## 3 Results and Discussion
### Effects of the Power-Law Index on the Sound Radiation Fields of P-FGM Plate  
Figure 6 depicts the effect of the power-law index, \(k\), on the sound power level and rms velocity of the FGM plate. For different values of the power-law index, the pattern variation in sound power level with frequency is similar. Furthermore, the peaks of the sound power level shift to higher frequencies.  
The influence on the SPL peaks is further investigated for the plate's normal V_rms peaks, different power-law indexes, \(k\), and plate modal indices (\(m\), \(n\)), as shown in Figures 7 and 8. It was discovered that sound power level peaks at the same frequencies correlate well with normal velocity peaks. The velocity profiles, also known as mobility response curves, depict the relationship between how sound is emitted at the FGM plate's surface. In Figure 7, the velocity profile peaks contribute significantly to the sound power level peaks. Furthermore, as illustrated in Figure 8, axisymmetric modes for various power-law indexes significantly contribute to sound power level. The modes shapes are only shown for \(k=1\) for brevity.  
Figure 5: Sound power level with present approach and ANSYS for \(k=0\), \(1\), \(5\), and \(10\).  
Figure 6: (a) SPL and (b) V_rms of clamped P-FGM plates for different \(k\)’s.  
As illustrated in Figure 7a, the modal density contribution to plate velocity is greatest for the investigated frequency range up to 2000 Hz for the power-law index \(k=0\). As the power-law index, k, increases, the modal density's contribution to plate velocity decreases, as shown in Figure 7b,c.  
Figure 9 illustrates sound power peaks variation with '\(k\)' and rms velocity. It illustrates that peak sound power level does not monotonically increase with the increase in power-law indexes. As per Figure 9, a critical value of the power-law index exists, for which the corresponding sound power level peak is minimum [29]. Because each of these peaks corresponds to a different frequency, a proper combination of the power-law index, \(k\), and frequency range will aid in proper material tailoring for the P-FGM plate's sound radiation behavior. In the low-frequency band around 231 Hz, for example, a power-law index of around five will help lower the first sound power level peak, as shown in Figure 9a. This type of information is especially important for designing for sound radiation from the P-FGM plate. It is noteworthy that as the power-law index changes, the modal frequency changes that leads to change in the contributing modes. These contributing modes have significant effects on the sound radiation fields radiated by the P-FGM plates. Due to continuous variation in contributing modes, with the change in power-law indexes, the sound power level does not always increase with increase in power-law index.  
Figure 7: Illustration of SPL and V_rms of P-FGM plates on a single plot for power-law indexes \(k=(\textbf{a})\) 0, (**b**) 1, (**c**) 5, and (**d**) 10.  
Figure 8: P-FGM plates contributing modes for \(k=0\). (**a**) Mode [1,1] 256.6 Hz; (**b**) Mode [1,3] 768.74 Hz; (**c**) Mode [3,1] 1102.2 Hz; (**d**) Mode [3,2] 1069.5 Hz; (**e**) Mode [3,3] 1561.9 Hz. (**f**) Mode [5,1] 1754.5 Hz; (**g**) Mode [1,5] 1832.2 Hz.