## 4 Numerical Modeling: Mass-Spring Waveguides
### Coupled Three-chain Mass-Spring Waveguides  
For three coupled mass-spring chains, the discrete elastic equations of motion are given by:  
\[m\ddot{u}_{n}-k_{nn}(u_{n+1}-2u_{n}+u_{n-1})-k_{c}(v_{n}-u_{n})+\eta\dot{u}_{n}=0 \tag{23}\]  
\[m\ddot{v}_{n}-k_{nn}(v_{n+1}-2v_{n}+v_{n-1})-k_{c}(u_{n}-v_{n})-k_{c}(w_{n}-v_{ n})+\eta\dot{v}_{n}=0, \tag{24}\]  
\[m\ddot{w}_{n}-k_{nn}(w_{n+1}-2w_{n}+w_{n-1})-k_{c}(v_{n}-w_{n})+\eta\dot{w}_{n}=0 \tag{25}\]  
In Equations (23)-(25), \(u_{n}\), \(v_{n}\), and \(w_{n}\) are the displacements of \(n^{th}\) mass of chain 1, 2, and 3, respectively. Here, we take \(k_{c}\) to be the same for all coupled chains, which is justified since the experimental sample ensures uniform epoxy fillings between the rods. To model the excitation applied by the transducer, the prescribed base periodic excitation \(F^{u}(t)=F^{u}_{0}\sum_{\omega_{i}}\sin(\omega_{i}t)\), \(F^{v}(t)=F^{v}_{0}\sum_{\omega_{i}}\sin(\omega_{i}t+\phi^{v}_{0})\) and \(F^{w}(t)=F^{w}_{0}\sum_{\omega_{i}}\sin(\omega_{i}t+\phi^{w}_{0})\) are applied to the first mass of each chain, where \(F^{w}_{0}\) and \(\phi^{w}_{0}\) are the amplitude and phase of the applied force to chain 3. Figure 9a reveals the eigen frequencies of the coupled three-chain mass-spring waveguides for the three spatial eigen modes, and Figure 9b shows the resultant dispersion relations. Similarly to the experimental result of Figure 5, the second and third spatial eigen modes differ from the first eigen mode. Below the cutoff frequencies, the transmissions are negligible. The resonances are clearly defined and closely spaced as one approaches the cutoff frequencies. Again, the wave numbers for all the bands are multiples of \(2N_{m}a\). We also use the coupling spring stiffness as \(k_{c}=33.5\times 10^{6}\) N/m, obtained from the model of two coupled chains. With this choice, we obtain the cutoff frequencies of  
Figure 8: (**a**) Power spectral density of the displacement of \(N_{m}\) mass in the coupled two-chain mass-spring waveguides showing the eigen frequency for mode: 1 1 and mode: 1 –1. (**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 anti-symmetric resonant modes and the solid line is a fit to these identifiable resonances using the dispersion relation \(\omega_{k}^{2}=\left(\beta k\right)^{2}+2(\alpha)^{2}\) with cutoff frequency \(\sqrt{2}\alpha=19.91\) kHz. System parameters: \(N_{m}=48\), \(\eta=6.08\) Ns/m, \(\omega_{i}\in\left(0,60\right.\)kHz\(\right)\) and \(\left(F^{u}_{0},F^{v}_{0},\phi^{v}_{0}\right)=\left(1\) N, 1 N, 0\(\right)\) for mode: 1 1 excitation and \(\left(F^{u}_{0},F^{v}_{0},\phi^{v}_{0}\right)=\left(1\) N, 1 N, \(\pi\right)\) for mode: 1 –1 excitation. 14.08 kHz and 24.43 kHz in excellent agreement with experimental values. This model of two and three coupled chains can therefore be used to interpret the experimentally observed behaviors. We identified an isofrequency (33.78 kHz) corresponding to nearly overlapping (1 0 -1) and (1 -2 1) resonances and to a trough between two resonances frequencies in the (1 1 1) eigen mode. We can therefore use this frequency to numerically realize a nonseparable superposition of states. In accord with the experiment, we excite such a nonseparable state by driving the three-chain mass-spring  
Figure 9: **(a)** Power spectral density of the displacement of the \(N_{m}\) mass in the coupled three-chain mass-spring waveguides showing the eigen frequency for mode: 1 1 1, mode: 1 0 –1, and mode: 1 –2 1. **(b)** Band structure determined and calculated from **(a)**. The asterisks in **(b)** are obtained from the resonances of mode: 1 1 1 of **(a)**, and the solid line associated with the asterisks is a fit to the low frequency resonances. The open and closed circles in **(b)** correspond to resonant modes associated with the \(e_{2}\).