"transonic" was defined to mean "across the speed of sound" and was invented by NACA director Hugh Dryden and Theodore von Kármán of the California Institute of Technology. Changes in aircraft Initially, NACA designed "dive flaps" to help stabilize the plane when reaching transonic flight. This small flap on the underside of the plane slowed the plane to prevent shock waves, but this design only delayed finding a solution to aircraft flying at supersonic speed. Newer wind tunnels were designed, so researchers could test newer wing designs without risking test pilots' lives. The slotted-wall transonic tunnel was designed by NASA and allowed researchers to test wings and different airfoils in transonic airflow to find the best wingtip shape for sonic speeds. After World War II, major changes in aircraft design were seen to improve transonic flight. The main way to stabilize an aircraft was to reduce the speed of the airflow around the wings by changing the chord of the plane wings, and one solution to prevent transonic waves was swept wings. Since the airflow would hit the wings at an angle, this would decrease the wing thickness and chord ratio. Airfoils wing shapes were designed flatter at the top to prevent shock waves and reduce the distance of airflow over the wing. Later on, Richard Whitcomb designed the first supercritical airfoil using similar principles. Mathematical analysis Prior to the advent of powerful computers, even the simplest forms of the compressible flow equations were difficult to solve due to their nonlinearity. A common assumption used to circumvent this nonlinearity is that disturbances within the flow are relatively small, which allows mathematicians and engineers to linearize the compressible flow equations into a relatively easily solvable set of differential equations for either wholly subsonic or supersonic flows. This assumption is fundamentally untrue for transonic flows because the disturbance caused by an object is much larger than in subsonic or supersonic flows; a flow speed close to or at Mach 1 does not allow the streamtubes (3D flow paths) to contract enough around the object to minimize the disturbance, and thus the disturbance propagates. Aerodynamicists struggled during the earlier studies of transonic flow because the then-current theory implied that these disturbances– and thus drag– approached infinity as local Mach number approached 1, an obviously unrealistic result which could not be remedied using known methods. One of the first methods used to circumvent the