PhD student in management, UPPA Eurydice Bled

Dr Igor Alvarez Etxeberria

Dr Jacques Jaussaud

Dr David Carassus

MAPPING THE FACTORS WHICH CONDITIONED THE ADOPTION OF SUSTAINABLE POLICIES BY FRENCH AND SPANISH LOCAL AUTHORITIES

Mapping the factors wich conditioned the adoption of sustainable policies by frech and spanish local autorithies

Bled Eurydice, Igor Alvarez, Jacques Jaussaud, David

INTRODUCTION

The paper focuses on sustainable innovation factors and practices in the French and Spanish local authorities. Concretely this paper tries to provide link from different theories, in order to understand the differences in the implementation in those countries local administrations. We have chosen, in objectivity and neutrality, to also mention the scientific look criticism from some authors on these concepts we will use in the following parts.

Firstly, to introduce the theoretical seeking, we consider that is important to understand the concepts of sustainability, accountability, ethics and democracy. Then we will study analyzes various fields of application in management sciences (social responsibility, social accounting, sustainable business, alternative management) and finally the interests of the local public application (Agenda 21,etc.). The concept of sustainable development emerged in the 80s, at the same time that the concept of Human Development at the United Nations, to counter the principles of the Development of the 50s, very associated with the notion of economic growth, all-powerful belief of the western people first, as developed countries, to find happiness.

The most common definition of sustainable development appears in the Brundtland Report: meeting the needs of the present without compromising the ability of future generations to meet theirs. As the maxim "Think globally, act locally". The concept of sustainable development meets the philosophical, scientific and political thought of those who are interested in the impact of humans on the environment and the beings that compose it, thinking broadly called ecology. This sustainable development is both criticized by [START_REF] Theys | Les contradictions de la ville durable[END_REF] for his Jacobin origin, and [START_REF] Naro | Management, mondialisation, écologie. Regards critiques en sciences de gestion[END_REF] who sees a contradiction between sustainable growth implied by sustainable development and the notion of decay underpinned by ecology.

Local applications emanating from international summits on climate and sustainable development, they are valid both for private organizations that public organizations. Thus, [START_REF] Freeman | Stakeholders, social responsability and performance : empirical evidence and theoretical perspectives[END_REF] develop prospects for Social Accountability of Enterprises, measures, according to them, inseparable from the economic success, while the action plan for local authorities "Agenda 21" was adopted by 178 heads of state at the Earth Summit in Rio. It calls for cross-application according to the 3 pillars: economic, social and environmental. [START_REF] Naro | Management, mondialisation, écologie. Regards critiques en sciences de gestion[END_REF] and Jempa Kanko Mutombo, E. ( 2006) respectively formulate critical of these intentions, with the fear that these devices confirm neoliberalism, head of an unsustainable economic system development, rather than revolutionize it, that seems to be more ideological than scientific.

For communities, which is our research topic, but for firms too, democracy and ethics are inseparable from the concept of sustainable development, as explained Depaquit, S., search better synergy between sustainability requirements and involvement of citizens. Similarly, the fight against corruption is an integral part of the democratic and ethical principles. Although for critical, democracy can be involved in the popularization of the rulers or participates to slow down the decisions on sustainability, as explained [START_REF] Theys | Les contradictions de la ville durable[END_REF] and [START_REF] Van Den Hove | Approches participatives pour la gouvernance en matière de développement durable: une analyse en termes d'effets[END_REF].

Management, which is our science study, can also convert some of its practices, towards sustainability. The social economy advocates a system of redistribution of wealth created, and as alternative management and participatory governance, democratization of decision making. Social accounting proposes to transform into numerical value the human and environmental costs, as well as social performance plans to integrate new criteria other than economic. Again, [START_REF] Naro | Management, mondialisation, écologie. Regards critiques en sciences de gestion[END_REF] denounces "management by the numbers", which for him is more carrying values than objectivity, thus odorless persuasion.

The interests of a local scale application is based largely on proximity between citizens and local governments, on the porosity of mutual information. The community can organize citizen approaches to sustainable development, awareness and lead by example. However, as Jempa Kanko Mutombo, E. ( 2006) accurate, if not to favor the top-down approach. This paper will now discuss the practices of sustainable development in the first part, and the factors of innovation for sustainable development in the second part, and try to put them in touch.

PART 1: LOCAL PRACTICES FOR SUSTAINABLE DEVELOPMENT

The first part explores the theoretical practices for sustainable development of European cities, through various books and exploratory research, and allows to group, as has done some authors, in four categories: environmental, social, economy, governance, as proposed [START_REF] Brodhag | Les enjeux de l'information en langue française sur le développement durable[END_REF]. The variables in local practices of sustainable development is the dependent variable. We have chosen to work on a very local analysis of environmental practices, that is to say at the municipal level, which is easily comparable internationally, since the French and Spanish municipalities practically have the same skills and same sustainable practices. The administrative division and the division of powers between the larger authorities scale differs between France and Spain, and is therefore not comparable in terms of practices or factors. Specifically, many authors, as Charlot-Valdieu, C., Outrequin, P. ( 2009), [START_REF] Theys | L'approche territoriale du développement durable, condition d'une prise en compte de sa dimension sociale[END_REF], [START_REF] Carlier | Les agendas 21, outils de développement durable, Territorial éditions[END_REF], explore a number of sub-variables related to social, first dependent variable, such as the North-South cooperation, access to housing, urban planning and insersion in which the community can play an important role. Others, like [START_REF] Angelini | Les pratiques environnementales au sein du CAC40, un regard critique sur les entreprises visionnaires[END_REF], evoke other variables, which we attribute to the economic field (second dependent variable) as CSR practices, encouraging new forms of savings, like social economy, cooperative societies, fair trade, encouraging integration and promotion of 'environmental work', in the field of energy , insulation , waste recycling, agro-ecology , food short circuits for examples, possible levers for communities.

On the other hand, authors like Lepoutre, J., Heene, A. ( 2006 2009) evoke the various options in participatory governance (third dependent variable), the local democracy, grid integration, transparency, and social accounting, integration of stakeholders and power struggles with the higher authorities. They may take the form of public forums, local referendums , publication of communities budgets on the internet, non -accumulation of for elected officials and publication heritage and taken interest, for examples. Finally, numerous works list possible actions in our fourth and often more important dependent variable: the preservation of the environment, divided into 12 sub-variables that are : the heritage, urban planning, transportation, energy , air, waste, water and sanitation, landscape and biodiversity, noise, public spaces, risks and yards. Certain actions and sub-variables concern several variables, such as support for the social economy, which comes in both economic variable by the nature of its title, that of participatory governance through its new democratic practices , the social for its efforts in professional insersion, and the environment by promoting best practices.

These four variables are deployed in hundreds of concrete actions by the authorities to engage in a more sustainable development, which they call Agenda 21. We will analyze in the next section, the factors that are responsible for these actions, the source of innovations in sustainable development.

PART 2: THE FACTORS OF SUSTAINABLE INNOVATION

The part explores the general innovation factors in institutions, next the factors relating specifically communities, and after the factors relating in particular to sustainable development.

For the first part of the research on institutional innovation, Weber, M. (1904Weber, M. ( -1905)), to Di Maggio, Paul J., Powell, Walter W. (1983), through Biot, V. (2003Biot, V. ( -2004)), authors working on the evolution of institutions offer us the pattern of neo-institutional isomorphism, corresponding to a progressive harmonization of institutions, some to bureaucracy, for others to democracy. It will represent our first explanatory subvariable. Di Maggio, Paul J., Powell, Walter W. (1983) offer us to divide these isomorphisms into three categories. The first is the coercive category, legislated by the top institutions. The second is the mimetic group, inspired by the other institutions of the same level. The third is normative dimension that is inspired by social norms. The is one for [START_REF] Rowan | Institutionalized organizations, formal structure as myth and ceremony[END_REF] that is the environnemental one, that ment the influence of institutional environnement. This is confirmed by more specific research to sustainable development in the communities who see specific influences in these three categories, the largest number related to the coercive isomorphism, originating from Europe, state, local government located at the top, as shown by the research [START_REF] Carlier | Les agendas 21, outils de développement durable, Territorial éditions[END_REF], Albareda, L., Lozano, J. M., Ysa, T. ( 2007), [START_REF] Bento | De la Corporate Social Responsability à la Responsabilité Sociale des Organisations, quelques apports à la réflexion[END_REF]. This is also confirmed that the work of researchers in public managerial innovation in general, as Jempa Kanko Mutombo, E. ( 2006), [START_REF] Brundtland | Factors that determine on influence managerial innovation in public contexts : the case of local performance management[END_REF].

But the authors that focus on sustainable development in particular, like Franklin-Johnson, E., Richomme-Huet, K. ( 2012), [START_REF] Rowan | Institutionalized organizations, formal structure as myth and ceremony[END_REF], [START_REF] Scott | The adolescence of institutional theory[END_REF], [START_REF] Gray | New accounts: toward a reframing of social accounting[END_REF] also foresee other types of factors, not institutional, influencing the practices of local communities in sustainable development. The study of these factors allowed us to group them into a new predictorin the global context. Many sub-variables are described by the authors, about the context of the local authority: economic, social, cultural, geographical, geopolitical, scientific, political, and environmental.

So here we have identified two sub-variables, the isomorphism and the global context, which can be grouped in the first explanatory variable: external factors in the local community. This includes variables that are not related to community activities, internal changes in its own situation, its decision-making choices. On the contrary, the factors of this category cannot be controlled, influenced by the community.

For the second part, many authors describe the influence of other local factors such as the guidance of local officials (other explanatory sub-variable) as Theys, J. ( 2002 This influence associated with political figures is confirmed by researchers from the public innovation such as Nutt, P.C., Backoff, R.N. (1992), Berman, E.M., Wang, X. (2000), [START_REF] Carassus | Les déterminants de l'adoption et mise en oeuvre d'un Management Par la Performance : application aux collectivités locales françaises[END_REF], as well as the influence of technical leadership is confirmed by [START_REF] Fernandez | Managing Successful Organizational Change in Public Sector[END_REF], [START_REF] Walker | Innovation Type and Diffusion: An Empirical Analysis of local Government[END_REF], [START_REF] Rusaw | Changing public organizations: four approaches[END_REF]. These two categories of factors are the explanatory variable called leaderships.

Research on sustainable innovations by [START_REF] Freeman | Stakeholder Theory and "the corporate objective revisited[END_REF], [START_REF] Lepoutre | Investigating the impact of firm size on small business social responsability : a critical review[END_REF] indicates the existence of other local factors that are not related to leadership. The categories in which we have grouped them are material or organizational context, local economic conditions, local social context, local political context. These factors can be summarized in the explanatory sub-variable: local context.

According to this research, two categories of factors, leadership and local contexts, can be grouped into a new explanatory variable: internal factors to the community. This research confirms the existence of two categories of factors: first, the internal factors, here with two sub-categories, which are leadership (technical and political) and local contexts (economic, geographic ...) and then external factors, in our case also with two sub-categories, which are isomorphisms (including the neo-institutional theory) and global contexts (geopolitical, cultural and social ...). Our work allows to relates the sustainable development practices, and its four sub-categories, with sustainable innovation inputs and four sub-categories.

CONCLUSION

In conclusion, the paper was used to study the contours of sustainable development and the internal contradictions in its content and its origins. He also identified the benefits of its application to public and private institutions, and disadvantages, as well as the strengths and brake application to local authorities. We have stressed that, locally, four pillars, including participatory governance, and the dangers and opportunities that it can represent. We also discussed opportunities for implementing sustainable development in management science, and criticism that can be made to these attempts. Thus, based on the observation that there are different interpretations of the relevance of Agenda 21, the paper has to understand that the local authorities, especially the municipalities can initiate sustainable development actions in four categories: the field social, environmental, economic, and governance. They represent the dependent variables.

The paper then served to link certain categories of factors that are behind these innovations in sustainable development in common: the internal factors and external factors. They represent the explanatory variables of our research. This theoretical study of the influence of lasting political factors could be pursued in the context of empirical research to verify, on the one hand the application of the public policy in the communities listed in a process of Agenda 21 and on anorther hand the relevance of the factors identified by this theoretical study. This field of operation could be conducted internationally, in order to see the similarities and differences between the practices and factors for different states. It would also allow us to understand whether there is a link between the four dependent variables and the two variables.

The interest of this research is to offer local authorities, especially the municipalities, the proper tools to implement sustainable development initiatives, since we now know the levers available to them. In order to apply different management strategies for different categories of sustainable development practices. It allowed us to understand that these scientific initiatives are not the only possible approaches to sustainable development, that all levels of our society are involved, and that we can question the overall managerial proposals from the noeliberal system, which management is party involved.

  ),[START_REF] Martinet | Formes de RSE et entreprises sociales, une hybridation des stratégies[END_REF],[START_REF] Tahri | Les effets psychologiques des pratiques socialement responsables sur le comportement des salariés au travail, étude exploratoire[END_REF],Gray, R. (2002), Charlot-Valdieu, C., Outrequin, P. (

  ),[START_REF] Charlot-Valdieu | Ecoquartier : mode d'emploi[END_REF][START_REF] Van Den Hove | Approches participatives pour la gouvernance en matière de développement durable: une analyse en termes d'effets[END_REF], decisions taken by the technical or administrative staff (new explanatory sub-variable) communities, as also explain Naro, G., Palpacuer, F., & Leroy, M. (2010) and Albareda, L., Lozano, J. M., Ysa, T. (2007).


Keywords: Modèle de Biot, ondes poroélastiques, ondes acoustiques, couplage acoustique/poroelastique, solution analytique, technique de Cagniard de Hoop

Solution analytique pour la propagation d'ondes en milieu stratifié hétérogène acoustique/poroélastique. Partie II : en dimension 3

Résumé : Nous nous intéressons à la modélisation de la propagation d'ondes dans les milieux infinis bicouche acoustique/poroélastique. Nous considérons le modèle bi-phasique de Biot dans la couche poroélastique. La première partie est consacrée au calcul de la solution analytique en dimension deux à l'aide de la technique de Cagniard-De Hoop. Dans cette deuxième partie nous considérons le cas de la dimension 3.

Introduction

The computation of analytical solutions for wave propagation problems is of high importance for the validation of numerical computational codes or for a better understanding of the reflexion/transmission properties of the media. Cagniard-de Hoop method [START_REF] Cagniard | Reflection and refraction of progressive seismic waves[END_REF][START_REF] De Hoop | The surface line source problem[END_REF] is a useful tool to obtain such solutions and permits to compute each type of waves (P wave, S wave, head wave...) independently. Although it was originally dedicated to the solution of elastodynamic wave propagation, it can be applied to any transient wave propagation problem in stratified media. However, as far as we know, few works have been dedicated to the application of this method to poroelastic medium. In [START_REF] Ezziani | Modélisation mathématique et numérique de la propagation d'ondes dans les milieux viscoélastiques et poroélastiques[END_REF] the analytical solution of poroelastic wave propagation in an homogeneous 2D medium is provided and in [START_REF] Feng | High-frequency acoustic properties of a fluid/porous solid interface. ii. the 2d reflection green's function[END_REF] the authors compute the analytical expression of the reflected wave at the interface between an acoustic and a poroelastic layer in two dimension but they do not explicit the expression of the transmitted waves.

In order to validate computational codes of wave propagation in poroelastic media, we have implemented the codes Gar6more 2D [START_REF] Diaz | Gar6more 2d[END_REF] and Gar6more 3D [START_REF] Diaz | Gar6more 3d[END_REF] which provide the complete solution (reflected and transmitted waves) of the propagation of wave in stratified 2D or 3D media composed of acoustic/acoustic, acoustic/elastic, acoustic/poroelastic or poroelastic/poroelastic The 2D code and the 3D code are freely downloadable at http://www.spice-rtn.org/library/software/Gar6more2D. and http://www.spice-rtn.org/library/software/Gar6more3D. In previous studies [START_REF] Diaz | Analytical solution for wave propagation in stratified acoustic/porous media. part I: the 2D case[END_REF][START_REF] Diaz | Analytical solution for wave propagation in stratified poroelastic medium. part I: the 2D case[END_REF] we have presented the 2D acoustic/poroelastic and poroelastic/poroelastic cases and we focus here on the 3D acoustic/poroelastic case,the 3D poroelastic/poroelastic case will be the object of forthcoming papers. We first present the model problem we want to solve and derive the Green problem from it (section 1). Then we present the analytical solution of wave propagation in a stratified 3D medium composed of an acoustic and a poroelastic layer (section 2) and we detail the computation of the solution (section 3). Finally we illustrate our results through numerical applications (section 4).

The model problem

We consider an infinite three dimensional medium (Ω = R 3 ) composed of an homogeneous acoustic layer Ω + = R 3 ×] -∞, 0] and an homogeneous poroelastic layer Ω -= R 3 × [0, +∞[ separated by an horizontal interface Γ (see Fig. 1). We first describe the equations in the two layers ( 1.1 and 1.2 ) and the transmission conditions on the interface Γ ( 1.3), then we present the Green problem from which we compute the analytical solution ( 1.4).

Ω + Ω - Acoustic Layer Poroelastic Layer z = 0

The equation of acoustics

In the acoustic layer we consider the second order formulation of the wave equation with a point source in space, a regular source function f in time and zero initial conditions:

                 P + -V + 2 ∆P + = δ x δ y δ z-h f (t), in Ω + ×]0, T ],
Ü + = -1 ρ + ∇P + , in Ω + ×]0, T ],

P + (x, y, 0) = 0, Ṗ + (x, y, 0) = 0, in Ω + U + (x, y, 0) = 0, U+ (x, y, 0) = 0, in Ω +

(1) where P + is the pressure;

U + is the displacement field;

V + is the celerity of the wave;

ρ + is the density of the fluid.

INRIA

Biot's model

In the second layer we consider the second order formulation of the poroelastic equations [START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. I. low-frequency range[END_REF][START_REF] Biot | Theory of propagation of elastic waves in a fluid-saturated porous solid. II. higher frequency range[END_REF][START_REF] Biot | Mechanics of deformation and acoustic propagation in porous media[END_REF]]

                                   ρ -Ü - s + ρ - f Ẅ --∇ ó Σ -= 0, in Ω -×]0, T ], ρ - f Ü - s + ρ - w Ẅ -+ 1 K - Ẇ -+ ∇P -= 0, in Ω -×]0, T ], Σ -= λ -∇ ó U - s I 3 + 2µ -ε(U - s ) -β -P -I 3 , in Ω -×]0, T ], 1 m -P -+ β -∇ ó U - s + ∇ ó W -= 0, in Ω -×]0, T ], U - s (x, 0) = 0, W -(x, 0) = 0, in Ω -, U - s (x, 0) = 0, Ẇ -(x, 0) = 0, in Ω -, (2)
with

(∇ ó Σ -) i = 3 j=1 ∂Σ - ij ∂x j ∀ i = 1, 3, I 3 is the usual identity matrix of M 3 (IR),
and ε(U - s ) is the solid strain tensor defined by:

ε ij (U ) = 1 2
∂U i ∂x j + ∂U j ∂x i .
In (2), the unknowns are:

U - s the displacement field of solid particles;

W -= φ -(U - f -U - s )
, the relative displacement, U - f being the displacement field of fluid particle and φ -the porosity; P -, the fluid pressure; Σ -, the solid stress tensor.

The parameters describing the physical properties of the medium are given by:

ρ -= φ -ρ - f + (1 -φ -)ρ -
s is the overall density of the saturated medium, with ρ - s the density of the solid and ρ - f the density of the fluid;

ρ - w = a -ρ - f /φ -,
where a -the tortuosity of the solid matrix;

K -= κ -/η -, κ -is the permeability of the solid matrix and η is the viscosity of the fluid; m -and β -are positive physical coefficients:

β -= 1 -K - b /K - s and m -= φ -/K - f + (β --φ -)/K - s -1
, where K - s is the bulk modulus of the solid, K - f is the bulk modulus of the fluid and K - b is the frame bulk modulus;

µ -is the frame shear modulus, and λ -= K - b -2µ -/3 is the Lamé constant.

Transmission conditions

Let n be the unitary normal vector of Γ outwardly directed to. The transmission conditions on the interface between the acoustic and porous medium are [START_REF] Carcione | Wave Fields in Real Media : Wave propagation in Anisotropic, Anelastic and Porous Media[END_REF] :

     W -ó n = (U + -U - s ) ó n, P -= P + , Σ -n = -P + n. (3)

The Green problem

We won't compute directly the solution to (1-2-3) but the solution to the following Green problem:

p+ -V + 2 ∆p + = δ x δ y δ z-h δ t , in Ω + ×]0, T ], ( 4a
) ü+ = - 1 ρ + ∇p + , in Ω + ×]0, T ], (4b)
ρ -ü- s + ρ - f ẅ--∇ ó σ -= 0, in Ω -×]0, T ], (5a) ρ
- f ü- s + ρ - w ẅ-+ 1 K - ẇ-+ ∇p -= 0, in Ω -×]0, T ], (5b)
σ -= λ -∇ ó u - s I 3 + 2µ -ε(u - s ) -β -p -I 3 , in Ω -×]0, T ], ( 5c
) 1 m -p -+ β -∇ ó u - s + ∇ ó w -= 0, in Ω -×]0, T ], (5d)
w -ó n = (u + -u - s ) ó n, on Γ, (6a)
p -= p + , on Γ, (6b) σ -n = -p + n, on Γ. ( 6c
)
The solution to (1-2-3) is then computed from the solution of the Green Problem thanks to a convolution by the source function. For instance we have :

P + (x, y, t) = p + (x, y, .) * f (.) = t 0 p + (x, y, τ )f (t -τ ) dτ

INRIA

(we have similar relations for the other unknowns). We also suppose that the poroelastic medium is non dissipative, i.e the viscosity η -= 0. Using the equations (5c, 5d) we can eliminate σ -and p -in (5) and we obtain the the equivalent system:

   ρ -ü- s + ρ - f ẅ--α -∇(∇ ó u - s ) + µ -∇ × (∇ × u - s ) -m -β -∇(∇ ó w -) = 0, z < 0 ρ - f ü- s + ρ - w ẅ--m -β -∇(∇ ó u - s ) -m -∇(∇ ó w -) = 0, z < 0 (7) with α -= λ -+ 2µ -+ m -β -2 .
And using the equation (4b) the transmission conditions (6) on z = 0 are rewritten as:

ü- s z + ẅ- z = - 1 ρ + ∂ z p + , (8a)
-m -β -∇ ó u - s -m -∇ ó w -= p + , (8b)
∂ z u - sx + ∂ x u - sz = 0, (8c)
∂ z u - sy + ∂ y u - sz = 0, (8d)
(λ -+ m -β -2 )∇ ó u - s + 2µ -∂ z u - sz + m -β -∇ ó w = -p + . ( 8e
)
We split the displacement fields u - s and u - f into irrotationnal and isovolumic fields (P-wave and S-wave):

u - s = ∇Θ - u + ∇ × Ψ - u ; w -= ∇Θ - w + ∇ × Ψ - w . (9)
The vectors Ψ u and Ψ w are not uniquely defined since:

∇ × (Ψ ℓ + ∇C) = ∇ × Ψ ℓ , ∀ ℓ ∈ {u, w}
for all scalar field C. To define a unique Ψ ℓ we impose the gauge condition:

∇ ó Ψ ℓ = 0
The vectorial space of Ψ ± ℓ verifying this last condition is written as:

Ψ ℓ =    ∂ y -∂ x 0    Ψ ℓ,1 +    ∂ 2 xz ∂ 2 yz -∂ 2 xx -∂ 2 yy    Ψ ℓ,2 ,
where Ψ ℓ,1 and Ψ ℓ,2 are two scalar fields. The displacement fields u - s and w -are written in the form:

u - s = ∇Θ - u +    ∂ 2 xz ∂ 2 yz -∂ 2 xx -∂ 2 yy    Ψ - u,1 -    ∂ y -∂ x 0    ∆Ψ - u,2 w -= ∇Θ - w +    ∂ 2 xz ∂ 2 yz -∂ 2 xx -∂ 2 yy    Ψ - w,1 -    ∂ y -∂ x 0    ∆Ψ - w,2 (10)
RR n 6595

We can then rewrite system [START_REF] Diaz | Approches analytiques et numériques de problèmes de transmission en propagation d'ondes en régime transitoire. Application au couplage fluide-structure et aux méthodes de couches parfaitement adaptées[END_REF] in the following form:

                     A -Θ--B -∆Θ -= 0, z < 0 Ψ- u,1 -V - S 2 ∆Ψ - u,1 = 0, z < 0 Ψ- u,2 -V - S 2 ∆Ψ - u,2 = 0, z < 0 Ψ- w = - ρ - f ρ - w Ψ- u , z < 0 ( 11
)
where Θ -= (Θ - u , Θ - w ) t , A -and B -are 2 × 2 symmetric matrices:

A -= ρ -ρ - f ρ - f ρ - w ; B -= λ -+ 2µ -+ m -(β -) 2 m -β - m -β - m - , and
V - S = µρ - w ρ -ρ - w -ρ - f 2
is the S-wave velocity.

We multiply the first equation of the system (11) by the inverse of A. The matrix A -1 B is diagonalizable:

A -1 B = PDP -1
, where P is the change-of-coordinate matrix, D = diag(V -

P f 2 , V - P s
2 ) is the diagonal matrix similar to A -1 B, V - P f and V - P s are respectively the fast P-wave velocity and the slow P-wave velocity (V P s < V P f ).

Using the change of variables

Φ -= (Φ - P f , Φ - P s ) t = P -1 Θ -, (12)
we obtain the uncoupled system on fast P-waves, slow P-waves and S-waves:

             Φ--D∆Φ -= 0, z < 0 Ψ- u,i -V - S 2 ∆Ψ - u,i = 0, i = 1, 2, z < 0 Ψ - w = - ρ - f ρ - w Ψ - u . z < 0 ( 13
)
Using the transmission conditions (8c)-(8d), we obtain:

2∂ 2 xz Θ - u + ∂ x (∂ 2 zz -∆ ⊥ )Ψ - u,1 -∂ 2 yz ∆Ψ - u,2 = 0, on Γ, ( 14a
) 2∂ 2 yz Θ - u + ∂ y (∂ 2 zz -∆ ⊥ )Ψ - u,1 + ∂ 2 xz ∆Ψ - u,2 = 0, on Γ, ( 14b
) with ∆ ⊥ = ∂ 2 xx + ∂ 2 yy .
Applying the derivative ∂ y to the equation (14a), ∂ x to the equation (14b) and subtracting the first obtained equation from the second one, we get:

(∂ z ∆ ⊥ )∆Ψ - u,2 = 0, on Γ, (15)
INRIA moreover, using that Ψ - u,2 satisfies the wave equation:

Ψ- u,2 -V - S 2 ∆Ψ - u,2 = 0, z < 0
and that u - s and w -satisfy, at t = 0, u - s = us = w -= ẇ-= 0, we obtain:

Ψ - u,2 = 0, z ≤ 0,
and from ( 14) we deduce the transmission condition equivalent to (8c) and(8d):

2∂ 2 z Θ - u + (∂ 2 zz -∆ ⊥ )Ψ - u,1 = 0, on Γ. ( 16
)
Finally, we obtain the Green problem equivalent to [START_REF] Cagniard | Reflection and refraction of progressive seismic waves[END_REF][START_REF] Carcione | Wave Fields in Real Media : Wave propagation in Anisotropic, Anelastic and Porous Media[END_REF][START_REF] De Hoop | The surface line source problem[END_REF]:

         p+ -V + 2 ∆p + = δ x δ y δ z-h δ t , z > 0 Φ- i -V - i 2 ∆Φ - i = 0, i ∈ {P f, P s, S} z < 0 B(p + , Φ - P f , Φ - P s , Φ - S ) = 0, z = 0 (17)
where we have set Φ - S = Ψ - u,1 in order to have similar notations for the P f , P s and S waves. The operator B represents the transmission conditions on Γ:

B     p + Φ - P f Φ - P s Φ - S     =              1 ρ + ∂ z (P 11 + P 21 ) ∂ 3 ztt (P 12 + P 22 ) ∂ 3 ztt ( ρ - f ρ - w -1) ∂ 2 tt ∆ ⊥ 1 m -(β -P 11 + P 21 ) V - P f 2 ∂ 2 tt m -(β -P 12 + P 22 ) V - P s 2 ∂ 2 tt 0 0 2P 11 ∂ z 2P 12 ∂ z ∂ 2 zz -∆ ⊥ 1 B 42 B 43 -2µ -∂ z ∆ ⊥                         p + Φ - P f Φ - P s Φ - S           
where P ij , i, j = 1, 2 are the components of the change-of-coordinates matrix P, B 42 and B 43 are given by:

B 42 = (λ -+ m -β -2 )P 11 + m -β -P 21 V - P f 2 ∂ 2 tt + 2µ -P 11 ∂ 2 zz , B 43 = (λ -+ m -β -2 )P 12 + m -β -P 22 V - P s 2 ∂ 2 tt + 2µ -P 12 ∂ 2 zz .
To obtain this operator we have used the transmission conditions (8a,8b,16,8e), the change of variables ( 9) and the uncoupled system [START_REF] Feng | High-frequency acoustic properties of a fluid/porous solid interface. ii. the 2d reflection green's function[END_REF].

Moreover, we can determine the solid displacement u - s by using the change of variables (9) and the fluid displacement u + by using (4b).

Expression of the analytical solution

Since the problem is invariant by a rotation around the z-axis, we will only consider the case y = 0 and x > 0, so that the y-component of all the displacements are zero. The solution for y = 0 or x ≤ 0 is deduced from the solution for y = 0 by the relations p(x, y, z, t) = p( x 2 + y 2 , 0, z, t) (18)

u s x (x, y, z, t) = x x 2 + y 2 u s x ( x 2 + y 2 , 0, z, t) (19)
u s y (x, y, z, t) = y x 2 + y 2 u s x ( x 2 + y 2 , 0, z, t) (20)
u s z (x, y, z, t) = u s z ( x 2 + y 2 , 0, z, t) (21)
To state our results, we need the following notations and definitions:

1. Definition of the complex square root. For q x ∈ C\IR -, we use the following definition of the square root g(q x ) = q 1/2

x :

g(q x ) 2 = q x and ℜe[g(q x )] > 0.

The branch cut of g(q x ) in the complex plane will thus be the half-line defined by {q x ∈ IR -} (see Fig. 2). In the following, we'll use the abuse of notation g(q x ) = i √ -q x for q x ∈ IR -.

ℑm(q x ) ℜe(q x ) -π +π Figure 2: Definition of the function x → (x) 1/2
2. Definition of the fictitious velocities For a given q ∈ IR, we define the fictitious velocities V + (q) and V - i (q) for i ∈ {P f, P s, S} by

V + := V + (q) = V + 1 1 + V + 2 q 2 and V - i := V - i (q) = V - i 1 1 + V - i 2 q 2 .

INRIA

These fictitious velocities will be helpful to turn the 3D-problem into the sum of 2Dproblems indexed by the variable q. Note that V + (0) and V - i (0) correspond to the real velocities V + and V - i . 3. Definition of the functions κ + and κ - i . For i ∈ {P f, P s, S} and (q x , q y ) ∈ C × IR, we define the functions

κ + := κ + (q x , q y ) = 1 V + 2 + q 2 x + q 2 y 1/2 = 1 V + 2 (q y ) + q 2 x 1/2
. and

κ - i := κ - i (q x , q y ) = 1 V - i 2 + q 2 x + q 2 y 1/2 = 1 V - i 2 (q y ) + q 2 x 1/2 .
4. Definition of the reflection and transmission coefficients. For a given (q x , q y ) ∈ C × IR, we denote by R(q x , q y ), T P f (q x , q y ), T P s (q x , q y ) and T S (q x , q y ) the solution of the linear system

A(q x , q y )          R(q x , q y ) T P f (q x , q y ) T P s (q x , q y ) T S (q x , q y )          = - 1 2κ + (q x , q y )V + 2           κ + (q x , q y ) ρ + 1 0 1           , (22)
where the matrix A(q x , q y ) is defined by:

A(q x , q y ) =             - κ + (q x , q y ) ρ + (P 11 + P 21 )κ - P f (q x , q y ) (P 12 + P 22 )κ - P s (q x , q y ) 1 - ρ - f ρ - w (q 2 x + q 2 y ) 1 m - V - P f 2 β -P 11 + P 21 m - V - P s 2 β -P 12 + P 22 0 0 2P 11 κ - P f (q x , q y ) 2P 12 κ - P s (q x , q y ) κ - S 2 (q x , q y ) + q 2 x + q 2 y 1 A 4,2 (q x , q y ) A 4,3 (q x , q y ) 2µ -(q 2 x + q 2 y )κ - S (q x , q y )             , with A 4,2 (q x , q y ) = λ -+ m -β -2 P 11 + m -β -P 21 V - P f 2 + 2µ -κ - P f 2 (q x , q y )P 11 , A 4,3 (q x , q y ) = λ -+ m -β -2 P 12 + m -β -P 22 V - P s 2 + 2µ -κ - P s 2 (q x , q y )P 12 .
We also denote by V max the greatest velocity in the two media:

V max = max(V + , V - P f , V - P s , V - S ).
We can now present the expression of the solution to the Green Problem :

RR n 6595

Theorem 2.1. The pressure and the displacement in the top medium are given by

p + (x, 0, z, t) = p + inc (x, z, t)+ dξ + ref dt (x, z, t) and u + (x, 0, z, t) = u + inc (x, z, τ ) dτ +u + ref (x, y, τ ) dτ,
and the displacement in the bottom medium is given by

u - s (x, 0, z, t) = u - P f (x, z, t) + u - P s (x, z, t) + u - S (x, z, t)
where p + inc and u + inc are respectively the pressure and the displacement of the incident wave and satisfy :

                     p + inc (x, z, t) := δ(t -t 0 ) 4πV + 2 r u inc,x (x, z, t) := xtH(t -t 0 ) 4πV + 2 r 3 ρ + u inc,z (x, z, t) := (z -h)tH(t -t 0 ) 4πV + 2 r 3 ρ + ,
where δ and H respectively denote the usual Dirac and Heaviside distributions. Moreover we set r = (x 2 + (z -h) 2 ) 1/2 and t 0 = r/V + denotes the time arrival of the incident wave at point (x, 0, z).

ξ + ref and u + ref are respectively the primitive of the pressure with respect to the time and the displacement of the reflected wave and satisfy :

                           ξ + ref (x, z, t) = - q 1 (t) 0 ℑm κ + (υ(t, q))R(υ(t, q)) π 2 r q 2 + q 2 0 (t) dq, u + ref,x (x, z, t) = - q 1 (t) 0 ℑm i υ(t, q)κ + (υ(t, q))R(υ(t, q)) π 2 rρ + q 2 + q 2 0 (t) dq, u + ref,z (x, z, t) = - q 1 (t) 0 ℑm κ + 2 (υ(t, q))R(υ(t, q)) π 2 rρ + q 2 + q 2 0 (t) dq, INRIA if t h 1 < t ≤ t 0 and x r > V + V max ,                                                                      ξ + ref (x, z, t) = - q 1 (t) q 0 (t)
ℑm κ + (υ(t, q))R(υ(t, q)) π 2 r q 2 -q 2 0 (t)

dq + q 0 (t) 0 ℜe κ + (γ(t, q))R(γ(t, q)) π 2 r q 2 0 (t) -q 2 dq, u + ref,x (x, z, t) = - q 1 (t) q 0 (t)
ℑm (i υ(t, q)κ + (υ(t, q))R(υ(t, q))

π 2 rρ + q 2 -q 2 0 (t) dq + q 0 (t) 0 ℜe i γ(t, q)κ + (γ(t, q))R(γ(t, q)) π 2 rρ + q 2 0 (t) -q 2 dq, u + ref,y (x, z, t) = - q 1 (t) q 0 (t) ℑm κ + 2 (υ(t, q))R(υ(t, q)) π 2 rρ + q 2 -q 2 0 (t) dq + q 0 (t) 0 ℜe κ + 2 (γ(t))R(γ(t)) π 2 rρ + q 2 0 (t) -q 2 dq, if t 0 < t ≤ t h 2 and x r > V + V max ,                            ξ + ref (x, z, t) = q 0 (t) 0 ℜe κ + (γ(t, q))R(γ(t, q)) π 2 r q 2 0 (t) -q 2 dq, u + ref,x (x, z, t) = q 0 (t) 0 ℜe i γ(t, q)κ + (γ(t, q))R(γ(t, q)) 2π 2 rρ + q 2 0 (t) -q 2 dq, u + ref,y (x, z, t) = q 0 (t) 0 ℜe κ + 2 (γ(t))R(γ(t)) π 2 rρ + q 2 0 (t) -q 2 dq, if t h 2 < t and x r > V + V max or if t 0 < t and x r ≤ V + V max and ξ ref (x, y, t) = 0 and u ref (x, y, t) = 0 else .
We set here r = (x 2 +(z +h) 2 ) 1/2 and t 0 = r/V + denotes the arrival time of the reflected volume wave at point (x, 0, z),

t h 1 = (z + h) 1 V + 2 - 1 V 2 max + |x| V max (23)
denotes the arrival time of the reflected head-wave at point (x, 0, z) and

t h 2 = r z + h 1 V + 2 - 1 V 2 max ( 24
)
RR n 6595 denotes the time after which there is no longer head wave at point (x, 0, z), (contrary to the 2D case, this time does not coincide with the arrival time of the volume wave). We also define the functions γ, υ, q 0 and q 1 by

γ : {t ∈ IR | t > t 0 } × IR → C := γ(t, q y ) = i xt r 2 + z + h r t 2 r 2 - 1 V + 2 (q y ) υ : {t ∈ IR | t h 1 < t < t h 2 }×IR → C := υ(t, q y ) = -i z + h r - 1 V + 2 (q y ) - t 2 r 2 + x r 2 t , q 0 : IR → IR := q 0 (t) = t 2 r 2 - 1 V + 2
and

q 1 : IR → IR := q 1 (t) = 1 x 2 t -(z + h) 1 V + 2 - 1 V 2 max 2 - 1 V 2 max .
Remark 2.1. For the practical computation of the pressure, we won't have to explicitly compute the derivative of the function ξ + ref (which would be rather tedious), since

p + ref * f = ∂ t ξ + ref * f = ξ + ref * f ′ .
Therefore, we'll only have to compute the derivative of the source function f . u - P f (x, z, t) is the displacement of the transmitted P f wave and satisfies:

           u - P f,x (x, z, t) = - P 11 π 2 q 1 (t) 0
ℜe i υ(t, q)T P f (υ(t, q)) ∂υ ∂t (t, q) dq,

u - P f,z (x, z, t) = P 11 π 2 q 1 (t) 0 ℜe κ - P f (υ(t, q))T P f (υ(t, q)) ∂υ ∂t (t, q) dq, if t h 1 < t ≤ t 0 and |ℑm [γ(t 0 , 0)]| < 1 V max ,                                      u - P f,x (x, z, t) = - P 11 π 2 q 0 (t) 0 ℜe i γ(t, q)T P f (γ(t, q)) ∂γ ∂t (t, q) dq - P 11 π 2 q 1 (t) q 0 (t)
ℜe i υ(t, q)T P f (υ(t, q)) ∂υ ∂t (t, q) dq,

u - P f,z (x, z, t) = P 11 π 2 q 0 (t) 0 ℜe κ - P f (γ(t, q))T P f (γ(t, q)) ∂γ ∂t (t, q) dq + P 11 π 2 q 1 (t) q 0 (t)
ℜe κ - P f (υ(t, q))T P f (υ(t, q)) ∂υ ∂t (t, q) dq,

INRIA if t 0 < t ≤ t h 2 and |ℑm [γ(t 0 , 0)]| < 1 V max ,            u - P f,x (x, z, t) = - P 11 π 2 q 0 (t) 0
ℜe i γ(t, q)T P f (γ(t, q)) ∂γ ∂t (t, q) dq,

u - P f,z (x, z, t) = P 11 π 2 q 0 (t) 0 ℜe κ - P f (γ(t, q))T P f (γ(t, q)) ∂γ ∂t (t, q) dq, if t h 2 < t and |ℑm [γ(t 0 , 0)]| < 1 V max or if t 0 < t and |ℑm [γ(t 0 , 0)]| ≥ 1 V max
and u - P f (x, z, t) = 0 else. t 0 denotes here the arrival time of the P f volume wave at point (x, 0, z) (we recall in appendix the computation of t 0 ),

t h 1 = h 1 V + 2 - 1 V 2 max -z 1
V - P f 2 - 1 V 2 max + |x| V max (25)
denotes the arrival time of the P f head wave at point (x, 0, z),

t h 2 = h 2 + z 2 -hz c 2 c 1 + c 1 c 2 + x 2 h c 1 -z c 2 (26)
denotes the time after which there is no longer head wave at point (x, 0, z), where

c 1 = 1 V + 2 - 1 V 2 max and c 2 = 1 V 2 P f - 1 V 2 max .
The function q 0 : [t 0 ; +∞] → IR + is the reciprocal function of t0 : IR + →: [t 0 , +∞], where t 0 (q) is the arrival time at point (x, 0, z) of the fictitious P f volume wave, propagating at a velocity V + (q) in the top layer and at velocity V - P f (q) in the bottom layer (we recall in appendix the computation of t 0 (q)).

The function q 1 : [t 1 ; t 0 ] → IR + is defined by

q 1 (t) = 1 x 2   t + z 1 V - P f 2 - 1 V 2 max -h 1 V + 2 - 1 V 2 max   2 - 1 V 2 max .
The function γ : {(t, q) ∈ IR + × IR + | t > t0 (q)} → C is implicitly defined as the only root of the function

F(γ, q, t) = -z 1
V - P f 2 (q) + γ 2 1/2 + h 1 V + 2 (q) + γ 2 1/2 + iγx -t
whose real part is positive. The function υ : E 1 ∪ E 2 → C is implicitly defined as the only root of the function

F(υ, q, t) = -z 1
V - P f 2 (q) + υ 2 1/2 + h 1 V + 2 (q) + υ 2 1/2 + iυx -t RR n 6595
such that ℑm [∂ t υ(t, q)] < 0, with E 1 = (t, q) ∈ IR + × IR + | t h 1 < t < t 0 and 0 < q < q 0 (t) and E 2 = (t, q) ∈ IR + × IR + | t 0 < t < t h 1 and q 0 (t) < q < q 1 (t) . u - P s (x, z, t) is the displacement of the transmitted P s wave and satisfies:

           u - P s,x (x, z, t) = - P 12 π 2 q 1 (t) 0
ℜe i υ(t, q)T P s (υ(t, q)) dυ dt (t, q) dq,

u - P s,z (x, z, t) = P 12 π 2 q 1 (t) 0
ℜe κ - P s (υ(t), q)T P s (υ(t, q)) dυ dt (t, q) dq,

if t h 1 < t ≤ t 0 and |ℑm [γ(t 0 , 0)]| < 1 V max ,                                      u - P s,x (x, z, t) = - P 12 π 2 q 0 (t) 0
ℜe i γ(t, q)T P f (γ(t, q)) ∂γ ∂t (t, q) dq,

- P 12 π 2 q 1 (t) q 0 (t)
ℜe i υ(t, q)T P s (υ(t, q)) dυ dt (t, q) dq u - P s,z (x, z, t) = P 12 π 2 q 0 (t) 0 ℜe κ - P s (γ(t, q))T P s (γ(t, q)) dγ dt (t, q) dq

+ P 12 π 2 q 1 (t) q 0 (t)
ℜe κ - P s (υ(t), q)T P s (υ(t, q)) dυ dt (t, q) dq,

if t 0 < t ≤ t h 2 and |ℑm [γ(t 0 , 0)]| < 1 V max ,            u - P s,x (x, z, t) = - P 12 π 2 q 0 (t) 0
ℜe i γ(t, q)T P s (γ(t, q)) dγ dt (t, q) dq,

u - P s,z (x, z, t) = P 12 π 2 q 0 (t) 0 ℜe κ - P s (γ(t, q))T P s (γ(t, q)) dγ dt (t, q) dq, if t h 2 < t and |ℑm [γ(t 0 , 0)]| < 1 V max or if t 0 < t and |ℑm [γ(t 0 , 0)]| ≥ 1 V max
and u - P s (x, z, t) = 0 else. t 0 denotes here the arrival time of the P s volume wave at point (x, 0, z),

t h 1 = h 1 V + 2 - 1 V 2 max -z 1
V - P s 2 - 1 V 2 max + |x| V max (27)
denotes the arrival time of the P s head wave at point (x, 0, z) and

t h 2 = h 2 + z 2 -hz c 2 c 1 + c 1 c 2 + x 2 h c 1 -z c 2 (28)
INRIA denotes the time after which there is no longer head wave at point (x, 0, z), where

c 1 = 1 V + 2 - 1 V 2 max and c 2 = 1 V 2 P s - 1 V 2 max .
The function q 0 : [t 0 ; +∞] → IR + is the reciprocal function of t0 : IR + →: [t 0 , +∞], where t 0 (q) is the arrival time at point (x, 0, z) of the fictitious P s volume wave, propagating at a velocity V + (q) in the top layer and at velocity V - P s (q) in the bottom layer. The function q 1 : [t 1 ; t 0 ] → IR + is defined by

q 1 (t) = 1 x 2 t + z 1 V - P s 2 - 1 V 2 max -h 1 V + 2 - 1 V 2 max 2 - 1 V 2 max .
The function γ : {(t, q) ∈ IR + × IR + | t > t0 (q)} → C is implicitly defined as the only root of the function

F(γ, q, t) = -z 1
V - P s 2 (q) + γ 2 1/2 + h 1 V + 2 (q) + γ 2 1/2 + iγx -t whose real part is positive. The function υ : E 1 ∪ E 2 →
C is implicitly defined as the only root of the function

F(υ, q, t) = -z 1
V - P s 2 (q) + υ 2 1/2 + h 1 V + 2 (q) + υ 2 1/2 + iυx -t such that ℑm [∂ t υ(t, q)] < 0, with E 1 = (t, q) ∈ IR + × IR + | t h 1 < t < t 0 and 0 < q < q 0 (t)
and E 2 = (t, q) ∈ IR + × IR + | t 0 < t < t h 1 and q 0 (t) < q < q 1 (t) .

u - S (x, z, t) is the displacement of the transmitted S wave and satisfies:

           u - S,x (x, z, t) = - 1 π 2 q 1 (t) 0 ℜe i υ(t, q)κ - S (υ(t, q))T S (υ(t, q)) dυ dt (t, q) dq, u - S,z (x, z, t) = 1 π 2 q 1 (t) 0 ℜe (υ 2 (t, q) + q 2 )T S (υ(t, q)) dυ dt (t, q) dq, if t h 1 < t ≤ t 0 and |ℑm [γ(t 0 , 0)]| < 1 V max ,
The function γ : {(t, q) ∈ IR + × IR + | t > t0 (q)} → C is implicitly defined as the only root of the function

F(γ, q, t) = -z 1 V - S 2 (q) + γ 2 1/2 + h 1 V + 2 (q) + γ 2 1/2 + iγx -t whose real part is positive. The function υ : E 1 ∪ E 2 →
C is implicitly defined as the only root of the function

F(υ, q, t) = -z 1 V - S 2 (q) + υ 2 1/2 + h 1 V + 2 (q) + υ 2 1/2 + iυx -t such that ℑm [∂ t υ(t, q)] < 0, with E 1 = (t, q) ∈ IR + × IR + | t h 1 < t < t 0 and 0 < q < q 0 (t)
and E 2 = (t, q) ∈ IR + × IR + | t 0 < t < t h 1 and q 0 (t) < q < q 1 (t) .

Proof of the theorem

To prove the theorem, we use the Cagniard-de Hoop method (see [START_REF] Cagniard | Reflection and refraction of progressive seismic waves[END_REF][START_REF] De Hoop | The surface line source problem[END_REF][START_REF] Van Der Hijden | Propagation of transient elastic waves in stratified anisotropic media[END_REF][START_REF] Pao | The generalized ray theory and transient response of layered elastic solids[END_REF][START_REF]Etude dans le domaine temporel de la propagation d'ondes elastiques en milieux stratifiés ; modèlisation de la reponse du thorax a un impacts[END_REF]), which consists of three steps:

1. We apply a Laplace transform in time, ũ(x, y, z, s) = +∞ 0 u(x, y, z, t) e -st dt, and a Fourier transform in the x and y variables,

û(k x , k y , z, s) = +∞ -∞ +∞ -∞
ũ(x, y, s) e i (kxx+kyy) dx dy to (17) in order to obtain an ordinary differential system whose solution Ĝ(k x , k y , z, s) can be explicitly computed ( 3.1);

2. we apply an inverse Fourier transform in the x and y variables to G (we recall that we only need the solution at y = 0:

G(x, 0, z, s) = 1 4π 2 +∞ -∞ +∞ -∞ Ĝ(k x , k y , z, s) e -i kxx dk x dk y .
And, using tools of complex analysis, we turn the inverse Fourier transform in the x variable into the Laplace transform of some function

H(x, k y , z, t) ( 3.2): G(x, 0, z, s) = 1 4π 2 +∞ -∞ +∞ 0 H(x, k y , z, t) e -st dt dk y ; (31)
3. the last step of the method consists in inverting the order of integration in (31) to obtain

G(x, 0, z, s) = 1 4π 2 +∞ 0 +q(t) -q(t)
H(x, k y , z, t) dk y e -st dt.

Then, using the injectivity of the Laplace transform, we identify G(x, 0, z, t) to

1 4π 2 +q(t) -q(t)
H(x, k y , z, t) dk y (see 3.3).

The solution in the Laplace-Fourier plane

Let us first apply a Laplace transform in time and a Fourier transform in the x and y variables to (17) to obtain

                   s 2 V + 2 + k 2 x + k 2 y p+ - ∂ 2 p+ ∂z 2 = δ(z -h) V + 2 , y > 0, s 2 V - i 2 + k 2 x + k 2 y Φ- - ∂ 2 Φ- i ∂z 2 = 0, i ∈ {P f, P s, S} y < 0, B(p + , Φ- P f , Φ- P f , Φ- S ) = 0 y = 0, (32)
where B is the Laplace-Fourier transform of the operator B.

From the two first equations of (32), we deduce that the solution (p + , ( Φ-

i ) i∈{P f,P s,S} ) is such that                    p+ = p+ inc + p+ ref , p+ inc = 1 sV + 2 κ + kx s , ky s e -s|z-h|κ + kx s , ky s , p+ ref = R(k x , k y , s)e -szκ + kx s , ky s , Φ- i = T i (k x , k y , s)e -s zκ - i kx s ky s , i ∈ {P f, P s, S}, (33)
where the coefficients R and T i are computed by using the last equation of (32):

B(p + ref , Φ- P f , Φ- P s , Φ- S ) = -B(p + inc , 0, 0, 0),
or, from( 33):

B [R(k x , k y , s), T P f (k x , k y , s), T P s (k x , k y , s), T S (k x , k y , s)] = - B   e -shκ + kx s , ky s sκ + kx s , ky s
, 0, 0, 0   .

INRIA

After some calculations that we don't detail here, we obtain that R(k x , k y , s), T P f (k x , k y , s), T P s (k x , k y , s), and T S (k x , k y , s)) are solution to

A k x s , k y s          R(k x , k y , s) s 2 T P f (k x , k y , s) s 2 T P s (k x , k y , s) s 3 T S (k x , k y , s)          = - e -shκ + s , ky s 2sκ + kx s , ky s V + 2            κ + kx s , ky s ρ + 1 0 1            . ( 34
)
From the definition of the reflection and transmission coefficients we deduce that

         R(k x , k y , s) s 2 T P f (k x , k y , s) s 2 T P s (k x , k y , s) s 3 T S (k x , k y , s)          = 1 s            R kx
           e -shκ + ( kx s ) . (35)
Finally, we obtain:

                             p+ = p+ inc +
                                        û+ = û+ inc + û+ ref , û+ inc,x = i k x ρ + s 2 p+ inc , û+ inc,z = sign(h -z) κ + kx s , ky s ρ + s p+ inc , û+ ref,x = i k x ρ + s 2 p+ ref , û+ ref,z = κ + kx s , ky s ρ + s p+ ref , û- sx = -i k x P 11 ΦP f -i k x P 12 Φ- P s -isk x κ - S k x s Φ - S û- sz = sκ - P f k x s , k y s P 11 Φ- P f + sκ - P s k x s , k y s P 12 Φ- P s + (k 2 x + k 2 y )Φ - S (37)
In the following we only detail the computation of ûsx,P s = -i k x P 12 Φ-P s , since the computation of the other terms is very similar.

The Laplace transform of the solution

We apply an inverse Fourier transform in the x and y variable to ûsx,P s and we set k x = q x s and k y = q y s to obtain (we recall that we consider y = 0) ũsx,P s (x, 0, z, s) = -

+∞ -∞ +∞ -∞
i q x P 12 4π 2 T P s (q x , q y )e -s(-zκ - P s (qx,qy)+hκ + (qx,qy))+i qxx) dq x dq y

= - P 12 4π 2 +∞ -∞ +∞ -∞
Ξ(q x , q y ) dq x dq y , with Ξ(q x , q y ) = i q x T s (q x , q y )e -s(-zκ - P s (qx,qy)+hκ + (qx,qy))+i qxx) .

Let us now focus on the integral over q x for a fixed q y +∞ -∞ Ξ(q x , q y ) dq x = +∞ -∞

i q x T P s (q x , q y )e -s(-zκ - P s (qx,qy)+hκ + (qx,qy))+i qxx) dq x (38)

This integral is very similar to the one we have obtained in 2D [START_REF] Diaz | Analytical solution for wave propagation in stratified acoustic/porous media. part I: the 2D case[END_REF], therefore, using the same method, we have:

if |γ(q y , t 0 )| > 1 V max (q y ) +∞ -∞ Ξ(q x , q y ) dq x = 2 +∞ t0 (qy)
ℜe i γ(q y , t)T P s (q y , γ(q y , t)) ∂γ(q y , t) ∂t e -st dt if |γ(q y , t0 (q y ))| < 1 V max (q y ) +∞ -∞ Ξ(q x , q y ) dq x = 2 t0 (qy) th (qy) ℜe i υ(q y , t)T P s (q y , υ(q y , t)) ∂υ(q y , t) ∂t e -st dt

+ 2 +∞ t0 (qy)
ℜe i γ(q y , t)T P s (q y , γ(q y , t)) ∂γ(q y , t) ∂t e -st dt where V max is greatest fictitious velocity defined by:

V max := V max (q) = V max 1 1 + V max 2 q 2 ,
t0 is the fictitious arrival time of the P s volume wave we have defined in the theorem, th (q y ) the fictitious arrival time of the P s head wave defined by

th := th (q) = h 1 V + 2 (q) - 1 V 2 max (q) -z 1
V - P s 2 (q) - 1 V 2 max (q) + |x| V max (q) .
Let us recall [START_REF] Diaz | Approches analytiques et numériques de problèmes de transmission en propagation d'ondes en régime transitoire. Application au couplage fluide-structure et aux méthodes de couches parfaitement adaptées[END_REF] that the condition |γ(q y , t0 (q y ))| < 1 V max (q y ) is equivalent to

|γ(0, t 0 )| < 1 V max and |q y | ≤ q max , INRIA with q max = r 2   h 1 V + 2 -1 Vmax 2 - z 1 V - P s 2 -1 Vmax 2   - 1 V max 2 .
Moreover, th is bijective from [0 ; q max ] to [t 0 ; th (q max )] and we denote its inverse by q h :

q h (t) = 1 x 2 t + z 1 V - P s 2 - 1 V 2 max -h 1 V + 2 - 1 V 2 max 2 - 1 V 2 max .
Let us also recall that for q y = q max , the arrival times of the fictitious head and volume waves are the same: th (q max ) = t0 (q max ). As an illustration, we represent the functions t0 (the red dotted line) and th (the blue solid line) in Fig. 3. We then deduce that

t q y q max t 0 t h 1 t0 (q y ) th (q y ) t0 (q max ) = th (q max ) = t h 2
if |γ(0, t 0 )| ≥ 1 Vmax IR 2 Ξ(q x , q y ) dq x dq y = 2 IR +∞ t0 (qy)
ℜe i γ(q y , t)T P s (q y , γ(q y , t)) ∂γ(q y , t) ∂t e -st dtdq y ;

if |γ(0, t 0 )| < 1 Vmax IR 2 Ξ(q x , q y ) dq x dq y = 2 ℜe i υ(q y , t)T P s (q y , υ(q y , t)) ∂υ(q y , t) ∂t e -st dtdq y + 2 IR +∞ t0 (qy)

ℜe i γ(q y , t)T P s (q y , γ(q y , t)) ∂γ(q y , t) ∂t e -st dtdq y .

Inversion of the integrals

The key point of the method is the inversion of the integral with respect to q y with the integral with respect to t. For the volume wave we have (see Figs. 4 and5), after having remark that the integrand is even with respect to q y : +∞ -∞ +∞ t0 (qy)

ℜe i γ(q y , t)T P s (q y , γ(q y , t)) ∂γ(q y , t) ∂t e -st dtdq y = 2 +∞ t 0 q 0 (t) 0

ℜe i γ(q y , t)T P s (q y , γ(q y , t)) ∂γ(q y , t) ∂t e -st dq y dt;

and for the head wave (see Figs. 6 and7): ℜe i υ(q y , t)T P s (q y , υ(q y , t)) ∂υ(q y , t) ∂t e -st dtdq y = 2

t 0 t h 1 q h (t) 0
ℜe i υ(q y , t)T P s (q y , υ(q y , t)) ∂υ(q y , t) ∂t e -st dq y dt

+ 2 t h 2 t 0 q h (t) q 0 (t)
ℜe i υ(q y , t)T P s (q y , υ(q y , t)) ∂υ(q y , t) ∂t e -st dq y dt. t q y t 0 t0 (q y ) Figure 4: Integration first over q y then over t for the volume wave t q y t 0 q 0 (t) Figure 5: Integration first over t then over q y for the volume wave t q y q max t 0 t h 1 t0 (q y ) th (q y ) Figure 6: Integration first over q y then over t for the head wave t q y t 0

t h 1 q 0 (t) q h (t) t0 (q max ) = th (q max ) = t h 2

Numerical illustration

To illustrate our results, we have computed the green function and the analytical solution to the following problem: we consider an acoustic layer with a density ρ + = 1020 kg/m 3 and a celerity V + = 1500 m/s on top of a poroelastic layer whose characteristic coefficients are:

the solid density ρ - s = 2500 kg/m 3 ; the fluid density ρ - f = 1020 kg/m 3 ; the porosity φ -= 0.4; the tortuosity a -= 2;

the solid bulk modulus K - s = 16.0554 GPa; the fluid bulk modulus K - f = 2.295 GPa;

the frame bulk modulus K - b = 10 GPa; the frame shear modulus µ -= 9.63342 GPa; so that the celerity of the waves in the poroelastic medium are: for the fast P wave, V - P f = 3677 m/s for the slow P wave, V - P s = 1060 m/s for the ψ wave, V - S = 2378 m/s. The source is located in the acoustic layer, at 500 m from the interface. It is a point source in space and a fifth derivative of a Gaussian of dominant frequency f 0 = 15 Hz:

f (t) = 2 π 2 f 2 0 3 + 12 π 2 f 2 0 t - 1 f 0 2 + 4 π 4 f 4 0 t - 1 f 0 4 e -π 2 f 2 0 t-1 f 0 2 .
We compute the solution at two receivers, the first one is in the acoustic layer, at 533 m from the interface; the first one is in the poroelastic layer, at 533 m from the interface; both are located on a vertical line at 400 m from the source (see Fig. 8). To compute the integrals over q and the convolution with the source function, we used a classical mid-point quadrature formula.

We represent the z component of the green function associated to the displacement from t = 0 to t = 1.

INRIA 5 Conclusion

In this paper we have provided the complete solution (reflected and transmitted wave) of the propagation of wave in a stratified 3D medium composed of an acoustic and a poroelastic layer. In a forthcoming paper we will extend the method to the propagation of waves in bilayered poroelastic medium in three dimensions.

Figure 1 :

 1 Figure 1: Configuration of the study

Figure 3 :

 3 Figure 3: Functions t0 (red, dotted line) and th (blue, solid line)

Figure 7 :

 7 Figure7: Integration first over t then over q y for the head wave

2 Figure 8 :Figure 9 :Figure 10 :

 28910 Figure 8: Configuration of the experiment

1 The model problem 3 1 . 1 6 2of the analytical solution 10 3

 11610 The equation of acoustics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2 Biot's model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.3 Transmission conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 The Green problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expression Proof of the theorem 19 3.1 The solution in the Laplace-Fourier plane . . . . . . . . . . . . . . . . . . . . 20 3.2 The Laplace transform of the solution . . . . . . . . . . . . . . . . . . . . . . 22 3.3 Inversion of the integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . .

RR n 6595

Diaz & Ezziani

ℜe i γ(t, q)κ - S (γ(t, q))T S (γ(t, q)) dγ dt (t, q) dq,

q 0 (t)

ℜe i υ(t, q)κ - S (υ(t, q))T S (υ(t, q)) dυ dt (t, q) dq u - S,z (x, z, t) =

ℜe (υ 2 (t, q) + q 2 )T S (υ(t, q)) dυ dt (t, q) dq,

or if t 0 < t and |ℑm [γ(t 0 , 0)]| ≥ 1 V max and u - P s (x, z, t) = 0 else. t 0 denotes here the arrival time of the S volume wave at point (x, 0, z) (we recall in appendix the computation of t 0 ),

denotes the arrival time of the S head-wave at point (x, 0, z) and

denotes the time after which there is no longer head wave at point (x, 0, z), where

The function q 0 : [t 0 ; +∞] → IR + is the reciprocal function of t0 : IR + →: [t 0 , +∞], where t 0 (q) is the arrival time at point (x, 0, z) of the fictitious S volume wave, propagating at a velocity V + (q) in the top layer and at velocity V - S (q) in the bottom layer (we recall in appendix the computation of t 0 (q)).

The function q 1 : [t 1 ; t 0 ] → IR + is defined by

INRIA

A Definition of the fictitious and real arrival times of the volume waves.

We detail in this section the computation of the fictitious and real arrival times of the transmitted P s wave at point (x, 0, z). For a given q y ∈ IR, we first determine fastest path of the wave from the source to the point (x, 0, z), travelling at a velocity V + (q y ) in the upper layer and at a velocity V - P s (q y ) in the bottom layer: we search a point ξ 0 on the interface between the two media which minimizes the function

(see Fig. 11). This leads us to find ξ 0 such that

From a numerical point of view, the solution of this equation is done by computing the roots of the following fourth degree polynomial

2 (q y ) , ξ 0 is thus the only real root of this polynomial located between 0 and x which is also solution of (39). Once ξ 0 is computed, we can define

and t 0 = t0 (0)

Let us remark that Property A.1. Since the fictitious velocities are smaller than the real one, the fictitious arrival times are greater than the real one. Moreover, since the fictitious velocities are even functions decreasing on IR + , t0 is an even function, increasing on IR + .

Corollaire A.1. The function t0 is bijective from IR + to IR + .

Source Medium 1

Medium 2 (0, 0, h) (ξ, 0, 0) V + (q y ) V - P s (q y ) (x, 0, z)