# Analysis of the Effect of Temperature on the Sound Transmission Loss of a Curved Plate
## 3 Model Validation
Note that the theoretical calculation results of both the modal frequency and STL agree well with the simulation results under different temperature conditions, indicating that the theoretical model can be used to analyze the modal frequency and STL of a curved plate. Furthermore, to compare the differences between the present study and previous solutions, the parameters in [13] were used. Thus, the results of the theoretical calculations in this study and those in [13] are given in Figure 6. The figure shows that the present study agrees well with the calculation results in [13]. Figure 4: Comparison of the theoretically calculated and simulated values of the modal frequencies. (a) \(T_{1}=0\)\({}^{\circ}\)C, \(T_{2}=0\)\({}^{\circ}\)C. (b) \(T_{1}=-10\)\({}^{\circ}\)C, \(T_{2}=0\)\({}^{\circ}\)C.  
The frequency ratio is defined as \(\gamma=f_{T}\!/\!f_{0}\), where \(f_{T}\) is the modal frequency under various temperature gradients, and \(f_{0}\) is the modal frequency of each order at the temperature of \(T_{1}=0\)\({}^{\circ}\)C. The variation in the frequency ratio of each order of modal frequency with temperature is shown in Figure 8. The figure demonstrates that each order of modal frequency varies approximately linearly with decreasing temperature. The frequency ratio of the (2, 1)-order modal frequency changes the most, with the modal frequency increasing on average by 13.7 Hz per 10 \({}^{\circ}\)C temperature decrease. The frequency ratio of the (1, 1)-order modal frequency decreases the least, with the modal frequency increasing on average by 2.8 Hz per 10 \({}^{\circ}\)C temperature decrease. The curves of the STL with the frequency of the curved plate at different temperature gradients when the acoustic wave is vertically incident are shown in Figure 9. The figure shows that the trough position of the STL gradually shifts toward high frequencies as the temperature decreases. Moreover, these trough positions and the modal frequencies are con  
Figure 3.0 Variation in modal frequency with temperature for each order. Figure 3.1 Variation in modal frequencies of each order at different temperatures. sistent. In particular, the frequencies corresponding to each trough are modal frequencies. The first four trough frequencies of the STL at \(T_{1}=0\), \(-10\), \(-20\), and \(-30\)\({}^{\circ}\)C correspond to the (3, 1)-, (1, 1)-, (3, 3)-, and (5, 1)-order modal frequencies at that temperature, and the first four trough frequencies of the STL at \(T_{1}=-40\) and \(-50\)\({}^{\circ}\)C correspond to the (1, 1)-, (3, 1)-, (3, 3)-, and (5, 1)-order modal frequencies at that temperature. The corresponding modal frequencies at different temperatures are (odd, odd)-order modal frequencies. This is because the (odd, odd)-order modes radiate more efficiently with increasing frequency and are more easily excited [36]. As shown in Table 2, when \(T_{1}\) is \(-40\) and \(-50\)\({}^{\circ}\)C, the change in temperature causes an array exchange between the (3, 1)-order mode and the (1, 1)-order mode. Therefore, the first trough position is the frequency position corresponding to the (1, 1)-order mode, and the temperature changes can cause significant changes in the STL performance of the curved plate. In low-temperature environments, the low-frequency locations, where the STL is weak at the vertical incidence of acoustic waves, shift to higher frequencies.