## 3 Experimental Results
### Coupled Three-Rod Waveguides, \(N=3\)  
\[\text{For}\,N=3,M_{3\times 3}=\left( \begin{array}{ccc}1&-1&0\\ -1&2&-1\\ 0&-1&1\end{array} \right)\text{and the three spatial eigen vectors corresponding to the eigen values }\lambda_{1}=0\), \(\lambda_{1}=1\), and \(\lambda_{3}=3\), are: \(e_{1}=\frac{1}{\sqrt{3}}\left( \begin{array}{c}1\\ 1\\ 1\end{array} \right)\), \(e_{2}=\frac{1}{\sqrt{2}}\left( \begin{array}{c}1\\ 0\\ -1\end{array} \right)\), \(e_{3}=\frac{1}{\sqrt{6}}\left( \begin{array}{c}1\\ -2\\ 1\end{array} \right)\). The associated dispersion relations are \(\omega_{k}^{2}=\left(\beta k\right)^{2}\), \(\omega_{k}^{2}=\left(\beta k\right)^{2}+\left(\alpha\right)^{2}\) and \(\omega_{k}^{2}=\left(\beta k\right)^{2}+3\left(\alpha\right)^{2}\). The frequency spectrum of the three spatial eigen modes are shown in Figure 5a. The spectra shows well-defined resonances of the standing waves. The second (\(e_{2}\)) and third (\(e_{3}\)) spatial eigen modes (Mode 1 0 -1: blue and Mode 1 -2 1: green of Figure 5a) transmission spectra differ from \(e_{1}\) (first) eigen mode (Mode 1 1: red). \(e_{2}\) and \(e_{3}\) eigen modes show a significant depression in the transmission amplitude below a cutoff frequency and the resonances are spaced more closely near the cutoff. Figure 5b shows the resultant dispersion relation determined and calculated from spectra of Figure 5a. The speed of sound, \(\beta=4467\) m/s, is extracted from the first eigen vector. By comparing the speed of sound of the coupled waveguides to the single waveguide, we see that the speed of sound decreases from 4571 m/s to 4467 m/s as the number of rods in the waveguides increases from \(N=1\) to \(N=3\). This can be attributed to the added mass in the coupled waveguides coming from the epoxy coupling medium, as well as an effect of the different cross-sectional geometry of the coupled rods compared to the single rod. Finally, the lsqcurvefit function of MATLAB is used again to obtain the dispersion relation of the second and third bands with cutoff frequencies 14.23 kHz for \(e_{2}\) and 24.27 kHz for \(e_{3}\). Figure 4: **(a)** Transmission spectrum of the two coupled rod system (\(N=2\)) for the two spatial eigen modes, namely the symmetric (1 1) mode and anti-symmetric (1 -1) mode. The inset in **(a)** identifies the first two resonances of the symmetric mode and negligible transmission of the anti-symmetric mode at frequency \(<\)10 kHz, identifying the first two resonances. The transmission amplitude is in arbitrary units. **(b)** Band structure determined and calculated from **(a)**. The asterisks in **(b)** are obtained from the resonances of mode: 1 1 of **(a)** and the solid line associated with the asterisks is a fit to the low frequency resonances. The open circles in **(b)** correspond to resonant modes associated with the anti-symmetric spatial eigen mode and the solid line is a fit to these identifiable resonances using the dispersion relation with a cutoff frequency of 20.24 kHz. The resonances shown in Figure 5 correspond to separable states, i.e., states expressible as tensor products of spatial and spinor degrees of freedom. To realize a nonfactorizable superposition of elastic states, i.e., an acoustic Bell state that is not separable, we construct the superposition of isofrequency states of the three coupled rods system corresponding to the two non-zero spatial eigen values: \(\Psi_{6\times 1}=Ae_{2}\otimes s_{2\times 1}(k_{2})\ e^{ik_{2}x}e^{i\omega t}+ Be_{3}\otimes s_{2\times 1}(k_{3})\ e^{ik_{3}x}e^{i\omega t}\). Here, \(A\) and \(B\) are the Bell state coefficients, which maybe complex, and we have chosen \(\omega_{2}=\omega_{3}=\omega\). The wave numbers \(k_{2}\) and \(k_{3}\) are in the \(e_{2}\) and \(e_{3}\) bands with isofrequency \(\omega\). In this superposition, both the eigen vectors of the spatial mode and the spinors are different. This superposition of states cannot be written in the form of a tensor product of one spatial eigen vector and one spinor.