= Induced angle of attack
{title2=$\alpha_i$}
The \b[induced angle of attack] $\alpha_i$ is the change between the geometric flow direction and the local effective flow direction caused by the wing's induced velocity. In the usual small-angle <lifting-line theory> convention, downward <downwash> is positive and
$$
\alpha_{\rm eff}=\alpha-\alpha_i,\qquad
\alpha_i\simeq\frac{w}{U}.
$$
Here $U$ is the reference streamwise <speed> and $w$ the local downward speed. The originating lecture notes explicitly distinguish geometric, induced and effective incidence.<Prandtl's Lifting Line Introduction (16.100) by MIT OpenCourseWare>[Pages 2–3, geometry and induced-angle convention]{ref}
If the local velocity has only these two components, its geometric inclination is $\arctan(w/U)$. The ratio $w/U$ is the small-angle approximation, with angles in radians. An original example uses $U=40$ m/s and $w=1.0$ m/s: $\alpha_i=\arctan(0.025)\simeq1.432\,{}^\circ$. A geometric incidence of $5\,{}^\circ$ then gives effective incidence about $3.568\,{}^\circ$. This does not subtract the <zero-lift angle of attack>, which is a separate term in the section lift law.
For ideal <elliptic span loading>, the angle is constant across the span and approximately
$$
\alpha_i=\frac{C_L}{\pi AR},\qquad
C_{D_i}\simeq C_L\alpha_i=\frac{C_L^2}{\pi AR}.
$$
The latter relation is the <lift-induced drag> contribution, with unit <span efficiency factor>.<Force Calculations for Lifting Line (16.100) by MIT OpenCourseWare>[Pages 2–3, induced drag and elliptic loading]{ref} For stipulated $C_L=0.60$ and $AR=8$, the approximation gives $\alpha_i\simeq0.02387$ radians and $C_{D_i}\simeq0.01432$.
A nonuniform wing generally has different local induced angles. Its total induced drag involves the loading-weighted integral $D_i\simeq\int L'(y)\alpha_i(y)\,dy$; an unweighted arithmetic mean of wake angles need not reproduce that integral. A wake velocity traverse and sectional <surface pressure integration for airfoil lift> test different pieces of this prediction.
The angle concerns induced flow at the modeled lifting line. A velocity measurement far downstream requires the corresponding wake model before being substituted into this formula. The small-angle, attached-flow approximation also does not account for all sources of drag or for strong crossflow.
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