## 3 Results
### Algorithm Performance
#### 3.4.4 Propagation Time  
In previous sections, it was already explained that the 2D FDTD calculation needs a sinusoidal input signal due to the periodic boundaries. One main problem of this configuration is the time that the signal needs to spread completely at high incidence angles, as can be seen in Figure 15, where space-time plots are presented. Before signal stabilization, a ripple effect is visible in the plots.  
With a view to avoid this ripple, and possible pressure voids, simulation time can be increased in order to adequately stabilize the pressure response.  
Figure 16 shows the error made at 250 Hz, 500 Hz and 1 kHz absorption calculation at different angles as a function of the simulation time. It can be seen that for low angles, the absorption coefficient becomes accurate in most cases, after a simulation time below 50 ms. For higher angles, the ripple effect becomes noticeable, but increasing propagation time helps to stabilize the pressure so that convergence to the final absorption coefficient can be reached within a given tolerance. To help in the analysis, the reception signal may be averaged for the last milliseconds of each simulation, avoiding the initial time where the signal did not reach the NEM interface, and to prevent possible spatial pressure variations. In the Figure 16 plots, mean squared error (MSE) variation in the absorption coefficient estimation is shown as a function of the simulation time, clearly revealing that the error is progressively reduced and that the absorption coefficient converges to the measured value. For higher angles, these curves are clearly more oscillatory, indicating that the convergence process is more irregular, although convergence is still reached. To allow a better evaluation of a possible convergence criterion, avoiding the effect of those oscillations, a smoothed MSE curve was also computed (figure green lines). Making use of this approach, even for the more challenging case presented (1 kHz and 80\({}^{\circ}\)), it is possible to  
Figure 14: The 60 mm melamine sheet sound absorption at random incidence using different domain widths (Y-direction) and a constant value of X-space set to \(\lambda_{0}/\cos(\theta)\).  
evaluate a convergence criterion with simplicity and select the stop point in the simulation process based on this criterion.  
Figure 15: Space–time plots for a 60 mm melamine sheet at 100 Hz and different incidence angles. Absolute pressure values are represented with [0–300] ms time interval and different incidence angles, (**a**) 40\({}^{\circ}\), (**b**) 70\({}^{\circ}\), (**c**) 80\({}^{\circ}\). White continuous line represents the first NEAM layer, dashed line \(C_{0}\), and dotted-dash \(C_{0}\)-\(\cos(\theta)\). Note that the tube length increases with the incidence angle.  
Figure 17 shows the propagation time needed for an entire simulation with an MSE of \(10^{-3}\) as a stop criterion. In this color plot, in which white and black shades correspond, respectively, to smaller and higher simulation times, it is clear that, except for very high incidence angles and low frequencies, simulation times below 50 ms are sufficient to obtain the required tolerance.  
Figure 16: Each figure, computed for a 60 mm melamine sheet, is divided in three rows: (1) Absolute space–time pressure. (2) Diffuse absorption coefficient calculated using different true values. (3) Mean Squared Error (MSE) with the black line representing the error using the last time pressure to calculate the true value of the absorption coefficient, and dotted green line a 10% sample smooth with an average of the last 150 ms as the true value. The different cases shown correspond to (**a**) 250 Hz at 20\({}^{\circ}\), (**b**) 500 Hz at 40\({}^{\circ}\), (**c**) 1 kHz at 80\({}^{\circ}\).