Scattering of waves by impurities in precompressed granular chains

We study scattering of waves by impurities in strongly precompressed granular chains. We explore the linear scattering of plane waves and identify a closed-form expression for the reflection and transmission coefficients for the scattering of the waves from both a single impurity and a double impurity. For single-impurity chains, we show that, within the transmission band of the host granular chain, high-frequency waves are strongly attenuated (such that the transmission coefficient vanishes as the wavenumber $k\to \pm \pi$), whereas low-frequency waves are well-transmitted through the impurity. For double-impurity chains, we identify a resonance—enabling full transmission at a particular frequency—in a manner that is analogous to the Ramsauer–Townsend (RT) resonance from quantum physics. We also demonstrate that one can tune the frequency of the RT resonance to any value in the pass band of the host chain. We corroborate our theoretical predictions both numerically and experimentally, and we directly observe almost complete transmission for frequencies close to the RT resonance frequency. Finally, we show how this RT resonance can lead to the existence of reflectionless modes in granular chains (including disordered ones) with multiple double impurities.

Figure 1

Schematic of a homogeneous granular chain with (a) one impurity and (b) two contiguous impurities (i.e., a double impurity). The incident wave is characterized by $I$, the reflected wave is characterized by $R$, and the transmitted wave is characterized by $T$. We label the identities of the particles with integers. We calculate the parameters $B_{ij}=\frac{3}{2}{A}_{ij}^{2/3}{F}_0^{1/3}$ from the static precompression and the interactions between consecutive particles.

Figure 2

(Left) Transmission and (right) reflection coefficients for the scattering of a plane wave in a chain with impurities as a function of the wave number $k$ and the radius ratio $\alpha$. We show examples for (top) a single impurity and (bottom) a double impurity. We describe the physical parameters of the particles in the chain in Table 1 (see Sec. 4).

Figure 3

Reflection coefficient $|R_{\left(ii\right)}{|}^2$ for the double impurity. The red dashed curves indicate the points at which the reflection coefficient is exactly 0. The resonant wave number $k_r$ is given by Eq. (13). The blue dashed line highlights the critical value ${\alpha }_c$; the system has a Ramsauer–Townsend (RT) resonance at wave number $k=\pm k_r$ for $\alpha >{\alpha }_c$.

Figure 4

(a) Space-time contour plot of particle velocity profiles in a host 63-particle chain in which a double impurity has been inserted between particles $-2$ and $+1$. We use $\alpha =1.5$ and the parameters in Table 1 for this numerical simulation. Arrows (1) and (2) indicate the regions that we consider for the calculation of the transmission coefficient. These regions are not affected by the plane waves that reflect from the left or right walls. We also show velocity profiles for particles (b) $n=-27$ and (c) $n=+3$. The dots indicate the maximum and minimum peaks of oscillatory velocity profiles, and the domains of (1) and (2) correspond to the temporal regions marked with (1) and (2) in panel (a).

Figure 5

Schematic of the experimental setup for a granular chain with a single impurity. In the inset, we show an image of the experimental setup.

Figure 6

Transmission of plane waves in a granular chain with impurities. The ratio of the impurity radius to the host-particle radius is (left) $\alpha =0.7$ and (right) $\alpha =1.5$. We show results for chains with (top) a single impurity and (bottom) a double impurity.

Figure 7

Schematic of a host homogeneous granular chain with multiple double impurities. The incident wave is $\left(I\right)$, the reflected wave is $\left(R\right)$, and the transmitted wave is $\left(T\right)$. We highlight impurities in solid turquoise boxes, and we indicate the scattering region with the dashed box.

Figure 8

(a) Space-time contour plot of the normalized velocity for a homogeneous chain with $N=200$ particles and an excitation frequency of $3.0\mathrm{kHz}$. (b) The same plot as in panel (a), but with five double impurities located at positions $n=50,n=72,n=90,n=96$, and $n=118$. The radius ratio is $\alpha =1.5$, and arrows indicate the position of the impurities. Panels (c)–(e) show the velocity at particle $n=130$ for excitation frequencies of (c) $2.0\mathrm{kHz}$, (d) $3.0\mathrm{kHz}$, and (e) $3.8\mathrm{kHz}$. The black (large-amplitude) curves are associated with the homogeneous chain [panel (a)], and the red (smaller-amplitude) curves are associated with the chain with the impurities [panel (b)].

Figure 9

Space-time contour plot of normalized velocity for experiments with (top) a single-impurity chain with $\alpha =1.5$ and (bottom) a double-impurity chain with $\alpha =0.7$. In each case, we normalize the measured velocities with respect to the maximum velocity. We use excitation frequencies of (left) 2 kHz and (right) 6 kHz. The arrows point to incidents of wave localization.