## 1 Introduction  
In agrochemical applications on orchards and vineyards, the idea of producing droplets by using nozzles and transporting them to vegetation, by using an air jet generated by a fan, is now the generally accepted method. Some studies [1; 2; 3; 4; 5; 6] have demonstrated that stream speed strongly influences the effectiveness of spray treatment. If the speed is too low, the distribution of chemicals on vegetation tends to be non-uniform. On the contrary, if the speed is too high, the leaves align with the air stream, and much of the spray is not captured by the vegetation. Spray drift increases with velocity until approximately 50% of the spray does not land on the vegetation [3; 4] and, consequently, causes both economic loss and environmental pollution.  
Therefore, to help understand the experimental results, it is useful to find an equation that relates the air jet velocity vs. the distance from the output of the sprayer fan when air passes through the canopy. This equation could also be helpful to equip sprayers with an automatic controller that sets the fan speed in order to maintain the optimum air jet velocity at exit of the canopy, both for the row width [1] and for the growth stage [7].  
In recent work [8], an equation was developed for the air velocity decay when the air jet crosses the canopy of tree rows. The equation was obtained by applying the momentum theorem under three assumptions:  
1. The foliage is uniformly distributed in space, and subsequently, the LAD (leaf area density) is constant.
2. The forward motion of the sprayer is ignored since its velocity is one order of magnitude lower than the air velocity from fan exit, and no effect of travel speed on spray deposition in the canopy was experimentally observed [9].
3. The inverse relationship between the drag coefficient and air velocity is considered.  
Assumption (c), also proposed by other authors [10; 11], was verified by wind tunnel tests [8]. All three works confirmed this hypothesis for air jet velocity \(v_{x}\), which is higher than a fixed value (inferior limit value).  
Due to the third assumption, (c), the result of applying the momentum theorem was a linear ordinary differential equation (ODE). By integrating the ODE, an equation to describe air speed decay through the vegetation was found.  
If the air speed is under an inferior limit value, the decay equation is not valid because the third assumption, (c), is no longer valid.  
However, air jet velocity was observed during experiments [8]; in some situations, when the vegetation is highly developed, the air jet velocity decreases under the limit set by the model. Consequently, the aim of the present work was to find a new equation for the decay speed that would also be valid under the limit when \(v_{x}\) approaches zero. In order to obtain the equation, we started from the non-linear ODE and found analytic solutions by approximating the non-linearity with piecewise linear terms. The method to approximate non-linearity by linear pieces has previously been conducted in other situations [12; 13; 14; 15; 16]. We used the approach here to obtain a series of analytical solutions. Finally, to obtain a unique equation for the \(v_{x}\) decay, a combination of piecewise solutions was proposed here, which used multipliers consisting of rectangle functions.