Wave drag is caused by the formation of shock waves around the aircraft. Shock waves radiate away a considerable amount of energy, energy that is experienced by the aircraft as drag. Although shock waves are typically associated with supersonic flow, they can form at much lower speeds at areas on the aircraft where, according to Bernoulli's principle, local airflow accelerates to supersonic speeds over curved areas. The effect is typically seen at transonic speeds above about Mach 0.8, but it is possible to notice the problem at any speed over that of the critical Mach of that aircraft's wing. The magnitude of the rise in drag is impressive, typically peaking at about four times the normal subsonic drag. It is so powerful that it was thought for some time that engines would not be able to provide enough power to easily overcome the effect, which led to the concept of a "sound barrier".
When the problem was being studied, wave drag came to be split into two – wave drag caused by the wing as a part of generating lift, and that caused by other portions of the plane. In 1947, studies into both problems led to the development of "perfect" shapes to reduce wave drag as much as theoretically possible. For a fuselage the resulting shape was the Sears-Haack body, which suggested a perfect cross-sectional shape for any given internal volume. The von Kármán ogive was a similar shape for bodies with a blunt end, like a missile. Both were based on long narrow shapes with pointed ends, the main difference being that the ogive was pointed on only one end.
However a number of new techniques developed during and just after World War II were able to dramatically reduce the magnitude of the problem, and by the early 1950s most fighter aircraft could reach supersonic speeds without too much trouble. If the problem of wave drag is caused by the acceleration of air over curves on the aircraft, the solution is, obviously, to reduce the curves. However this is not always easy, for instance, a wing generates lift at subsonic speeds primarily due to the curvature on the leading edge of the wing. Things are somewhat better for fuselage shaping, but simple things like a cockpit canopy or smoothing off the metal around an air intake can create additional "hot spots".
These research projects were quickly put to use by aircraft designers. One common solution to the problem of wave drag due to the wings was to use a swept-wing, which had actually been developed before WWII and used on some German wartime designs such as the Me-262. Sweeping the wing to the rear makes it appear thinner and longer in the direction of the airflow, making a "normal" wing shape closer to that of the von Kármán ogive, while still remaining useful at lower speeds where curvature and thickness are important.
The wing need not be swept as it is possible to build a wing that is extremely thin. This solution was used on a number of designs, perhaps the most obvious being the F-104 Starfighter. The downside to this approach is that the wing is so thin it is no longer possible to use it for fuel storage or landing gear.
Fuselage shaping was similarly changed with the introduction of the Whitcomb area rule. Whitcomb had been working on testing various airframe shapes for transonic drag when, after watching a presentation by Adolf Busemann in 1952, he realized that the Sears-Haack body had to apply to the entire aircraft. This meant that the fuselage needed to be made considerably skinnier where the wings met it, so that the cross-section of the entire aircraft matched the Sears-Haack body, not just the fuselage itself.
Application of the area rule can also be seen in the use of anti-shock bodies on subsonic aircraft, such as jet airliners. Anti-shock bodies, which are pods along the trailing edges of the wings, serve the same role as the narrow waist fuselage design of other transonic aircraft.
Several other attempts to reduce wave drag have been introduced over the years, but have not become common. The supercritical airfoil is a new wing design that results in reasonable low speed lift like a normal planform, but has a profile considerably closer to that of the von Kármán ogive. All modern civil airliners use forms of supercritical aerofoil and have substantial supersonic flow over the wing upper surface.
Busemann's Biplane avoids wave drag entirely, but is incapable of generating lift, and has never flown.