# Development and Testing of an Active Noise Control System for Urban Road Traffic Noise
## 3 Development of the Active Noise Control Prototype System  
### Principles of Active Noise Control  
Active noise control (ANC) is a potent noise reduction technology, operating on the principle of sound wave interference; that is, when two sound waves overlap at the same spatial location, they interact with each other based on the principle of superposition [47]. ANC achieves noise reduction by generating a secondary noise source with equal amplitude and opposite phase to the original noise source, thereby causing them to cancel each other out and effectively reducing the noise level.  
This process can be expressed mathematically. If the phases of two sound waves of the same frequency at a certain location in space are \(\varphi_{1}\) and \(\varphi_{2}\), their sound pressure amplitude values are:  
\[\left\{\begin{aligned} p_{1}&=p_{1a}\text{cos}(\omega t -\varphi_{1})\\ p_{2}&=p_{2a}\text{cos}(\omega t-\varphi_{2})\end{aligned}\right. \tag{1}\]  
According to the principle of superposition, the total sound pressure is the sum of their individual sound pressures:  
\[p=p_{1}+p_{2}=p_{1a}\text{cos}(\omega t-\varphi_{1})+p_{2a}\text{cos}(\omega t -\varphi_{2})\text{= }p_{a}\text{cos}(\omega t-\varphi) \tag{2}\]  
where,  
\[\left\{\begin{aligned} p_{a}^{2}=p_{1a}^{2}+p_{2a}^{2}+2p_{1a}p_{2a }\text{cos}(\varphi_{2}-\varphi_{1})\\ \varphi=arctan\frac{p_{1a}sin\varphi_{1}+p_{2a}sin\varphi_{2}}{p_{1a }\text{cos}\varphi_{1}+p_{2a}cos\varphi_{2}}\end{aligned}\right. \tag{3}\]  
The sound pressure amplitude of the composite sound wave depends on the phase difference \(\varphi_{2}-\varphi_{1}\) of the two sound waves, and the square of the sound pressure amplitude reflects the average sound energy density \(\overline{z}\) in the sound field.  
\[\overline{z}=\frac{p_{a}^{2}}{2\rho_{0}c_{0}^{2}} \tag{4}\]  
Taking the time average of the above equation yields the average energy density of the composite sound wave, i.e.,  
\[\overline{z}=\overline{z_{1}}+\overline{\varepsilon_{2}}+\frac{p_{1a}p_{2a}} {\rho_{0}c_{0}^{2}}\text{cos}(\varphi_{2}-\varphi_{1}) \tag{5}\]  
Here, if \(p_{1a}=p_{2a}\), and \(\varphi_{2}-\varphi_{1}=\pm\pi,\pm 3\pi\ldots\) equals any odd multiple of \(\pi\), then the composite sound pressure amplitude and the average sound energy density are zero, with the two sound waves canceling each other out:  
\[\left\{\begin{aligned} p_{a}=p_{1a}-p_{2a}=0\\ \overline{z}=\overline{\varepsilon_{1}}+\overline{\varepsilon_{2}} -\frac{p_{1a}p_{2a}}{\rho_{0}c_{0}^{2}}=\frac{p_{1a}^{2}}{2\rho_{0}c_{0}^{2} }+\frac{p_{2a}^{2}}{2\rho_{0}c_{0}^{2}}-\frac{p_{1a}p_{2a}}{\rho_{0}c_{0}^{2} }=0\end{aligned}\right. \tag{6}\]  
When two sound waves have equal amplitude and their phase difference is \(\pi\) (or any odd multiple of \(\pi\)), the two sound waves completely interfere with each other, resulting in a total sound pressure of 0. This means that the total sound energy density at this location is also 0, i.e., the sound energy density of the composite sound field at this location is zero.  
To eliminate the noise at a certain location, a sound wave opposite to the noise can be emitted so that their phase difference is \(\pi\), thereby obtaining a virtually silent zone at that location. As low-frequency sound waves have long wavelengths, ANC systems can accurately detect and cancel out such noise, giving ANC a natural advantage in dealing with low-frequency noise. By contrast, high-frequency noise, with its shorter wavelength and rapid spatial changes, poses challenges to the practical application of ANC. Since road traffic noise is primarily low-frequency, ANC technology has shown obvious advantages in this field.