Reply to “Comment on ‘Renormalization group analysis and numerical simulation of propagation and localization of acoustic waves in heterogeneous media’ ”

Based on extensive new numerical simulations, we show that Chu’s arguments and objections against our previous results are invalid. In addition, we explain the origin of the differences between our results and the previous ones, obtained based on a simple model of one-dimensional disordered materials.

Figure 1

(Color online) Dependence of the minimum allowed frequency of the system on the coupling constant $g$. The inset shows the collapse of the numerical results onto a single curve.

Figure 2

(Color online) Same as in Fig. 1 but for the maximum allowed frequency.

Figure 3

(Color online) Frequency dependence of the localization length $\xi$ in a 1D lattice of size $N=500$ for three values of the coupling constant $g$. Inset: the same results but up to the maximum frequency.

Figure 4

(Color online) Frequency dependence of the localization length $\xi$ of a 1D lattice of size $N$, up to maximum frequency, with a lattice constant, $\Delta =0.1$. The extended modes exhibit oscillatory behavior. For larger $N$, the oscillations become much smaller, hence indicating that they are an artifact of finite value of $N$.

Figure 5

(Color online) Frequency dependence of the localization length $\xi$ in a 1D lattice of size $N$ and lattice constant $\Delta$, where $L=N\Delta =\text{const}$.

Figure 6

(Color online) Frequency dependence of the localization length $\xi$ in a 1D lattice of 1D “spheres” with short-range correlations.