A Third Drag Component the Basic Model Doesn't Capture
The standard drag polar (C_D = C_D0 + C_L²/(π·e·AR)) is a low-speed, incompressible-flow model — it works well for most flight below roughly Mach 0.6-0.7, but it doesn't account for a third drag source that appears specifically as an aircraft approaches and exceeds certain local flow speeds: wave drag, caused by shock waves forming on the aircraft as local airflow over curved surfaces (like the upper wing surface) reaches and exceeds the speed of sound, even while the aircraft's overall airspeed is still subsonic.
Why Local Flow Can Be Supersonic Before the Aircraft Is
Air accelerates as it flows over a curved surface like the top of a wing (the same principle that generates lift). At a high enough aircraft speed, this local acceleration can push the airflow over the highest-curvature point of the wing to exactly Mach 1, even though the aircraft's true airspeed is still below Mach 1. The aircraft speed at which this first occurs is called the critical Mach number (M_crit) — it depends on the wing's shape, thickness, and sweep, not on any single universal value.
The Drag-Divergence Mach Number
Beyond M_crit, further increases in aircraft speed cause the region of locally supersonic flow to grow, and shock waves form where that supersonic region transitions back to subsonic flow — these shocks are a genuinely new, additional drag mechanism (wave drag) layered on top of the parasite and induced drag the low-speed polar already accounts for. The drag-divergence Mach number (M_dd) is commonly defined as the speed at which wave drag causes a sharp, sudden rise in total drag coefficient — practically, the point where the smooth low-speed drag curve breaks upward dramatically rather than continuing its gentle trend.
Why Aircraft Design Pushes M_dd as High as Possible
Since cruising near or above the drag-divergence Mach number causes a large drag (and therefore fuel-burn) penalty, transonic aircraft (most commercial jets cruise in the high-subsonic transonic range) are designed specifically to push M_dd as high as practical, primarily through wing sweep (a swept wing effectively "sees" a lower component of the freestream velocity perpendicular to its leading edge, delaying the onset of local supersonic flow) and supercritical airfoil sections (specially shaped to flatten the upper-surface pressure distribution and delay shock formation). This is why swept-wing shapes are near-universal on jet airliners — it's a direct, deliberate response to wave drag, not a stylistic choice.
Practical Implication for the Drag Equation
For flight well below M_crit (most propeller aircraft, low-speed general aviation), the basic drag equation and polar are accurate on their own. For any aircraft cruising in the high subsonic or transonic regime, wave drag has to be added as a Mach-dependent term, and a constant-C_D drag calculation (like a simple drag calculator using a single input C_D value) is only valid as a snapshot at one specific flight condition — it does not capture how C_D changes as the aircraft accelerates through the transonic drag rise.