# Research on the Processing Method of Acoustic Focusing Cavities Based on the Temperature Gradient
## 3 Numerical Results and Discussion  
Acoustic Pressure and Intensity Distribution of Transmitted Wave Focusing under Discrete Temperature Gradient  
Based on the parameters and boundary conditions set in Section 2.4, the acoustic pressure distribution of the plane wave in the transmission unit at different temperatures was simulated, using the multi-physics simulation software COMSOL, as shown in Figure 3.  
Figure 2: Schematic diagram of transmission unit. It can be seen from the numerical simulation results that when the acoustic wave was transmitted into the constant-temperature transmission unit, the sound of speed increased with the increase in temperature; furthermore, when the frequency was constant, the wavelength of the acoustic wave increased. According to the wavelength formula and the sound of speed in air being 343 m/s, the distance of the acoustic wave before transmission was greater than one wavelength, but the distance of the acoustic wave in the transmission unit with constant temperature was obviously less than one wavelength. It can be concluded that the same frequency of the acoustic source, when launched at the same time, would not result in the same distance, reflecting the influence of different temperatures on the acoustic phase. Based on the propagation law of the plane wave in the transmission units at different temperatures, the transmission waves were focused by arranging the transmission units with different temperatures. The model is shown in Figure 4.  
Taking \(y=80\) cm, the speed of sound at different temperatures was introduced into Equations (1) and (5), along with the frequencies of acoustic waves and a focal length of 30 cm, to obtain the phase change corresponding to different temperatures. Temperatures of 430 k, 530 k, 650 k, 760 k, 850 k, 970 k and 1190 k corresponded to phase delays of \(7\pi/4\),\(3\pi/2\),\(5/4\pi\),\(\pi\),\(3\pi/4\),\(\pi/2\), and \(\pi/4\), respectively, as shown in Figure 5. It can be seen that the transmission wave produced a phase delay when the acoustic wave passed through the thermoacoustic phase control unit. Figure 4: Distribution of the metasurface on the \(y\)-axis. Figure 3: Acoustic pressure distribution of transmission unit at different temperatures. The function of Equation (5) was drawn, using the Origin drawing software. Seven kinds of transmission units with different temperatures corresponded to seven kinds of phase delays of acoustic waves. The distribution of transmission units is shown in Figure 6.  
As can be seen in Figure 6, from \(y=40\) cm to \(y=0\) cm, the fluctuating temperature led to the transmission angle having a positive value before gradually decreasing. From \(y=0\) cm to \(y=-40\) cm, the fluctuating temperature led to the opposite effect. On the \(y\)-axis, the transmission angle at both ends was larger, and the transmission angle in the middle was smaller, forming a gradient distribution of transmission angle on the \(y\)-axis. By establishing a model in COMSOL software and setting the parameters and boundary conditions, the acoustic pressure and acoustic intensity distribution of the transmitted wave focusing under a discrete temperature gradient could be obtained, as shown in Figures 7 and 8. Figure 5: The delay of acoustic phase corresponding to temperature. Figure 6: Distribution of transmission units in the \(y\)-direction. From Figure 8, it can be concluded that the focal length was about 30 cm, thus verifying the previously set parameters. Compared with the numerical simulation, the acoustic focusing model obtained by combining the discrete temperature gradient and metasurface showed great advantages. It facilitated the focusing of the acoustic wave [32; 33], as well as expanded its formation during inner cavity processing. In order to overcome the limitation of a single-sound source frequency on the acoustic focusing effect, temperature control was used to solve the problem of acoustic impedance matching so as to achieve a wideband acoustic focusing effect.