## Preliminary Investigations
### Simulations  
The goal of the simulations was to estimate the zones of quiet around the error microphones assuming a direct sound field. Considering broadband control, the zone of quiet can be estimated by applying tonal noise of different frequencies, covering the entire bandwidth. As the zone shrinks when increasing the frequency of the sound, the most careful simulations had to be carried out near the cut-off frequency of the designed bandwidth, i.e., at a frequency of 1 kHz. The calculations have been carried out with two important simplifications. First, the sound field has been calculated assuming far field, i.e., the sound pressure level is inversely proportional to the distance from the source. This assumption is not correct for the secondary sources at low frequencies, but fair near 1 kHz. On the other hand, it is expected that the zones of quiet are large enough at low frequencies, and their extent is crucial only at higher frequencies. Hence, the simulations are informative in the most important cases. The second simplification is that the influence of the user's head is ignored. This choice is based on the fact that the head starts to shadow the ear farther away from the sound only at frequencies above 1500 Hz [12]. Thus, simple simulations for active noise control can neglect the effect of the head. As there are two secondary sources and two error microphones, the sound field generated by the secondary loudspeakers perfectly cancels the primary noise at the error microphones. Our noise-canceling chair is based on the X Rocker Gaming Chair [13]. The relative positions of the secondary sources and the user's ears, as well as the error microphones have been set according to the size of the chair. The dimensions of the head and the positions of the loudspeakers can be seen in Figure 5. The loudspeakers on the left are indicated by two co-axial circles. The positions of the sources and the microphones can be seen in Figure 6. The list of symbols can be seen in Figure 2.  
Plenty of simulations have been carried out, where the frequency of the noise, the position of the error microphones and the direction of the primary source were the variables. Here, only five simulations are presented. In all the cases introduced here, the distance of the primary source form the chair is 3 m. The simulations cover a 1 m \(\times\) 1 m area in the \(x\)-\(y\) and the \(x\)-\(z\) plane, respectively. In the first case the frequency of the noise is set to 500 Hz. The error microphones are \(d=2\) cm away in the \(x\) direction from the ears, while the angle of the primary source is \(\alpha=0^{\circ}\). The result of the simulation can be seen in Figure 7. The pictures show the level of the suppression in the \(x\)-\(y\) and the \(x\)-\(z\) plane, and the color bar on the right helps to evaluate the level in dB. The zone of quiet is spread to the entire headrest, so the positions of the error microphones are not crucial. Figure 5: Position of the secondary sources relative to the head. In the following simulations, the frequency of the noise is set to 1 kHz. In the second case, the other settings are the same as above: \(d=2\) cm, \(\alpha=0^{\circ}\). The result of the simulation can be seen in Figure 8. The suppression map clearly shows the reduced zone of quiet. Figure 8: Result of the second simulation. \(f=1\) kHz, \(d=2\) cm, \(\alpha=0^{\circ}\). Left plot: \(x\)-\(y\) view. Right plot: \(x\)-\(z\) view. Figure 6: Positions of the sources and the microphones relative to the head. Figure 7: Result of the first simulation. \(f=500\) Hz, \(d=2\) cm, \(\alpha=0^{\circ}\). Left plot: \(x\)-\(y\) view. Right plot: \(x\)-\(z\) view. The third simulation deals with a similar arrangement, and the only difference is that the error microphones are \(d=4\) cm away from the ears. The result of the simulation can be seen in Figure 9.  
In the fourth case, the distance between the error microphones and the ears remains \(d=4\) cm, but the angle of the primary source has been changed to \(\alpha=45^{\circ}\).