# Exploration of Resonant Modes for Circular and Polygonal Chladni Plates

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Resonant Mode Chladni Patterns

#### 2.1. Impedance Experiment to Determine Resonant Modes

_{f}, was calculated as a function of frequency from the variable voltage output, V

_{f}, and the base current, I, via the formula Z

_{f}= V

_{f}/I. The effective impedance was then determined by subtracting the impedance, found in the case where no plate was attached, from the impedance found when the plate was attached. Figure 2a–e shows the results for each plate, where the impedance is shown as a function of frequency and the peaks indicate the resonant frequencies in each case. The resonant frequencies found from the impedance peaks are shown in Table 1.

#### 2.2. Chladni Plate Vibration at Resonant Mode Frequencies

## 3. Theoretical Determination of the Nodal Line Patterns

_{1}is the outward wave; z

_{2}is the reflected wave; z

_{r}is the resultant wave; A is the amplitude; ω is the angular frequency; and ϕ is the phase difference between the outward and reflected wave.

_{r}in Equation (3) to equal 0, i.e.,

_{r}= 0, φ = π/2 (note the polar coordinate φ is not the same as the phase difference ϕ defined earlier). We can, therefore, assume that r only exists in the x, y plane, which physically represents the spatial dimension of the plate. If we then consider the spatial dimension of the plate in terms of time, we can define r as,

_{θ}is the spatial dimension given as a function of θ; t is the time it takes for the wave to traverse the distance L

_{θ}; and v is the velocity of the wave. Note, the velocity of a sound wave is ~340 ms

^{−1}in air (at average room temperature); ~1500 ms

^{−1}in water; and, for solids, it is given as $v=\sqrt{\frac{Y}{\rho}}$, where $Y=\frac{\sigma \left(\epsilon \right)}{\epsilon}=\frac{F/A}{\Delta L/{L}_{0}}$ is the Young’s Modulus given in pascals (Pa); σ(ε) is a measure of stress; ε is a measure of strain; and ρ is the density. However, in the case of the Chladni plate, we are not interested in the velocity of the sound wave. Instead, we are interested in the velocity of the mechanical wave which lifts the sand grains up and down, which can be given as v = λf. For a circular plate of known diameter, the wavelength, and hence velocity, can be determined from the observed nodal line patterns. For the circular plate, the resulting velocity as a function of frequency was thus found to be,

_{R}, such that Equation (5) becomes,

#### 3.1. Circular Chladni Plate

_{R}, as a simple coefficient, which we call

**C**(see Table 2 for the list of coefficients). The nodal points making up the nodal line patterns are then determined from Equation (8) and by setting the velocity v, as given by Equation (7) and the boundary condition in terms of L = a/2 (where a is the diameter of the circular plate). In each case, the coefficient,

**C**, was first determined by manually adjusting the value until the theoretical nodal line patterns match those of the experiment. This was performed both visually and by using a fitting function that adjusts the coefficients according to the fit. Note that all equations and boundary conditions were implemented using a custom software application created in Python 3.10.

**C**coefficient, which increases with increasing frequency and decreases with an increasing number of nodal lines (see Table 2 for the full list of coefficients).

#### 3.2. Square Chladni Plate

_{θ}varies as,

_{R}, for the square plate, or any other polygon, will thus depend on the variability in the length of the plate, with respect to the centre, θ, as well as the distance from the centre, r. To investigate this further, and as a first approximation, we set the response function in the form of a trigonometric function, e.g., a rose function,

_{R}as given by Equation (11), the boundary condition in terms of L

_{θ}as given by Equation (10), and the velocity v, as given by Equation (7), we recreated the nodal line patterns in a good approximation with the experimental results (see Figure 15 and Figure 16). It is interesting to note that, depending on the position of the nodal line relative to the plate centre, we see lesser or greater effects of the response function variability. For example, at a critical closeness to the centre, we see no variability in the response function, and it acts in the same way as for a circular plate. Then, as we move further away, it exhibits more variability, until reaching a critical distance away from the centre, where again we again see no variability in the response function. For a specific plate, this is clearly expressed in the value of the

**A**coefficient, which appears to oscillate between positive and negative values with the increasing number of nodal lines. As well as the circular plate, the value of the

**C**coefficient increases with increasing frequency for a specific nodal line (defined as

**C**

_{r}) and decreases with an increasing number of nodal lines for a specific frequency (see Table 2 for the full list of coefficients). For the higher frequencies, we can see the beginnings of mode mixing, not yet fully effecting. As we move into increasingly higher frequencies, this will be more evident.

#### 3.3. Triangular Chladni Plate

_{θ}varies as,

_{R}as given by Equation (11) with N = 3, the boundary condition in terms of L

_{θ}as given by Equation (13), and the velocity v, as determined from the experimental analysis, as given by Equation (7), we recreate the nodal line patterns in a good approximation with the experimental results (see Figure 17 and Figure 18). It is interesting to note that, as with the square plate, we see the same progression of pattern complexity, with the value of the coefficients following a similar relationship to that of both the circle and the square plate.

#### 3.4. Pentagon Chladni Plate

_{θ}varies as,

_{R}as given by Equation (11) with N = 5, the boundary condition in terms of L

_{θ}as given by Equation (15), and the velocity v, as given by Equation (7), we recreate the nodal line patterns in a good approximation with the experimental results (see Figure 19 and Figure 20). It is interesting to note that, again, like the other polygon plates, we see the same progression of pattern complexity, with the value of the coefficients following a similar relationship to that of both the circle and the other polygon plates.

#### 3.5. Hexagon Chladni Plate

_{θ}varies as,

_{R}as given by Equation (11) with N = 6, the boundary condition in terms of L

_{θ}as given by Equation (17), and the velocity v, as given by Equation (7), we recreate the nodal line patterns in a good approximation with the experimental results (see Figure 21 and Figure 22). It is interesting to note that, again, like the other polygon plates, we see the same progression of pattern complexity, with the value of the coefficients following a similar relationship to that of both the circle and the other polygon plates.

## 4. Summary

**C**coefficients, for a specific nodal line (defined as

**C**

_{r}), increases with increasing frequency and, for a specific frequency, decreases with an increasing number of nodal lines. For the polygonal plates, the value of the

**A**coefficient, for a specific nodal line (defined as

**A**

_{r}), appears to oscillate between positive and negative values (or between higher and lower values), tending to zero as the frequency increases. Moreover, for a specific frequency, the

**A**coefficient decreases with an increasing number of nodal lines. However, at the higher frequencies, there are a couple of exceptions to this relationship (i.e., the hexagon a = 12 cm, f = 1635 Hz; and the pentagon a = 14.5 cm, f = 1071 Hz). These exceptions are most likely due to the effects of mode mixing. Note, with the range of frequencies investigated in this study, we can only see the beginnings of mode mixing, which will become more evident as we investigate increasingly higher frequencies. For these effects to be accounted for and a quantitative relationship to be made, more investigations yielding a larger dataset are required and will be carried out in a future study.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Plots of the effective impedance of a mechanical oscillator with a plate attached shown as a function of frequency for the following plates: (

**a**) large circle; (

**b**) small circle; (

**c**) large square; (

**d**) small square; (

**e**) large triangle; (

**f**) small triangle; (

**g**) large pentagon; (

**h**) small pentagon; (

**i**) large hexagon; and (

**j**) small hexagon. The peaks indicate the resonance frequencies which are listed in Table 1 below.

Shape | Circle | Square | Triangle | Pentagon | Hexagon | |||||
---|---|---|---|---|---|---|---|---|---|---|

Size (cm) | 24 | 18 | 24 | 18 | 24 | 18 | 14.5 | 9.5 | 12 | 9 |

f_{1} (Hz) | 158 | 160 | 107 | 118 | 152 | 176 | 112 | 228 | 174 | 174 |

f_{2} (Hz) | 582 | 616 | 283 | 326 | 426 | 477 | 381 | 895 | 630 | 682 |

f_{3} (Hz) | 1378 | 511 | 592 | 1159 | 1402 | 604 | 1137 | |||

f_{4} (Hz) | 744 | 1071 | 1635 | |||||||

f_{5} (Hz) | 1181 |

**Table 2.**Coefficients

**A**and

**C**of the response function for each of the 10 plates and for each frequency.

Coefficients | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

Shape (cm) | Frequency (Hz) | Velocity (m/s) | A1 | A2 | A3 | A4 | C1 | C2 | C3 | C4 |

circle 24 | 158 | 35.6 | 0.95 | |||||||

582 | 68.2 | 1.35 | 1 | |||||||

1378 | 105.0 | 2.1 | 1.15 | 0.75 | ||||||

circle 18 | 160 | 35.8 | 1.65 | |||||||

616 | 70.2 | 2.4 | 2.25 | |||||||

square 24 | 107 | 29.3 | −0.1 | 0.95 | ||||||

283 | 47.6 | 0.05 | −1.85 | 1.25 | 0.6 | |||||

511 | 63.9 | 0 | 1.7 | 1.6 | 0.7 | |||||

744 | 77.1 | 0 | −1.2 | −1.3 | 2.7 | 1.1 | 0 | |||

1181 | 97.2 | 0 | 0.2 | −0.55 | −5.8 | 2.7 | 1.45 | 1.35 | −3.65 | |

square 18 | 118 | 30.7 | −0.05 | 1.55 | ||||||

326 | 51.1 | 0 | −0.9 | 2.3 | 2.2 | |||||

592 | 68.8 | 0 | 0.6 | 2.5 | 2.4 | |||||

triangle 24 | 152 | 34.9 | −0.2 | 1.4 | ||||||

426 | 58.4 | 0.1 | −1.4 | 1.6 | 0.8 | |||||

1159 | 96.3 | 0 | 0.7 | −2.3 | 2.7 | 1.6 | −0.8 | |||

triangle 18 | 176 | 37.5 | −0.2 | 1.75 | ||||||

477 | 61.8 | 0 | −0.95 | 2.15 | 2 | |||||

1402 | 105.9 | 0 | 0.6 | −2.8 | 2.8 | 2.6 | 0 | |||

pentagon 14.5 | 112 | 29.9 | 0 | 1.7 | ||||||

381 | 55.2 | 0 | −0.4 | 2.3 | 2.3 | |||||

604 | 69.5 | 0 | 0.7 | −2.9 | 2.8 | 2.4 | 2 | |||

1071 | 92.6 | 0 | −0.2 | 1.1 | 2.9 | 2.6 | 2.6 | |||

pentagon 9.5 | 228 | 42.7 | 0 | 2 | ||||||

895 | 84.6 | 0 | −0.3 | 2.7 | 2.35 | |||||

hexagon 12 | 174 | 37.3 | 0 | 1.1 | ||||||

630 | 71.0 | 0 | −2 | 1.85 | 1.25 | |||||

1137 | 95.4 | 0 | 0.5 | −2.5 | 2.55 | 1.4 | 1.3 | |||

1635 | 114.4 | 0 | −0.15 | 1 | 2.8 | 1.6 | 1.1 | |||

hexagon 9 | 174 | 37.3 | 0 | 1.9 | ||||||

682 | 73.9 | 0 | −0.1 | 2.65 | 2.45 |

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**MDPI and ACS Style**

Val Baker, A.; Csanad, M.; Fellas, N.; Atassi, N.; Mgvdliashvili, I.; Oomen, P.
Exploration of Resonant Modes for Circular and Polygonal Chladni Plates. *Entropy* **2024**, *26*, 264.
https://doi.org/10.3390/e26030264

**AMA Style**

Val Baker A, Csanad M, Fellas N, Atassi N, Mgvdliashvili I, Oomen P.
Exploration of Resonant Modes for Circular and Polygonal Chladni Plates. *Entropy*. 2024; 26(3):264.
https://doi.org/10.3390/e26030264

**Chicago/Turabian Style**

Val Baker, Amira, Mate Csanad, Nicolas Fellas, Nour Atassi, Ia Mgvdliashvili, and Paul Oomen.
2024. "Exploration of Resonant Modes for Circular and Polygonal Chladni Plates" *Entropy* 26, no. 3: 264.
https://doi.org/10.3390/e26030264