Wave Drag Due to Lift

The expression given for wave drag due to lift:

holds for wings of very low aspect ratio.

A more general expression is derived by R.T. Jones in "Minimum Drag of Airfoils at Supersonic Speeds", J. of Aero Sciences, Dec. 1952.

The combined vortex and wave drag may be written:

This expression approaches the correct limits for ellipses as M-> 1 and as AR -> 0 or infinity. The assumption here is that the lift distribution is elliptical in all directions, an assumption that is not realized exactly in practice.

Jones also gives an expression for the wave drag due to lift for a yawed ellipse, showing that there is an optimum sweep angle. At M = 1.4, a 10:1 yawed ellipse at 55° has less than 1/2 of the wave drag of the ellipse with 0° or 90° of yaw.

When the planform shape is not elliptical, it may be better to form an equivalent ellipse with the same area and length rather than one with the same aspect ratio as the real wing. In this case:

Here, S is the wing area and l is the overall length. This choice preserves the average wing pressure difference and agrees well with experimental data for well-designed supersonic wings.

Supersonic Drag Due To Lift Computed by Present Method (*) and Boeing Optimization Results