nonlinearity of transonic flow models was the hodograph transformation. This concept was originally explored in 1923 by an Italian mathematician named Francesco Tricomi, who used the transformation to simplify the compressible flow equations and prove that they were solvable. The hodograph transformation itself was also explored by both Ludwig Prandtl and O.G. Tietjen's textbooks in 1929 and by Adolf Busemann in 1937, though neither applied this method specifically to transonic flow. Gottfried Guderley, a German mathematician and engineer at Braunschweig, discovered Tricomi's work in the process of applying the hodograph method to transonic flow near the end of World War II. He focused on the nonlinear thin-airfoil compressible flow equations, the same as what Tricomi derived, though his goal of using these equations to solve flow over an airfoil presented unique challenges. Guderley and Hideo Yoshihara, along with some input from Busemann, later used a singular solution of Tricomi's equations to analytically solve the behavior of transonic flow over a double wedge airfoil, the first to do so with only the assumptions of thin-airfoil theory. Although successful, Guderley's work was still focused on the theoretical, and only resulted in a single solution for a double wedge airfoil at Mach 1. Walter Vincenti, an American engineer at Ames Laboratory, aimed to supplement Guderley's Mach 1 work with numerical solutions that would cover the range of transonic speeds between Mach 1 and wholly supersonic flow. Vincenti and his assistants drew upon the work of Howard Emmons as well as Tricomi's original equations to complete a set of four numerical solutions for the drag over a double wedge airfoil in transonic flow above Mach 1. The gap between subsonic and Mach 1 flow was later covered by both Julian Cole and Leon Trilling, completing the transonic behavior of the airfoil by the early 1950's. Condensation clouds At transonic speeds supersonic expansion fans form intense low-pressure, low-temperature areas at various points around an aircraft. If the temperature drops below the dew point a visible cloud will form. These clouds remain with the aircraft as it travels. It is not necessary for the aircraft as a whole to reach supersonic speeds for these clouds to form. Typically, the tail of the aircraft will reach supersonic flight while the bow of the aircraft is still in subsonic flight. A bubble of supersonic expansion fans terminating by a wake shockwave surround the tail. As the aircraft continues