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We are concerned with the acoustic scattering problem by many small rigid obstacles of arbitrary shapes. We give a sufficient condition on the number M and the diameter a of the obstacles as well as the minimum distance d between them under which the Foldy-Lax approximation is valid. Precisely, if we use single layer potentials for the representation of the scattered fields, as is done sometimes in the literature, then this condition is (M −1) a d 2 < c, with an appropriate constant c, while if we use double layer potentials, then a weaker condition of the form √ M − 1 a d < c is enough. In addition, we derive the error in this approximation explicitly in terms of the parameters M, a, and d. The analysis is based, in particular, on the precise scalings of the boundary integral operators between the corresponding Sobolev spaces. As an application, we study the inverse scattering by the small obstacles in the presence of multiple scattering. B 1 , B 2 , . . . , B M be M open, bounded, and simply connected sets in R 3 with Lipschitz boundaries 1 containing the origin. We assume that the Lipschitz constants of B j , j = 1, . . . , M, are uniformly bounded. We set D m := B m + z m to be the small bodies characterized by the parameter > 0 and the locations z m ∈ R 3 , m = 1, . . . , M. Let U i be a solution of the Helmholtz equation (Δ + κ 2 )U i = 0 in R 3 . We denote by U s the acoustic field scattered by the M small bodies D m ⊂ R 3 due to the incident field U i . We restrict ourselves to (1) the plane incident waves, U i (x, θ) := e ikx·θ , with the incident direction θ ∈ S 2 , with S 2 being the unit sphere, and (2) the scattering by rigid bodies. Hence the total field U t := U i + U s satisfies the following exterior Dirichlet problem of the acoustic waves: Introduction and statement of the results. Let

In this paper, we investigate the problem of reconstructing soundsoft acoustic obstacles using multifrequency far field measurements corresponding to one direction of incidence. The idea is to obtain a rough estimate of the obstacle's shape at the lowest frequency using the least-squares approach, then refine it using a recursive linearization algorithm at higher frequencies. Using this approach, we show that an accurate reconstruction can be obtained without requiring a good initial guess. The analysis is divided into three steps. Firstly, we give a quantitative estimate of the domain in which the least-squares objective functional, at the lowest frequency, has only one extreme (minimum) point. This result enables us to obtain a rough approximation of the obstacle at the lowest frequency from initial guesses in this domain using convergent gradient-based iterative procedures. Secondly, we describe the recursive linearization algorithm and analyze its convergence for noisy data. We qualitatively explain the relationship between the noise level and the resolution limit of the reconstruction. Thirdly, we justify a conditional asymptotic Hölder stability estimate of the illuminated part of the obstacle at high frequencies. The performance of the algorithm is illustrated with numerical examples.

We deal with the problem of reconstruction of the coefficient discontinuities (or supports) of scalar divergence form equations with lower order terms from the Dirichlet-to-Neumann map using complex geometrical optics (CGO) solutions. We consider both penetrable and impenetrable obstacles. The usual proofs for justifying this method assume, in addition to the smoothness of the coefficients and the interfaces, the following two conditions. The finiteness of the touching points of the phase’s level curves (or surfaces) of the used CGO solutions with the interface. The positivity of the lower bound of the relative curvature of the interface. In this paper, we show how we can remove these two conditions and justify the reconstruction method considering L∞ coefficients and Lipschitz interfaces of discontinuity.

In the first part of this paper, it is proved that a C 2 -regular rigid scatterer in R 3 can be uniquely identified by the shear part (i.e. S-part) of the far-field pattern corresponding to all incident shear waves at any fixed frequency. The proof is short and it is based on a kind of decoupling of the S-part of scattered wave from its pressure part (i.e. P-part) on the boundary of the scatterer. Moreover, uniqueness using the S-part of the far-field pattern corresponding to only one incident plane shear wave holds for a ball or a convex Lipschitz polyhedron. In the second part, we adapt the factorization method to recover the shape of a rigid body from the scattered S-waves (resp. P-waves) corresponding to all incident plane shear (resp. pressure) waves. Numerical examples illustrate the accuracy of our reconstruction in R 2 . In particular, the factorization method also leads to some uniqueness results for all frequencies excluding possibly a discrete set.

We provide a limiting absorption principle for the selfadjoint realizations of Laplace operators corresponding to boundary conditions on (relatively open parts Σ of) compact hypersurfaces Γ = ∂Ω, Ω ⊂ R n . For any of such self-adjoint operators we also provide the generalized eigenfunctions and the scattering matrix; both these objects are written in terms of operator-valued Weyl functions. We make use of a Kreȋn-type formula which provides the resolvent difference between the operator corresponding to self-adjoint boundary conditions on the hypersurface and the free Laplacian on the whole space R n . Our results apply to all standard examples of boundary conditions, like Dirichlet, Neumann, Robin, δ and δ ′ -type, either assigned on Γ or on Σ ⊂ Γ.

We deal with the linearized model of the acoustic wave propagation generated by small bubbles in the harmonic regime. We estimate the waves generated by a cluster of M small bubbles, distributed in a bounded domain Ω, with relative densities having contrasts of the order a β , β > 0, where a models their relative maximum diameter, a ≪ 1. We provide useful and natural conditions on the number M , the minimum distance and the contrasts parameter β of the small bubbles under which the point interaction approximation (called also the Foldy-Lax approximation) is valid.With the regimes allowed by our conditions, we can deal with a general class of such materials. Applications of these expansions in material sciences and imaging are immediate. For instance, they are enough to derive and justify the effective media of the cluster of the bubbles for a class of gases with densities having contrasts of the order a β , β ∈ ( 3 2 , 2) and in this case we can handle any fixed frequency. In the particular and important case β = 2, we can handle any fixed frequency far or close (but distinct) from the corresponding Minnaert resonance. The cluster of the bubbles can be distributed to generate volumetric metamaterials but also low dimensional ones as metascreens and metawires.2010 Mathematics Subject Classification. 35R30, 35C20.

We deal with the point-interaction approximations for the acoustic wave fields generated by a cluster of highly contrasted bubbles for a wide range of densities and bulk moduli contrasts. We derive the equivalent fields when the cluster of bubbles is appropriately distributed (but not necessarily periodically) in a bounded domain Ω of R 3 . We handle two situations.(1) In the first one, we distribute the bubbles to occupy a 3 dimensional domain. For this case, we show that the equivalent speed of propagation changes sign when the medium is excited with frequencies smaller or larger than (but not necessarily close to) the Minnaert resonance. As a consequence, this medium behaves as a reflective or absorbing depending on whether the used frequency is smaller or larger than this resonance. In addition, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above a certain threshold, then the medium behaves as a 'wall', i.e. allowing no incident sound to penetrate.(2) In the second one, we distribute the bubbles to occupy a 2 dimensional (open or closed) surface, not necessarily flat. For this case, we show that the equivalent medium is modeled by a Dirac potential supported on that surface. The sign of the surface potential changes for frequencies smaller or larger than the Minnaert resonance, i.e. it behaves as a smart metasurface reducing or amplifying the transmitted sound across it. As in the 3D case, if the used frequency is extremely close to this resonance, for a cluster of bubbles with density above an appropriate threshold, then the surface allows no incident sound to be transmitted across the surface, i.e. it behaves as a white screen.2010 Mathematics Subject Classification. 35R30, 35C20.