Acoustic gaps in a chain of magnetic spheres

Acoustic gaps are normally observed in granular inhomogeneous structures made of composite materials. The modulation of the elastic properties in such media creates the coherent effects of scattering and interference that ultimately lead to frequency intervals where sound propagation is forbidden. Contrastingly, we report here an experimental observation of acoustic gaps in homogeneous media; specifically, in granular chains. The beads used in our study are magnetic. Therefore, instead of modulating the elastic properties of the chain, we modulate the magnetization (i.e., the contact forces). We also observe that the propagation speed of acoustic signals through the magnetic chains used in this study is at odds with the speed predicted by Hertz’s law.

Figure 1

(Color online) (a) A chain of magnetic spheres. The dipoles are aligned in a head-to-tail fashion $\left(\to \to \to \to \right)$. (b) A two-dimensional lattice formed by magnetic spheres. Along one of the axis, the dipoles are aligned as shown in (a) and in the other they alternate orientations (left and right). (c) The experimental setup used in the experiments.

Figure 2

(Color online) (a) Contact forces for chains of different lengths, with $N=6,11,16,21$. The configuration of the dipoles is in the head-to-tail fashion. Note that the contact forces are smaller at the boundaries of the chains due to the nature of the magnetic interaction. (b) Contact forces for chains with $N=6,11,16,21$, where the dipoles alternate orientations. Note that the contact forces are smaller (greater) at the boundaries of the chains due to the nature of the dipole-dipole interaction.

Figure 3

(a) Acoustic pulse sent through a chain of magnetic spheres. The pulse has a width of $2\mu \text{s}$. (b) The signal measured at the end of the chain. $\Delta t$ is the time of flight of the pulse in the chain. Note that the accelerometer takes around $400\mu \text{s}$ to relax.

Figure 4

(Color online) Speed of sound as a function of the contact force in the middle of the chain. Data are well fitted by a power law with an exponent 1/3 (see also the log-log plot in the inset). Black squares were measured as indicated in the text and red dots were obtained by first measuring the cutoff frequencies (frequencies at which no signal propagates) and then using the expression $V=2\pi a{f}_c$, where $a$ is the radius of the beads (see [1]). We show, for comparison, the speed of sound predicted by Hertz’s law (exponent 1/6).

Figure 5

(Color online) Contact forces in short chains as a function of length. Each chain is formed by five and a half cells and each cell by five magnetic (yellow) and (a) no, (b) two, (c) three, and (d) four nonmagnetic spheres (black).

Figure 6

(a) Acoustic signal sent through a chain of magnetic spheres. The signal is composed by a 20 ms wave train, where the frequency increases with time from 1 kHz to 15 kHz. (b) Signal measured at the end of the chain.

Figure 7

(Color online) Power spectra of the transmitted pulse as the number of nonmagnetic spheres $\left(n\right)$ is increased; (a) $n=0$, $\eta =1$, $f_0=9.5\text{kHz}$, $\sigma =4.5\text{kHz}$, and $\lambda =0.95$; (b) $n=2$, $\eta =0.61$, $f_0=7.6\text{kHz}$, $\sigma =3.5\text{kHz}$, and $\lambda =0.22$; (c) $n=3$, $\eta =0.58$, $f_0=6.1\text{kHz}$, $\sigma =3.5\text{kHz}$, and $\lambda =0.14$; (d) $n=4$, $\eta =0.54$, $f_0=5.2\text{kHz}$, $\sigma =3.5\text{kHz}$, and $\lambda =0.025$. Black dots are experimental data and the (blue) curve the theoretical prediction.