# Exploiting Modern Chladni Plates to Analogously Manifest the Point Interaction
## 3 Exploring the Point Interactions in Modern Chladni Plates  
The transverse vibration of a plate is governed by the biharmonic equation [32]  
\[\left(\
abla^{4}_{2D}-k^{4}\right)\psi(\mathbf{r})=0, \tag{11}\]  
where \(\
abla^{4}_{2D}\) is the biharmonic operator given by  
\[\
abla^{4}_{2D}\ \psi=\frac{\partial^{4}\psi}{\partial x^{4}}+2\frac{\partial^{4} \psi}{\partial x^{2}\partial y^{2}}+\frac{\partial^{4}\psi}{\partial y^{4}}. \tag{12}\]  
Figure 4: (**a**) Eigenvalue with the coupling parameter \(\alpha\) for a specific case of \(k^{(p)}_{7,7}\). (**b**) Dependence of the wave patterns of the eigenfunctions on the parameter \(\alpha\). (**c**) Nodal-line patterns corresponding to the wave patterns in (**b**). (**d**) Contribution of original eigenfunctions \(\psi_{n,m}(\mathbf{r})\) in the perturbed eigenfunction \(\Psi(\mathbf{r};k)\). Equation (11) can be factorized as  
\[\Big{(}\
abla_{2D}^{2}+k^{2}\Big{)}\Big{(}\
abla_{2D}^{2}-k^{2}\Big{)}\psi(\mathbf{ r})=0, \tag{13}\]  
in which the first factor describes propagating dispersive waves, whereas the second one describes evanescent waves. As the aspect ratio of the thickness to the lateral dimension for the studied plate is generally less than 0.02, the vibrating eigenmodes can be approximated with the 2D Helmholtz equation [33]. Even so, the free edge boundary conditions make the problem particularly difficult, as noted by [PERSON] [34]. As a first approximation, one can model the plate as a tightly stretched thin elastic membrane by assuming the constant \(\mu\) (the ratio of lateral contraction to longitudinal elongation) to be zero, i.e., every point of the circumference is free to move along lines perpendicular to the plane of the plate. Under this approximation, the boundary condition can be simplified as the Neumann boundary conditions with \(\partial\psi_{n}/\partial n=0\) on \(\delta\Omega\). Strictly, the Neumann boundary condition is practically applicable to a stretched membrane, but not to a plate vibrating in virtue of rigidity. Nevertheless, it has been demonstrated [35] that a hypothetical free membrane can be used to deduce some of classical Chladni figures successfully. Based on this confirmation, a free membrane with a point scatter is proposed to model the modern Chladni figures. A square-shaped aluminum plate with a side length of \(L=32\) cm and a thickness of \(h=1.0\) mm was used to perform the modern Chladni experiment. The frequency response was measured by a sine function generator with an automatic scanning system with a resolution of 0.1 Hz. As the conductance of the vibrating system displayed a sharp drop at each resonant frequency, we accurately measured the resonant frequency by exceeding the change in conductance of the system [36]. Chladni patterns at resonant frequencies were exhibited with silica sand of 0.3 mm grain size and were stored with a digital camera. Fixing the center of the plate on the mechanical oscillator, several resonant patterns could be precisely observed by scanning the frequency between 200 and 3000 Hz, as shown in Figure 5a for the nodal-line patterns. All the observed nodal-line patterns are obviously different from the characteristics of eigenfunctions of free plate. From Equations (6) and (7), the eigenfunctions of the square plate with the point interaction at \(\mathbf{r}_{\mathrm{s}}\) can be directly expressed as the Green function:  
\[G(\mathbf{r},\mathbf{r}_{\mathrm{s}};k)=\sum_{n=0}^{N}\ \sum_{m=0}^{N}\ \frac{ \widetilde{\psi}_{n,m}(\mathbf{r}_{\mathrm{s}})\ \widetilde{\psi}_{n,m}(\mathbf{r})}{k^{2}-[(n^{2}+m^{2})\pi^{2}/L^{2}]}\, \tag{14}\]  
where the free eigenfunctions \(\widetilde{\psi}_{n,m}(\mathbf{r})\) are given by  
\[\widetilde{\psi}_{n,m}(\mathbf{r})=\frac{2}{L}\cos\Bigl{(}\frac{n\pi}{L}x \Bigr{)}\cos\Bigl{(}\frac{m\pi}{L}y\Bigr{)}, \tag{15}\]  
where \(n=0\),1,2,3\(\cdots\) and \(m=0\),1,2,3\(\cdots\). The eigenvalues are determined from the transcendental equation \(1+\alpha\ \xi(k)=0\) with \(\xi(k)=G(\mathbf{r}_{\mathrm{s}},\mathbf{r}_{\mathrm{s}};k)\).