Minimum induced drag condition
Thus comparing Eqn (5.50) with the induced-drag coefficient for the elliptic case (Eqn (5.43)) it can be seen that modifying the spanwise distribution away from the elliptic increases the drag coefficient by the fraction 6 that is always positive. It follows that for the induced drag to be a minimum 6 must be zero so that the distribution for minimum induced drag is the semi-ellipse. It will also be noted that the minimum drag distribution produces a constant downwash along the span whereas all other distributions produce a spanwise variation in induced velocity. This is no coincidence. It is part of the physical explanation of why the elliptic distribution should have minimum induced drag.
To see this, consider two wings (Fig. 5.33a and b), of equal span with spanwise distributions in downwash velocity w = wo = constant along (a) and w = f(z) along
(b) . Without altering the latter downwash variation it can be expressed as the sum of two distributions wo and wi = fi(z) as shown in Fig. 5.33c. Fig. 5.33 (a) Elliptic distribution gives constant downwash and minimum drag, (b) Non-elliptic distribution gives varying downwash. (c) Equivalent variation for comparison purposes