TY - JOUR

T1 - The equivalent refraction index for the acoustic scattering by many small obstacles

T2 - With error estimates

AU - Ahmad, Bashir

AU - Challa, Durga Prasad

AU - Kirane, Mokhtar

AU - Sini, Mourad

N1 - Publisher Copyright:
© 2014 Elsevier Inc.

PY - 2015/4/1

Y1 - 2015/4/1

N2 - 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 R3. 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(at) 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 as 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.

AB - 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 R3. 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(at) 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 as 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.

KW - Acoustic scattering

KW - Effective medium

KW - Multiple scattering

UR - http://www.scopus.com/inward/record.url?scp=84920626077&partnerID=8YFLogxK

U2 - 10.1016/j.jmaa.2014.11.020

DO - 10.1016/j.jmaa.2014.11.020

M3 - Article

AN - SCOPUS:84920626077

SN - 0022-247X

VL - 424

SP - 563

EP - 583

JO - Journal of Mathematical Analysis and Applications

JF - Journal of Mathematical Analysis and Applications

IS - 1

ER -