## 5 Conclusions  
The results of several experimental works have widely demonstrated the importance of the correct velocity of air carrying agrochemicals drops to the canopy of an orchard or vineyard, which must be protected. It has been known that a slow air jet compromises the uniformity of the spray deposition, while an excessive air speed produces significant losses of pesticide to the environment (\(\geq\)50%).  
An equation was obtained in previous work [8] to predict the decay of air velocity during tree row crossings in an attempt to furnish a mathematical instrument to improve the interpretation of previous experimental data and to foresee the development of a self-adjusting system for orchard and vineyard fan sprayers. The velocity decay equation that was proposed was not valid under a limit velocity of \(v_{x}=3\) m/s because the equation was obtained assuming an inverse relationship between the drag coefficient \(c_{r}\) and velocity \(v_{x}\).  
Figure 5: The maximum velocity (centerline) of the air jet in the foliage, predicted by simplified combination (11) (- - Q- - -), is shown in comparison to the predicted maximum velocity by Equation (6) (- - - -) and to the experimental value (\(*\)).  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \(\mathbf{n}^{o}\) **Linear Piecewise \(\mathbf{n}\)** & \(\Delta\mathbf{v_{x}}\) **(m/s)** & \(\mathbf{a}\) & \(\mathbf{b}\) **(m/s)** & \(v_{xin}\) **(m/s)** & \(\mathbf{x_{in}}\) **(m)** & \(v_{xen}\) **(m/s)** \\ \hline
1 & \(\geq\)2 & 0 & 1.10 & 7.70 & 0.71 & 2.00 \\
2 & \(2+0\) & 0.55 & 0 & 2.00 & 1.59 & 0 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Values of the constants \(a\) and \(b\) of the emulator curve and \(v_{xin}\), \(x_{in}\), and \(v_{xen}\) for each linear piecewise \(n\).  
However, a previous experiment [8] has shown that, in some situations when the vegetation is highly developed, the air speed decreases under this limit of 3 m/s.  
Therefore, finding a solution to the more general non-linear ODE is important in order to have a velocity decay equation for the entire domain of \(v_{x}\), namely, until its null value.  
Therefore, starting from the non-linear ODE, four analytic solutions were found by approximating the non-linearity with four piecewise linear terms. Subsequently, to obtain a single equation for decay \(v_{x}\) (10), a combination of these solutions was proposed by using multipliers consisting of rectangle functions built by the hyperbolic tangent function, tanh. Finally, the obtained equation (10) was verified in-field on a vineyard row by measuring air velocity \(v_{x}\) at exit of the canopy. The results showed good correspondence between the experimental mean value and the predicted one, with a relative error of 6.6%. In addition, there was no significant difference between the experimental and predicted curves.  
As the procedure to build a unique equation by combining the four solutions of the four differential linearized equations tends to be challenging, we simplified the procedure by reducing the number of linearized equations from four to two. Therefore, the combination of only two solutions produced a velocity decay equation (11) that was much more manageable. Again, there was no significant difference between the experimental value and the predicted one by combination (11), but the mean relative error (MRE) of Equation (11) vs. Equation (10) was 3.6%, and the maximum error was 10.7% at \(x=1.63\) m. This is an acceptable error, compared to the variation of experimental measurements of air velocity \(v_{x}\) at the exit of the foliage, which was confirmed by the wide standard deviation.  
The authors declare no conflicts of interest.