# Exploiting Modern Chladni Plates to Analogously Manifest the Point Interaction

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## Abstract

**:**

## 1. Introduction

## 2. Theoretical Model for the Point Interaction

## 3. Exploring the Point Interactions in Modern Chladni Plates

## 4. Applications

^{3}[38]. Using these material properties and h = 1 mm, the theoretical coefficient $C/2\pi $ can be calculated to be 0.248. We employed the parabolic formula in Equation (16) to make the best fit to the experimental results obtained in Figure 5 and Figure 6. The fitting coefficient $C/2\pi $ is approximately 0.228, as shown in Figure 8. The experimentally determined coefficient $C/2\pi $ can be found to agree with the theoretical value very well. This good agreement validates that the point-interaction model can be utilized not only to explore the formation of modern Chladni patterns, but also to determine the dispersion relation between the resonant frequency and the wave number for the vibration of plates. As the dispersion relation in Equation (16) includes the information of the flexural rigidity D as well as the Young modulus E, the proposed theoretical model is believed to be practically useful for measuring the acoustic properties of plates.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Calculated result for $\xi (k)$ in Equation (8) as a function of k with ${r}_{s}=(L/2,L/2)$ and N = 50.

**Figure 3.**Calculated results for ${\left[1+\alpha \xi (k)\right]}^{-1}$ as a function of k for different N increasing from 10 to 50 for (

**a**) $\alpha =5$ and (

**b**) $\alpha =10$.

**Figure 4.**(

**a**) Eigenvalue with the coupling parameter α for a specific case of ${k}_{7,7}^{(p)}$. (

**b**) Dependence of the wave patterns of the eigenfunctions on the parameter α. (

**c**) Nodal-line patterns corresponding to the wave patterns in (

**b**). (

**d**) Contribution of original eigenfunctions ${\mathsf{\psi}}_{n,m}(r)$ in the perturbed eigenfunction $\mathsf{\Psi}(r;k)$.

**Figure 5.**(

**a**) Experimental nodal-line patterns for fixing the plate at ${r}_{s}=(0.5L,0.5L)$(on the center). (

**b**) Numerical nodal-line patterns corresponding to the experimental results.

**Figure 6.**(

**a**) Experimental nodal-line patterns for fixing the plate at ${r}_{s}=(0.57L,0.57L)$ (off the center). (

**b**) Numerical nodal-line patterns corresponding to the experimental results.

**Figure 7.**Dependence of the coupling strength on the resonant frequency by means of the best reconstruction of experimental patterns.

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**MDPI and ACS Style**

Tseng, Y.-C.; Hsu, Y.-H.; Lai, Y.-H.; Yu, Y.-T.; Liang, H.-C.; Huang, K.-F.; Chen, Y.-F.
Exploiting Modern Chladni Plates to Analogously Manifest the Point Interaction. *Appl. Sci.* **2021**, *11*, 10094.
https://doi.org/10.3390/app112110094

**AMA Style**

Tseng Y-C, Hsu Y-H, Lai Y-H, Yu Y-T, Liang H-C, Huang K-F, Chen Y-F.
Exploiting Modern Chladni Plates to Analogously Manifest the Point Interaction. *Applied Sciences*. 2021; 11(21):10094.
https://doi.org/10.3390/app112110094

**Chicago/Turabian Style**

Tseng, Yu-Chen, Yu-Hsin Hsu, Yu-Hsiang Lai, Yan-Ting Yu, Hsing-Chih Liang, Kai-Feng Huang, and Yung-Fu Chen.
2021. "Exploiting Modern Chladni Plates to Analogously Manifest the Point Interaction" *Applied Sciences* 11, no. 21: 10094.
https://doi.org/10.3390/app112110094