# The Aerodynamic Mechanisms of the Formation Flight of Migratory Birds: A Narrative Review
## 3 Results
### Theoretical and Experimental Studies
Aerodynamic theory methods have been used to calculate the reduction in flight power for formations of arbitrary shapes with any number of birds. The total power reduction is strongly dependent on the lateral distance between the wings. In addition, he showed that a longitudinal displacement of the wings in the direction of flight has no influence on the total reduction in flight power, but only on its distribution among the individuals involved. He also addressed the communication aspects of formation flying. Still based on [PERSON] and [PERSON]'s initial theory [28], [PERSON] [37] also looked at the aerodynamics of formation flight in migratory birds. In this study, [PERSON] meticulously described the projective geometry technique used in his collaboration with [PERSON]. He applied this technique to his own photographs of Canadian geose formations and observed that the birds frequently changed lateral positions relative to the preceding bird. This observation contradicted the basic energy conservation model proposed by [PERSON] and [PERSON] [28], which predicted an optimal position for energy conservation. Using their model, [PERSON] concluded that photographed geose formations demonstrated only a 36% energy advantage over solitary flight, about half that predicted by [PERSON] and [PERSON]'s model [28]. He warned \begin{table}
\begin{tabular}{l l} \hline [PERSON] and [PERSON] [28] & 120\({}^{\circ}\) \\ \hline [PERSON] and [PERSON] [6] & 27.5\({}^{\circ}\)–44\({}^{\circ}\) \\ \hline [PERSON] et al. [34] & 38\({}^{\circ}\)–124\({}^{\circ}\) \\ \hline [PERSON] et al. [16] & 24\({}^{\circ}\)–124\({}^{\circ}\) \\ \hline [PERSON] and [PERSON] [35] & 24\({}^{\circ}\)–122\({}^{\circ}\) \\ \hline \end{tabular}
\end{table} 
Table 1: Formation angle range of V-formation from various authors. against adopting a simplistic engineering model to fully explain behavior that could be highly variable depending on environmental parameters. [PERSON] [38] also focused on the formation flight of Canadian geese and evaluated energetic advantages by testing various hypotheses on the geometric characteristics of the formation. [PERSON] examined the positioning of birds within different diagonal line formations (V, J, echelon, etc.) and proposed hypotheses on how geese maximize benefits. Positioning was defined in relation to the leader bird and could vary in three ways: along the flight path (depth), perpendicular to the flight path (wingtip spacing), or both (see Figure 4). Each hypothesis predicted a characteristic pattern of flight behavior in the form of a relationship between depth and wingtip spacing. He postulated that if the aerodynamic advantage was the main driver of line flight, there should be a specific geometric relationship between birds in a formation. [PERSON] proposed an energy advantage of around 10% for formation flight over solitary flight, a value significantly lower than the estimates of [PERSON] (who proposed saving of 30% saved energy for follower birds) or [PERSON] and [PERSON] [6] who estimated energy saving of 51%. Along with his predecessors, [PERSON] [39] investigated the energetic benefits of formation flight in Canadian geese (_Branta canadensis_). His study analyzed the variation in vortex position induced by wing movements in V-shaped formations. The researchers measured various parameters, such as wingtip spacing, formation depth, wingbeat frequency, and relative extreme wing positions, to assess the use of these variations in vortex positioning. The results showed that only 48% among 73 birds from eight different formations had wing-beat frequencies similar to those of the bird in front of them (difference \(\leq\) 0.1 beats per second). The study suggests that birds at the rear of a formation adjust wingtip spacing (WTS) in response to changes in the horizontal position of the bird in front of them. Some birds, around 20%, maintain positions leading to relatively low economies, highlighting individual variability in the use of aerodynamic strategies.