Wave drag

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In aeronautics, wave drag is a component of the aerodynamic drag on aircraft wings and fuselage, propeller blade tips and projectiles moving at transonic and supersonic speeds, due to the presence of shock waves. Wave drag is independent of viscous effects, [1] and tends to present itself as a sudden and dramatic increase in drag as the vehicle increases speed to the Critical Mach number. It is the sudden and dramatic rise of wave drag that leads to the concept of a sound barrier.

Aeronautics science involved with the study, design, and manufacturing of airflight-capable machines

Aeronautics is the science or art involved with the study, design, and manufacturing of air flight capable machines, and the techniques of operating aircraft and rockets within the atmosphere. The British Royal Aeronautical Society identifies the aspects of "aeronautical Art, Science and Engineering" and "the profession of Aeronautics ."

Shell (projectile) projectile

A shell is a payload-carrying projectile that, as opposed to shot, contains an explosive or other filling, though modern usage sometimes includes large solid projectiles properly termed shot. Solid shot may contain a pyrotechnic compound if a tracer or spotting charge is used. Originally, it was called a "bombshell", but "shell" has come to be unambiguous in a military context.

Transonic flight condition in which airflow speeds are concurrently above and below the speed of sound

In aeronautics, transonic flight is flying at or near the speed of sound 343 meters per second, relative to the air through which the vehicle is traveling. A typical convention used is to define transonic flight as speeds in the range of Mach 0.72 to 1.0.



Wave drag presents itself as part of pressure drag due to compressibility effects. It is caused by the formation of shock waves around a body. Shock waves create a considerable amount of drag, which can result in extreme drag on the body. Although shock waves are typically associated with supersonic flow, they can form at subsonic aircraft speeds on areas of the body where local airflow accelerates to supersonic speed. The effect is typically seen on aircraft at transonic speeds (about Mach 0.8), but it is possible to notice the problem at any speed over that of the critical Mach of that aircraft. It is so pronounced that, prior to 1947, it was thought that aircraft engines would not be powerful enough to overcome the enhanced drag, or that the forces would be so great that aircraft would be at risk of breaking up in midflight. It led to the concept of a sound barrier .

Compressibility measure of the relative volume change of a fluid or solid as a response to a pressure change

In thermodynamics and fluid mechanics, compressibility is a measure of the relative volume change of a fluid or solid as a response to a pressure change. In its simple form, the compressibility may be expressed as

Shock wave Propagating disturbance

In physics, a shock wave, or shock, is a type of propagating disturbance that moves faster than the local speed of sound in the medium. Like an ordinary wave, a shock wave carries energy and can propagate through a medium but is characterized by an abrupt, nearly discontinuous, change in pressure, temperature, and density of the medium.

Mach number Ratio of speed of object moving through fluid and local speed of sound

In fluid dynamics, the Mach number is a dimensionless quantity representing the ratio of flow velocity past a boundary to the local speed of sound.


In 1947, studies into wave drag 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.

Sears–Haack body

The Sears–Haack body is the shape with the lowest theoretical wave drag in supersonic flow, for a given body length and given volume. The mathematical derivation assumes small-disturbance (linearized) supersonic flow, which is governed by the Prandtl–Glauert equation. The derivation and shape were published independently by two separate researchers: Wolfgang Haack in 1941 and later by William Sears in 1947.

Reduction of drag

A number of new techniques developed during and just after World War II were able to dramatically reduce the magnitude of wave drag, and by the early 1950s the latest fighter aircraft could reach supersonic speeds.

World War II 1939–1945, between Axis and Allies

World War II, also known as the Second World War, was a global war that lasted from 1939 to 1945. The vast majority of the world's countries—including all the great powers—eventually formed two opposing military alliances: the Allies and the Axis. A state of total war emerged, directly involving more than 100 million people from more than 30 countries. The major participants threw their entire economic, industrial, and scientific capabilities behind the war effort, blurring the distinction between civilian and military resources. World War II was the deadliest conflict in human history, marked by 70 to 85 million fatalities, most of whom were civilians in the Soviet Union and China. It included massacres, the genocide of the Holocaust, strategic bombing, premeditated death from starvation and disease, and the only use of nuclear weapons in war.

Fighter aircraft Military aircraft designed particularly for air-to-air combat against other aircraft

A fighter aircraft is a military aircraft designed primarily for air-to-air combat against other aircraft, as opposed to bombers and attack aircraft, whose main mission is to attack ground targets. The hallmarks of a fighter are its speed, maneuverability, and small size relative to other combat aircraft. Although helicopters are sometimes used in similar roles, the term fighter is at present applied only to fixed-wing aircraft.

These techniques were quickly put to use by aircraft designers. One common solution to the problem of wave drag was to use a swept wing, which had actually been developed before World War II and used on some German wartime designs. Sweeping the wing makes it appear thinner and longer in the direction of the airflow, making a conventional teardrop wing shape closer to that of the von Kármán ogive, while still remaining useful at lower speeds where curvature and thickness are important.

Swept wing

A swept wing is a wing that angles either backward or occasionally forward from its root rather than in a straight sideways direction. Wing sweep has the effect of delaying the shock waves and accompanying aerodynamic drag rise caused by fluid compressibility near the speed of sound, improving performance. Swept wings are therefore often used on jet aircraft designed to fly at these speeds. Swept wings are also sometimes used for other reasons, such as structural convenience or visibility.

The wing need not be swept when it is possible to build a wing that is extremely thin. This solution was used on a number of designs, beginning with the Bell X-1, the first manned aircraft to fly at the speed of sound. The downside to this approach is that the wing is so thin it is no longer possible to use it for storage of fuel or landing gear. Such wings are very common on missiles, although, in that field, they are often referred to as "fins".

Bell X-1 Experimental rocket powered aircraft, first airplane to break the sound barrier in level flight

The Bell X-1,, is a rocket-engine–powered aircraft, designated originally as the XS-1, and was a joint National Advisory Committee for Aeronautics–U.S. Army Air Forces–U.S. Air Force supersonic research project built by Bell Aircraft. Conceived during 1944 and designed and built in 1945, it achieved a speed of nearly 1,000 miles per hour in 1948. A derivative of this same design, the Bell X-1A, having greater fuel capacity and hence longer rocket burning time, exceeded 1,600 miles per hour in 1954. The X-1, piloted by Chuck Yeager, was the first crewed airplane to exceed the speed of sound in level flight and was the first of the X-planes, a series of American experimental rocket planes designed for testing new technologies.

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, not just the fuselage. This meant that the fuselage needed to be made narrower where it joined the wings, so that the cross-section of the entire aircraft matched the Sears-Haack body.

The Convair 990 had particularly obvious anti-shock bodies; modern airliners usually have more subtle shaping for area ruling. Convair 990 on ramp EC92-05275-30.jpg
The Convair 990 had particularly obvious anti-shock bodies; modern airliners usually have more subtle shaping for area ruling.

Application of the area rule can also be seen in the use of anti-shock bodies on transonic aircraft, including some 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.

Other drag reduction methods

Several other attempts to reduce wave drag have been introduced over the years. The supercritical airfoil is a type that results in reasonable low speed lift like a normal airfoil, 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.

Mathematical formula

For flat plate aerofoil


For double-wedge aerofoil



  1. Clancy, L.J. (1975), Aerodynamics, Section 11.7
  2. 1 2 "How can I calculate wave drag in supersonic airfoil?". 2 February 2014. Retrieved 23 May 2019.

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Area rule rule in physics, relating to an aircrafts drag at transonic or supersonic speed

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Critical Mach number The lowest Mach number at which the airflow over some point of the aircraft reaches the speed of sound

In aerodynamics, the critical Mach number of an aircraft is the lowest Mach number at which the airflow over some point of the aircraft reaches the speed of sound, but does not exceed it. At the lower critical Mach number, airflow around the entire aircraft is subsonic. At the upper critical Mach number, airflow around the entire aircraft is supersonic.

Supercritical airfoil

A supercritical airfoil is an airfoil designed primarily to delay the onset of wave drag in the transonic speed range. Supercritical airfoils are characterized by their flattened upper surface, highly cambered ("downward-curved") aft section, and larger leading-edge radius compared with NACA 6-series laminar airfoil shapes. Standard wing shapes are designed to create lower pressure over the top of the wing. The camber of the wing determines how much the air accelerates around the wing. As the speed of the aircraft approaches the speed of sound, the air accelerating around the wing reaches Mach 1 and shockwaves begin to form. The formation of these shockwaves causes wave drag. Supercritical airfoils are designed to minimize this effect by flattening the upper surface of the wing.

Mach tuck

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Anti-shock body

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Supersonic airfoils

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Thickness-to-chord ratio

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