# Research on the Processing Method of Acoustic Focusing Cavities Based on the Temperature Gradient
## 2 Mathematical Model and Methods
### Temperature Control Focusing Theory  
In this section, different temperatures were set in the transmission unit, and the phase and transmission angle of the transmission wave were regulated by the temperature, thereby focusing the transmitted wave form. When the plane wave was incident perpendicular to the metasurfaces, the incident angle was zero. Therefore, the generalized Snell law of refraction [29] can be expressed as follows:  
\[\frac{d\varphi}{dx}=\frac{\omega}{c_{2}}\sin\theta_{4} \tag{3}\]  
Based on the inference of the refraction law in the generalized Snell law, it can be known that, when a plane wave is incident on a vertical metasurface at a normal angle [30], the transmission angle is not zero. According to Equation (3), the expression of the transmission angle can be obtained as follows:  
\[\theta_{4}=\arcsin(\frac{d\varphi}{dx}\frac{c_{2}}{\omega}) \tag{4}\]Since the arcsine function monotonically increases, when the temperature increases, the transmission angle increases; otherwise, it decreases. The transmission angle is changed by changing the speed of sound through temperature so as to realize the deflecting direction of the transmitted wave. The phase distribution formula of the metasurface is as follows:  
\[\varphi(y)=-\frac{\omega}{c}(\sqrt{y^{2}+b^{2}}-b) \tag{5}\]  
where \(\varphi(y)\) is the phase of the acoustic wave, and \(b\) is the focal length.  
According to the formula, the phase of acoustic wave can be changed by changing the speed of sound, and then the phase of the acoustic wave can be controlled by changing the temperature. Upon setting \(\mathrm{d}\varphi/\mathrm{d}x\) to \(\pi/14\) in Equation (4), Equation (5) dictates that the speed of sound increases with the increase of temperature. Thus, temperatures of 340 k, 420 k, 530 k, 650 k, 850 k, and 1230 k correspond to speeds of sound of approximately 360 m/s, 372 m/s, 383 m/s, 398 m/s, 409 m/s, and 422 m/s, respectively. When the frequency is 20 kHz, these speeds of sound introduce into Equation (4) yield transmission angles of approximately \(3^{\circ},9^{\circ},15^{\circ},21^{\circ},27^{\circ}\), and \(33^{\circ}\), respectively. When the corresponding coordinates are negative, \(\mathrm{d}\varphi/\mathrm{d}x\) is set to \(-\pi/14\), and the speeds of sound at the same temperature can be entered into Equation (4). Since the arcsine function is an odd function, this yields transmission angles of approximately \(-3^{\circ}\), \(-9^{\circ}\), \(-15^{\circ}\), \(-21^{\circ}\), \(-27^{\circ}\), and \(-33^{\circ}\), respectively. It is concluded that the projection angle of the acoustic wave under the effect of temperature control shows a gradient distribution in the \(y\)-direction; hence, combined with Equation (5), the corresponding phase delay is \(3\pi/2,5\pi/4\), \(\pi,3\pi/4\), \(\pi/2\), and \(\pi/4\), respectively. Through the superposition of acoustic waves, the acoustic d intensity at the focal point is maximized. The schematic diagram of the temperature-controlled focusing principle is shown in Figure 1.  
It can be seen from Figure 1 that the high-temperature area has a large acoustic wave transmission angle, a smaller phase delay, and a larger speed of sound, but the acoustic wave travels farther. The low-temperature area has a smaller acoustic wave transmission angle, a greater phase delay, and a slower speed of sound, but the distance is relatively short; thus, the acoustic waves eventually arrive at the same point in the same phase to form an acoustic wave gathering.  
Figure 1: Schematic diagram of temperature-controlled focusing principle.