0, +_{∞[) and δ}i ∈ H1(][0, +∞), we can

see that the L2 _{norm of derivatives of solution is uniformly bounded over time.}

3.1.3 Existence and uniqueness

According to ((33), (34))Jband (59), we can apply, for a given t ∈ R ∗ +, lemma 3.2with U = Uk_{(t,·), Φ =}∂2 Uk−2 ∂t2 (t,·) and gi= g k−1 i (t,·) with                        g_ik−1(t,_·) = ∂u k−1 i ∂si (t, 0) + k/2 X l=1 ∂2l+1_uk−2l−1 i ∂t2l_∂s i (t, 0) σ 2l i (2l)! + k/2 X l=1 ∂2l_uk−2l i ∂t2l (t, 0) σ_i2l−1 (2l− 1)! + k/2 X l=1 X p>1 ∂2l Di,p(Uk−2l(t,·)) ∂t2l e “ −pπσi_ci ”∂d2li,p ∂si (σi) ewp,i (64)

where the index k − 1 in gk−1

i is "justified" by the fact that gik−1 is known

explicitly when the uk−1

gives, once we made the change of index k = k + 1,                   N X i=1 ci ∂uk i ∂si (t, 0) = Z b J ∂2_Uk−1 ∂t2 − N X i=1 ci (k+1)/2_X l=1 ∂2l+1_uk−2l i ∂t2l_∂s i (t, 0) σ 2l i (2l)! − N X i=1 ci (k+1)/2_X l=1 ∂2l_uk+1−2l i ∂t2l (t, 0) σ_i2l−1 (2l_{− 1)!} (65)

Remark 3.2. One could see that if we take (σ′

i)16i6N such that σi′>σi, and if

we define bJ′ _as b J′ _{= bJ} ∪ N [ i=1 n x_{∈ R}2_{/ /σi}6 x_{· ti}< σ′ i and |x · ni| < ci 2 o ,

one should have                   N X i=1 ci ∂uk i ∂si (t, 0) = Z b J ∂2_Uk−1 ∂t2 − N X i=1 ci (k+1)/2_X l=1 ∂2l+1_uk−2l i ∂t2l_∂s i (t, 0) σ 2l i (2l)! − N X i=1 ci (k+1)/2 X l=1 ∂2l_uk+1−2l i ∂t2l (t, 0) σ_i2l−1 (2l_{− 1)!}

This point can be proved by using the modal expansion of Uk−1 _{on each}

rect-angle (bsi, bνi)∈]σi, σ′i[×] − ci/2, ci/2[.

Given 1 6 i 6 N, integrating Uk _{on Γ}

i by using modal expansion (56) and

"Dirichlet" condition (55-i) - "Neumann" condition (55-ii) gives                  1 ci Z Γi Uk_(t, ·) = uk i(t, 0) + ∂uk−1_i ∂si (t, 0)σi + k/2 X l=1 ∂2l+1_uk−2l−1 i ∂t2l_∂s i (t, 0) σ 2l+1 i (2l + 1)! + k/2 X l=1 ∂2l_uk−2l i ∂t2l (t, 0) σ2l i (2l)! (66)

Then using (62) with U = Uk_{, Φ =} ∂2Uk−2

∂t2 and g k−1

i defined in (64), one has,

for 1 6 i 6 N − 1                         uki(t, 0)− uki+1(t, 0) = Z b J ∂2_Uk−2 ∂t2 Wi+ n X j=1 Z Γj Wigjk−1 +∂u k−1 i+1 ∂si+1 (t, 0)σi+1− ∂uk−1i ∂si (t, 0)σi + k/2 X l=1 ∂2l+1_uk−2l−1 i+1 ∂t2l_∂s i+1 (t, 0)₋ k/2 X l=1 ∂2l+1_uk−2l−1 i ∂t2l_∂s i (t, 0) + k/2 X l=1 ∂2luk−2li+1 ∂t2l (t, 0) σ2l i+1 (2l)!− k/2 X l=1 ∂2l_uk−2l i ∂t2l (t, 0) σ2l i (2l)! (67)

For each k, we have succeeded to decouple the calculation of uk

i since jump

conditions (67) and average condition (65) written for k = k + 1 are sufficient, when associated to equations (63), to determine uk

i uniquely (lemma3.3).

As the solution of problem (60) with Φ =∂2Uk−2

∂t2 and gi = g k−1

i , Uk is defined

up to an additive constant. To fix this constant we can use again (55-i) (in a symmetric way with respect to i):

                  N X i=1 1 ci Z Γi Uk(t,_·) = N X i=1 uki(t, 0) + N X i=1 ∂uk−1_i ∂si (t, 0)σi + N X i=1 k/2 X l=1 ∂2l+1_uk−2l−1 i ∂t2l_∂s i (t, 0) σ2l+1_i (2l + 1)! + N X i=1 k/2 X l=1 ∂2l_uk−2l i ∂t2l (t, 0) σ2l i (2l)! (68)

Finally, we obtain a problem equivalent to ((31), (32)), ((33), (34))_Jb, (55-i) and (59) by replacing (55-i) by (65), (67) and (68). More precisely

Theorem 3.4. The following two propositions are equivalent (for the clarity of notation, we omit to mention again the functional setting):

(i) (uk1, . . . , ukN, Uk), k > 0 is solution of ((31), (32)), ((33), (34))_Jb, (55-i) and (59) (ii) (uk1, . . . , ukN, Uk), k > 0 is solution of ((31), (32), (65), (67)) and ((33), (34))_Jb, (59), (68) with gk−1 i defined as (64)

Proof. We just proved the implication (i)⇒ (ii). We will prove the implication (ii) ⇒ (i). Let (uk1, . . . , ukN, Uk), k > 0

be a solution of ((31), (32), (65), (67)) and ((33), (34))_Jb, (59), (68) with gk−1

i defined as (64). The only point we

on Γi, we can rely Di,0(Uk(t,·)) and the integration of Uk over Γi. However,

(68) gives some information about the sum of the integrals of Uk _{over the Γ} i’s,

and (62) gives some information about the difference of the integrals of Uk _over

two consecutive Γi’s. By taking some linear combination, we can retrieve the

integral of Uk _{over a given Γ}

i, and by identification, (55-i) is satisfied.

Next, we show that ((31), (32), (65), (67)) and ((33), (34))Jb, (59), (68), with

gik−1 defined as (64), admits a unique solution



(uk1, . . . , ukN, Uk), k > 0

, by induction on k ∈ N.

The case k = 0. With the convention of2.2, we see from ((31), (32), (65), (67)) that u0

i is, as expected, the solution of the 1D wave equation with classical

Kirchhoff conditions at the node (see (6), (7)). Moreover, we see from ((33), (34))Jb, (59) that U

0 _{solves (60) with Φ = 0 and}

gi= 0, which implies that U0(t,·) is constant. Next, (68) gives

U0(t,_{·) =} 1 N N X i=1 1 ci Z Γi U0(t,_{·) =} 1 N N X i=1 u0i(t, 0) on bJ

The general case k > 1. Assume that(ul1, . . . , ulN, Ul), l 6 k− 1

are known, then, according to theorem3.4,

• We compute gk−1

i thanks to (64),

• We determine (uk

i)16i6N as the unique solution of the 1D transmission

problem (63) with the transmission conditions (67,68) (cf. lemma3.3), • We determine Uk _{as the solution, cf. lemma3.2, of the boundary value}

problem ((33,34)_Jb, 59, 68). One must of course check the compatibility relation (61) (written for gi = gk−1i and Φ = ∂

2

Uk−2

∂t2 ), which is a

conse-quence of (65).

Finally, regrouping the above results with theorem3.1and3.4, we have proved the following theorem

Theorem 3.5. There exists a unique family

n

(uk1, . . . , ukn, Uk)∈ C1,2(]0, L1[)× · · · × C1,2(]0, LN[)× C1,2,loc(bJ), k > 0

o

satisfying (31,32,33,34), the matching conditions (55) and the growth condi-tion (54).

Remark 3.3. In fact, by construction, one can check that, for a given k 6 1, the

datas α and δi of problem (63) depends on ∂

2

uk−1

∂t2 and ∂u k−1

∂s . By recurrence,

one can check that if we want to buid uk_{, the real regularity on u}0 _{must be}

u0i ∈ C0(R∗+, Hk+1(]0, +∞[) ∩ C1(R∗+, Hk(]0, +∞[))

and since the problem satisfied by u0_{is a classical problem, one can check that}

we should have fε i ∈ Hk+1(]0, +∞[) giε ∈ H k (]0, +∞[)