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Let $M$ be the number of bounded and Lipschitz regular obstacles $D_j, j:=1,..., M$ having a maximum radius $a$, $a<<1$, located in a bounded domain$\Omega$ of $\mathbb{R}^3$. We are concerned with the acoustic scatteringproblem with a very large number of obstacles, as $M:=M(a):=O(a^{-1})$,$a\rightarrow 0$, when they are arbitrarily distributed in $\Omega$ with aminimum distance between them of the order $d:=d(a):=O(a^t)$ with $t$ in anappropriate range. We show that the acoustic farfields corresponding to thescattered 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 $a\rightarrow 0$.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 (preciselytheir capacitance) and the second one is related to the local distribution ofthese 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 samecapacitance, or the coefficient $C$ is piecewise constant) in $\Omega$ and thecoefficient $K$ is H$\ddot{\mbox{o}}$lder continuous. These approximations canbe applied, in particular, to the theory of acoustic materials for the designof refraction indices by perforation using either the geometry of the holes,i.e. the coefficient $C$, or their local distribution in a given domain$\Omega$, i.e. the coefficient $K$.

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