The equivalent refraction index for the acoustic scattering by many small obstacles: With error estimates

Bashir Ahmad, Durga Prasad Challa, Mokhtar Kirane, Mourad Sini

Research output: Contribution to journalArticlepeer-review

13 Scopus citations


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.

Original languageBritish English
Pages (from-to)563-583
Number of pages21
JournalJournal of Mathematical Analysis and Applications
Issue number1
StatePublished - 1 Apr 2015


  • Acoustic scattering
  • Effective medium
  • Multiple scattering


Dive into the research topics of 'The equivalent refraction index for the acoustic scattering by many small obstacles: With error estimates'. Together they form a unique fingerprint.

Cite this