**3. Comparison of Various Bending Plate Theories**

In this paper, the comparisons between the governing equation for the bending plate and the governing equations for various classical bending plates are presented in Table 1.


**Table 1.** Comparison of different theories of plate bending.

The dispersion equation based on the Lagrange–Germain plate theory (CPT) can be described as:

$$\frac{c}{c\_2} = \sqrt{\frac{4\pi^2}{6(1-\nu)}\frac{h}{\lambda}}.\tag{52}$$

where *λ* is the wavenumber of an elastic wave.

The dispersion equation in consideration of the moment of inertia can be described as:

$$\frac{c}{c\_2} = \sqrt{\frac{4\pi^2}{6(1-\nu)} \left[1 + \frac{\pi^2}{3} \left(\frac{h}{\lambda}\right)^2\right]^{-1}} \frac{h}{\lambda}. \tag{53}$$

When the moment of inertia and shear deformation are involved, the implicit dispersion equation can be described as:

$$
\frac{\pi^2}{3} \left( \frac{h}{\lambda} \right)^2 \left( 1 - \frac{c^2}{\kappa^2 c\_2^2} \right) \left( \frac{c\_p^2}{c^2} - 1 \right) = 1. \tag{54}
$$

The implicit dispersion equation based on the three-dimensional elasto-dynamics theory can be described as:

$$\frac{4c\_2^2\sqrt{\left(c\_2^2-\kappa c^2\right)\left(c\_2^2-c^2\right)}}{\left(2c\_2^2-c^2\right)^2} = \frac{\tanh\left(2\pi\frac{h}{\lambda}\sqrt{c\_2^2-\kappa c^2}\right)}{\tanh\left(2\pi\frac{h}{\lambda}\sqrt{c\_2^2-c^2}\right)}.\tag{55}$$

The dispersion equation based on the exact plate theory in this paper is:

$$a^4 - (2 - v)k\_2^2 a^2 + (1 - v)k\_2^2 \left[ \frac{k\_2^2 (7 - 8v)}{8(1 - v)} - \frac{6}{h^2} \right] = 0. \tag{56}$$

## **4. Discussion of the Exact Plate Theory**

In this paper, the derived plate bending vibration equation is compared with the classical corresponding equation. The comparison of the specific equation form is shown in Table 1. In the process of comparison, the equation form in the frequency domain is used. The bending vibration equation of plates presented in this paper is similar to other classical bending vibration equations of plates. When the statics problem is studied, the elastic vibration equation of plates derived in this paper degenerates into an exact equation of the static bending of plates.

According to those dispersion equations above, Figure 1 is drawn to compare the dispersion curves; Figure 2 is drawn to compare the scattering wave numbers. From Figure 1, we can see that the dispersion curves based on the classic thin plate theory and Mindlin plate theory are far apart from the three-dimensional elasto-dynamics theory, but the dispersion curves by the exact plate theory in this paper are very close to the dispersion curves based on the three-dimensional elasto-dynamics theory. By comparing those curves, the superiority of the exact plate theory to other plate theories is obvious.

As can be seen in Figure 2, the scattering wave number *α*<sup>1</sup> obtained by the exact plate theory and Mindlin plate theory is very close, but the scattering wave number *α*<sup>2</sup> obtained by the exact plate theory is quite different from that of the Mindlin plate theory. With the increase of the vibration frequency, the scattering wave number becomes greater. It can be seen that the scattering wave number *α*<sup>1</sup> at any frequency is greater than zero, so it can be concluded that the wave mode is in the propagation region. Nevertheless, when the frequency is low, the scattering wave number *α*<sup>2</sup> is less than zero, and the wave mode is in the cutoff frequency domain, which is called a localized standing wave. When the frequency is high, the scattering wave number *α*<sup>2</sup> is greater than zero, and the wave mode is in the propagation region, which is called a propagating wave. According to Reference [7], it can be seen that the applicable frequency interval of the Mindlin plate theory is *ω*/*ω*<sup>0</sup> < 1, i.e., *h*/*λ*<sup>2</sup> < 0.5, and the application of the Mindlin plate theory is limited. The dynamic model proposed in this paper is completely based on the three-dimensional elasto-dynamics theory, and consequently, its limitations of application are minor.

**Figure 1.** Dispersion relation by the different theories.

**Figure 2.** Wavenumbers by the different theories.
