# Research on the Processing Method of Acoustic Focusing Cavities Based on the Temperature Gradient
## 3 Numerical Results and Discussion
### Focusing of Reflected Waves under Discrete Temperature Gradients  
First, a plane wave propagation model was established in a single reflection unit as shown in Figure 11.  
It can be seen from Figure 11 that the plane wave propagates along the \(x\)-direction, before being transmitted into the reflecting unit. Upon meeting the rigid boundary, most of it is reflected back, before finally being transmitted from the reflecting unit to the air at normal temperature. Figure 11: Schematic diagram of plane wave propagation in reflection unit. Figure 10: Spatial distributions of acoustic intensity field through focusing lens at (**a**) 28 kHz and (**b**) 35 kHz. Boundaries 1-5 were all set as plane wave radiation. An incident pressure field was additionally added to boundary 1 to represent the sound source. The amplitude of sound pressure was 1 Pa. Boundary 6 was set as a rigid, hard sound field boundary. The acoustic source was a plane wave with a frequency of 20 kHz, and the medium was air. Seven reflection units with different temperatures were adopted (460 k, 590 k, 730 k, 850 k, 990 k, 1120 k, and 1260 k). The simulation results are shown in Figure 12.  
According to the numerical simulation and Equation (5), it can be seen that the speed of sound could be transformed into the phase difference of acoustic waves, and the temperature changed the phase of sound waves by changing the speed of sound, thus controlling the speed of sound wave propagation. Consider a focal length of 25 cm, y value of 80 cm, frequency of 20 kHz, and sound of speed corresponding to 460 k, 590 k, 730 k, 850 k, 990 k, 1120 k and 1260 k. Introducing these values into Equations (1) and (5) yields the phase delays corresponding to different temperatures, as shown in Figure 13.  
The reflected wave focusing model is shown in Figure 14.  
Figure 12: Acoustic pressure distribution of plane wave in seven reflection units with different temperatures. Figure 13: Phase delay corresponding to different temperatures. Combined with Equation (4), it can be further obtained that the temperature changed the speed of sound, thus changing the deflection angle of the sound wave. When the sound wave propagated to the rigid boundary, most of it was reflected back. When transmitted to the air at normal temperature, the transmission angle of the \(y\)-axis presented a gradient distribution, thereby focusing the acoustic wave forms. When the frequency was set to 20 kHz, the acoustic intensity distribution of the focused reflection wave in the simulation was as shown in Figure 15.  
It can be seen from Figure 15 that the focal length was about 25 cm, which agrees with the previously set focal length value. Compared with the acoustic intensity at the focal point formed by the transmitted wave, the acoustic intensity at the focal point of the reflected wave was significantly smaller [34]. Because the focus formed by the reflected wave was transmitted three times, there was some loss in the reflection of the sound wave at the rigid boundary. Furthermore, the influence of different frequencies on the acoustic intensity at the focal point was numerically simulated by a frequency sweep in COMSOL, and the simulation results are shown in Figure 16.  
Figure 11:0 Reflection wave focusing model diagram. Figure 11:1 Spatial distribution of acoustic intensity field through focusing lens at 20 kHz. It can be seen from Figure 16 that the acoustic intensity changed non-linearly as the frequency increased. This is because different frequencies resulted in different phases of the reflected waves, and the phase difference of the reflected waves of different phases at the focus point was not zero. As a result, the acoustic intensity at the focus point was in the 28 kHz to 35 kHz frequency band, and the acoustic intensity gradually decreased, whereas the acoustic intensity gradually increased between 40 kHz and 45 kHz. However, with the increase in frequency of acoustic waves, the focusing effect could not be formed upon reaching 50 kHz.