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
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.
We are concerned with the acoustic scattering problem, at a frequency κ, by many small obstacles of arbitrary shapes with impedance boundary condition. These scatterers are assumed to be included in a bounded domain Ω in R 3 which is embedded in an acoustic background characterized by an eventually locally varying index of refraction. The collection of the scatterers Dm, m = 1, ..., M is modeled by four parameters: their number M , their maximum radius a, their minimum distance d and the surface impedances λm, m = 1, ..., M . We consider the parameters M, d and λm's having the following scaling properties:≈ a t and λm := λm(a) = λm,0a −β , as a → 0, with non negative constants s, t and β and complex numbers λm,0's with eventually negative imaginary parts.We derive the asymptotic expansion of the farfields with explicit error estimate in terms of a, as a → 0. The dominant term is the Foldy-Lax field corresponding to the scattering by the point-like scatterers located at the centers zm's of the scatterers Dm's with λm|∂Dm| as the related scattering coefficients. This asymptotic expansion is justified under the following conditionsand the error of the approximation is C a 3−2β−s , as a → 0, where the positive constants a0, λ−, λ+ and C depend only on the a priori uniform bounds of the Lipschitz characters of the obstacles Dm's and the ones of M (a)a s and d(a) a t . We do not assume the periodicity in distributing the small scatterers. In addition, the scatterers can be arbitrary close since t can be arbitrary large, i.e. we can handle the mesoscale regime. Finally, for spherical scatterers, we can also allow the limit case β = 1 with a slightly better error of the approximation.
Let M be the number of bounded and Lipschitz regular obstacles Dj, j := 1, ..., M having a maximum radius a, a << 1, located in a bounded domain Ω of R 3 . We are concerned with the acoustic scattering problem with a very large number of obstacles, as M := M (a) := O(a −1 ), a → 0, when they are arbitrarily distributed in Ω with a minimum distance between them of the order d := d(a) := O(a t ) with t in an appropriate range. We show that the acoustic farfields corresponding to the scattered waves by this collection of obstacles, taken to be soft obstacles, converge uniformly in terms of the incident as well the propagation directions, to the one corresponding to an acoustic refraction index as a → 0. This refraction index is given as a product of two coefficients C and K, where the first one is related to the geometry of the obstacles (precisely their capacitance) and the second one is related to the local distribution of these obstacles. In addition, we provide explicit error estimates, in terms of a, in the case when the obstacles are locally the same (i.e. have the same capacitance, or the coefficient C is piecewise constant) in Ω and the coefficient K is Hölder continuous. These approximations can be applied, in particular, to the theory of acoustic materials for the design of refraction indices by perforation using either the geometry of the holes, i.e. the coefficient C, or their local distribution in a given domain Ω, i.e. the coefficient K.
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