# Derivation and Validation of Bandgap Equation Using Serepentine Resonator
## 2 Theory  
### Bandgap  
Figure 1 shows the plate on which the mass-spring resonators are uniformly distributed. Assuming that the resonators are infinitely periodic and smaller than the wavelength of interest, the equations of motion can be divided into two parts as:  
\[\rho h\ddot{w}_{p}+D\
abla^{4}w_{p}=f_{pr}, \tag{1a}\] \[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]  
where \(\rho\) is mass density, \(h\) is the thickness of the plate, \(w_{p}\) and \(w_{r}\) are the vertical displacements of the plate and resonator at the connection point, respectively, \(D\) is the bending stiffness of the plate, \(f_{rp}\) and \(f_{pr}\) are the forces of the resonator on plate and plate on resonator, and \(m_{r}\) and \(k_{r}\) are the lumped mass and stiffness of resonator, respectively. The vertical displacement of the plate for the harmonic wave is defined as  
\[w_{p}(x,y,t)=\overline{w}_{p}(x,y)\exp(j\omega t) \tag{2}\]  
where \(\omega\) is the angular frequency. By substituting acceleration with displacement in Equation (2), Equation (1a) can be modified as:  
\[\Big{(}\
abla^{4}-\omega^{2}\rho h/D\Big{)}\overline{w}_{p}(x,y)=\frac{f_{pr} }{D}, \tag{3}\]The wavenumber of the free bending wave in the plate \(k_{b}\), angular frequency, mass density, thickness of the plate, and bending stiffness of the plate are related as:  
\[k_{b}=\left(k_{x}^{2}+k_{y}^{2}\right)^{1/4}=\left(\omega^{2}\rho h/D\right)^{1/4} \tag{4}\]  
Using Equation (4), the equations of motion can be obtained as follows:  
\[\left(\
abla^{4}-k_{b}^{4}\right)\overline{w}_{p}(x,y)=\frac{f_{pr}}{D}. \tag{5}\]  
Using Equation (1b), the equations of motion can be obtained as Equation (6a). From the base excitation, the relation between vertical displacement of the plate and resonator can be expressed as Equation (6b):  
\[\left(-\omega^{2}m_{r}+k_{r}\right)\overline{w}_{r}=k_{r}\overline{w}_{p}, \tag{6a}\] \[\overline{w}_{r}=\overline{w}_{p}\frac{k_{r}}{m_{r}\left(\omega_{0}^{2}-\omega ^{2}\right)}. \tag{6b}\]  
where \(\omega_{0}\) is the natural resonance frequency of the mass-spring system. From the equilibrium of force between the resonator and plate, \(f_{pr}\) can be obtained as follows:  
\[f_{pr}=k_{r}(\overline{w}_{r}-\overline{w}_{p})=\omega^{2}m_{r}\overline{w}_{ r}. \tag{7}\]  
By substituting Equation (7) in Equation (5), \(f_{pr}\) can be eliminated to obtain Equation (8a). Substituting Equation (6b) in Equation (8a) and rearranging the terms, an equation containing only the vertical displacement of the plate \(w_{p}\) can obtained as follows:  
\[\left(\
abla^{4}-k_{b}^{4}\right)\overline{w}_{p}=\frac{1}{D}\omega^{2}m_{r} \overline{w}_{r}, \tag{8a}\] \[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]0 (8b) \[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]1  
\[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]2 (**a**) Bandgap structure and (**b**) mass–spring resonator represented by the blue circle. This equation is the modified Helmholtz equation for the plate on which the mass-spring resonators are uniformly distributed. From Equation (8c), the modified wavenumber can be obtained as follows:  
\[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]3  
Bandgap is defined as a frequency band where free waves do not propagate. To prevent wave propagation, the wavenumber has to be purely imaginary, which is the condition for an evanescent wave. Therefore, the modified wavenumber must satisfy the following condition:  
\[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]4 \[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]5  
Eliminating \(m_{r}\) in the numerator and denominator of the second term and rearranging the remaining terms, Equation (10b) can be expressed as follows:  
\[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]6 \[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]7 (11b) \[k_{r}(w_{p}-w_{r})=m_{r}\ddot{w}_{r}=f_{rp}, \tag{1b}\]8  
\(m_{p}\) is the mass of the plate, which is always positive.