Supersonic Flight Vehicles

J. Gordon Leishman

57 Supersonic Flight Vehicles

Introduction^[1]

Engineers have long sought to develop flight vehicles capable of ever higher speeds. While modern militaries maintain a continual “need for speed” for many missions, the demand for civil aircraft capable of rapidly transporting passengers and cargo over long distances is also significant. Aircraft are not inherently limited to subsonic or transonic flight, but sustained supersonic operation introduces challenges that extend beyond aerodynamics and propulsion. At high Mach numbers, kinetic heating of the airframe becomes an important design consideration, requiring structures and materials capable of withstanding elevated temperatures and thermal stresses.

One of the most remarkable aircraft developed for sustained supersonic flight was the SR-71 “Blackbird,” shown in Figure 1. Developed by Lockheed from the earlier A-12 and YF-12A aircraft, the SR-71 was designed as a long-range strategic reconnaissance platform capable of cruising at speeds above Mach 3 and altitudes exceeding 80,000 ft. Achieving such performance required innovative solutions in aerodynamics, propulsion, materials, and thermal management. Much of the aircraft structure was constructed from titanium alloys to withstand the severe aerodynamic heating encountered during high-Mach flight.

The SR-71 “Blackbird” strategic reconnaissance aircraft first flew in 1964 and remained the world’s fastest and highest-flying operational aircraft. Air-to-air front view of an SR-71A.

Supersonic military aircraft continue to be developed with emphasis on greater speed, maneuverability, survivability, and stealth. Modern designs employ advanced materials, refined aerodynamics, and highly integrated propulsion systems. Current research focuses on improving supersonic cruise efficiency, maintaining maneuverability at high Mach numbers, and reducing radar signatures. Programs such as the U.S. Air Force’s Next Generation Air Dominance initiative are intended eventually to succeed aircraft such as the F-22 Raptor. In 2025, Boeing received the engineering and manufacturing development contract for the F-47, which is intended to combine advanced stealth, extended range, and integration with uncrewed combat aircraft to form a networked air-combat system.

On the civil side, renewed efforts are underway to reintroduce commercial supersonic transport (SST). NASA and Lockheed Martin are developing quieter, more efficient designs, such as the X-59 QueSST, aimed at reducing sonic boom levels that limited earlier airplanes like the Concorde. These efforts seek to enable routine supersonic travel over land while meeting modern noise and environmental standards. In parallel, several companies are pursuing supersonic business jets (SSBJs) for high-speed point-to-point travel in the private aviation market. Nevertheless, certification requirements, economic viability, and environmental considerations remain significant challenges.

Learning Objectives

Be able to describe the key aerodynamic design features of supersonic airplanes, including the use of thin, highly swept, or delta-type wings.
Understand why increasing the Mach number influences shock wave formation and affects the aerodynamic performance of wings.
Use appropriate methods to calculate aerodynamic coefficients for supersonic airfoils and finite wings.
Know the difference between the aerodynamic characteristics of wings with subsonic versus supersonic leading edges.
Understand how to interpret the physical principles behind vortex lift on slender delta wings at high angles of attack.
Be aware of the significant engineering and operational challenges of designing and operating supersonic transport aircraft.
Discover the factors that influence sonic booms and learn how to mitigate their effects.

History

In 1887, Ernst Mach and Peter Salcher presented their seminal paper on the supersonic motion of a projectile, “Photographische Fixirung der durch Projectile in der Luft eingeleiteten Vorgänge” (“Photographic Recording of the Processes Initiated in Air by Projectiles”) in the journal Sitzungsberichte der Kaiserlichen Akademie der Wissenschaften [Wien], Mathematisch-Naturwissenschaftliche (Proceedings of the Imperial Academy of Sciences [Vienna], Mathematical and Natural Sciences Section). One of their famous schlieren photographs is shown in Figure 2. They theoretically predicted and experimentally confirmed the formation of a conical shock wave, commonly known as a Mach cone, at the nose of a supersonic projectile. This was accompanied by a trailing expansion wave, completing the characteristic wave pattern of supersonic motion. Note also the turbulent wake trailing behind the projectile.

An 1887 Schlieren image of the compression and expansion waves formed by a supersonic projectile.

In the early 1920s, Alexander Martin Lippisch proposed a series of delta-wing concepts, not for supersonic aircraft but for unconventional “tailless” aircraft. However, by 1930, considerable interest in supersonic flight had developed, and the aerodynamics associated with supersonic flows were being studied extensively in wind tunnels. ^[2] Parallel theoretical studies were also underway, with Jakob Ackeret publishing a pioneering paper on the lift and drag of a supersonic airfoil in 1925.^[3]

During WWII, aircraft designs were driven by the need for speed; examples include the P-51 Mustang and Spitfire, which could approach the speed of sound in a dive. However, the increased speeds introduced a new problem: a strong nose-down pitching moment, in which the airplane’s nose would pitch downward uncontrollably. This phenomenon was known as the “dive recovery problem” or “Mach tuck,” which sometimes made recovery from the dive impossible.^[4] Additionally, advancements in aerodynamic research and understanding the physics of supersonic flight led to the development of new airplane designs, such as swept-wing configurations, ultimately allowing airplanes to operate at higher airspeeds before encountering Mach tuck and other adverse effects resulting from the compressibility of the airflow.

The aerodynamic effects of using sweep at high speeds were first investigated in Germany as early as 1935 by Albert Betz and Adolph Busemann. After that, Lippisch designed the famous rocket-powered Messerschmitt Me 163 Komet, which had a swept, almost delta wing and was powered by a rocket engine. It was the first piloted airplane of any type to approach Mach 0.9 but not quite exceed the speed of sound in level flight. This aircraft was followed by the Messerschmitt Me 262, which featured a swept wing and became the world’s first operational turbojet-powered fighter aircraft. The Gloster Meteor was the first British jet-powered fighter and the first Allied jet aircraft to enter operational service during WWII. It first flew for the Royal Air Force in 1944, when the Luftwaffe deployed the Me 262, but saw limited combat.

The first airplane to fly supersonically in straight and level flight was the Bell X-1, a rocket-powered research aircraft, which conducted several test flights near the speed of sound immediately after WWII. The air-dropped Bell X-1, piloted by Chuck Yeager, first reached supersonic speeds in level flight in October 1947. The British were also developing supersonic aircraft. The Miles M.52 was designed in secrecy between 1942 and 1945 during WWII, with Frank Whittle‘s Power Jets being contracted to produce its turbojet engine. The M.52 design had a conical bullet-shaped nose, thin, mildly swept wings, and a tail with sharp leading edges. While this airplane never flew with a pilot, scaled models were flown supersonically in 1947.

In September 1948, the tailless swept-wing de Havilland D.H.108 Swallow, as shown in Figure 3, reached just over Mach 1.0 while in a shallow dive. Unfortunately, the airplane suffered a catastrophic structural failure that killed its designer and pilot, Geoffrey de Havilland Jr. The failure was associated with severe unsteady aerodynamic loads during high-speed flight. In the meantime, the U.S. continued to pursue supersonic flight with variants of the X-1, including the X-1A through X-1E, several of which exceeded Mach 2.

The British De Havilland D.H.108 Swallow was among the first swept-wing research aircraft designed to study transonic flight.

After the fundamental challenges of aerodynamics, stability, and control, as well as propulsion, were resolved near and beyond Mach 1.0 by the early 1950s, it was not long before new military aircraft were designed to operate routinely at supersonic speeds. The invention of the turbojet engine, along with the development of the afterburner (or reheat), which burns additional fuel in the exhaust stream to increase thrust, enabled aircraft to accelerate into the transonic regime and sustain supersonic flight.

Examples include the Convair F-102 Delta Dagger and its successor, the F-106 Delta Dart, both of which employed delta wings to improve supersonic performance. Aircraft such as the General Dynamics F-111 and the Grumman F-14 Tomcat utilized variable-sweep wings to optimize aerodynamic efficiency across a wide range of speeds. The English Electric Lightning was capable of speeds exceeding Mach 2 in level flight and was among the first supersonic interceptors to enter military service. The Dassault Mirage III was a highly successful delta-wing aircraft that saw widespread international use. More recently, the multinational Panavia Tornado achieved significant success with its variable-geometry wings, optimized for both low-speed and supersonic flight.

However, civil supersonic aircraft carrying fare-paying passengers have encountered additional challenges. The rise and fall of the Anglo-French Concorde (Figure 4) and the Soviet Tupolev Tu-144 provide compelling case studies on the intersection of technological capabilities and commercial viability. While the Concorde was a remarkable engineering achievement, the costs of developing a supersonic transport (SST) were enormous, and its economic viability was limited, as Boeing quickly discovered with their proposed Model 2707 SST. Furthermore, Concorde‘s loud “sonic booms” restricted its commercial flights to overwater routes, thereby limiting its ability to achieve its full economic potential and ultimately leading to the discontinuation of all supersonic passenger travel in 2003.

The Anglo-French supersonic *Concorde* was a remarkable engineering achievement that flew in airline service for over 20 years.

It seems unlikely that supersonic transport airplanes (SSTs) will return to mainstream aviation anytime soon, even if technically feasible. Any revival of supersonic passenger travel must address the aerodynamic and propulsion challenges of long-range, high-speed flight, as well as the economic, environmental, and regulatory constraints that accompany it. Upstarts like Boom Supersonic face a difficult path forward, as their technological optimism and savvy marketing must ultimately contend with the rigid constraints of sound engineering, market viability, and the rigorous demands of certification. Despite sonic boom mitigation and engine efficiency improvements, persistent issues, including high fuel consumption, high operating costs, limited demand, and stringent emissions regulations, remain unresolved. Unless SSTs achieve economic viability, environmental compliance, and the mitigation of sonic boom intensity, they are unlikely to progress to commercial reality, at least for now.

Fundamentals of Supersonic Flow

In supersonic aerodynamics, the behavior of the flow encountering a body and the changes in its geometry are fundamentally different from those of subsonic flow. A key distinction is that pressure disturbances in supersonic flow can propagate only downstream; upstream propagation is confined by wavefronts known as Mach waves and shock waves. Consequently, a supersonic flow adjusts locally to geometric changes in the body shape through compression waves, shock waves, or expansion waves.

Basic Flow Physics

When a supersonic flow turns upon itself, such as at a corner or an upstream-facing wedge, the result is compression and the formation of a shock wave. To help explain this behavior, the schlieren flow visualization in Figure 5 shows detailed features of the Mach 2 supersonic flow around a projectile. A shock wave is formed at the nose of the projectile, which, in three dimensions, is cone-shaped and sometimes called a shock cone. The effect of this shock wave, often referred to simply as a “shock,” is to decelerate the flow and reduce its Mach number, accompanied by an increase in static pressure and temperature. This compression through the shock is accompanied by entropy generation and a loss of total pressure, rendering the process inherently irreversible and resulting in wave drag.

Schlieren image of the flow around a supersonic projectile, illustrating the compression waves and expansion fans characteristic of supersonic flow.

Conversely, when a supersonic flow is deflected around a convex corner, it undergoes expansion, as shown in this detailed schlieren image in Figure 5. In this case, a shock wave cannot form. Instead, the supersonic flow adjusts smoothly through a continuous, isentropic expansion called a Prandtl-Meyer^[5] expansion fan, composed of an infinite number of infinitesimally weak Mach waves. These expansion waves allow the supersonic flow to turn and accelerate. Unlike compression shocks, expansion waves increase the Mach number while reducing the static pressure and temperature. Because this process is isentropic, there is no increase in entropy or loss of total pressure.

Notice also in the schlieren image in Figure 5 that a small bump near the projectile’s tail generates a Mach wave. Downstream of the projectile, the flow comprises a contracting, then expanding, turbulent wake, with the turbulence intensity generating a series of significant sound waves that coalesce along another set of Mach waves.

A proper understanding of flow compression by shock waves and expansion by Prandtl-Meyer fans is essential for designing and analyzing high-speed aerodynamic configurations. With a supersonic wing, for example, compression and expansion waves produce the changes in pressure that give rise to lift and drag. In supersonic inlets, multiple oblique shocks compress the flow before it enters the engine. In supersonic nozzles, expansion fans accelerate the flow to higher Mach numbers, thereby producing thrust. Understanding when and how these wave systems arise and their effects on flow is central to designing efficient supersonic vehicles and their propulsion systems, whose details can now be elucidated.

Mach Waves & Shock Waves

The terms Mach wave and shock wave (or shock) are used in many contexts of high-speed flows, so they require further distinction and elaboration. A Mach wave is an infinitesimally weak pressure disturbance that forms in supersonic flow at a specific angle known as the Mach angle, $\mu$ , which depends on the Mach number of the flow, $M$ , and is given by

(1) $\begin{equation*} \mu = \sin^{-1} \left( \frac{1}{M} \right) \end{equation*}$

Consider a case where a flow passes over an infinitely small bump on a flat plate, such as one located on the floor of a wind tunnel, as illustrated in Figure 6. In subsonic flow, the spherically spreading pressure disturbances generated by the bump can propagate both upstream and downstream. Consequently, the flow adjusts smoothly and continuously in response to the bump without producing shock waves or other flow discontinuities. However, in supersonic flow, the bump generates acoustic disturbances that cannot propagate upstream; therefore, the leading edges of the waves accumulate along a wavefront, known as a Mach wave.

In subsonic flow, pressure disturbances can propagate upstream, allowing the flow to adjust gradually to surface irregularities. In supersonic flow, disturbances are confined to wavefronts that propagate downstream behind the Mach wave.

In detail, as the surface rises over the forward portion of the bump, the flow turns toward itself, producing an infinitesimally weak compression wave. As the surface descends over the aft portion, the flow turns away from itself, generating an infinitesimally weak expansion wave. In the limit as the bump becomes infinitesimally small, the compression and expansion disturbances become separate, infinitesimally weak Mach waves inclined at the Mach angle $\mu$ .

This directional behavior is characteristic of linear acoustic wave propagation in supersonic flow. Notably, these weak waves are isentropic, meaning they involve no change in entropy and behave like sound waves, with a leading-edge boundary inclined relative to the flow direction. Therefore, Mach waves are considered reversible and non-dissipative because they do not cause significant changes in pressure, temperature, or entropy. Nevertheless, because they are associated with small but finite changes in density, they can still be visualized using schlieren or shadowgraph techniques.

In contrast, a shock wave is a finite-strength discontinuity in the flow where pressure, temperature, density, and entropy increase abruptly. Shock waves form when a finite deflection compresses the flow, typically when it turns strongly into itself. These waves are non-isentropic and irreversible, resulting in an increase in entropy, a loss of total pressure, and an increase in static temperature. The Mach wave limit represents the boundary between these two types of disturbances; it occurs when the flow deflection angle approaches zero, and the shock wave becomes vanishingly weak. The shock angle approaches the Mach angle in this limit, and the flow transition becomes isentropic. Therefore, a shock wave effectively turns into a Mach wave, and this limit defines the weakest possible compression wave that can exist in a supersonic flow.

To better illustrate these concepts, compare infinitesimal oblique compression and expansion waves, as shown in Figure 7. For the compression wave, the surface, or “wall,” turns upward through a small angle, $\delta$ , forming a concave corner. Suppose the upstream flow is supersonic at $M_1$ as it approaches the corner. After crossing the compression wave, the flow Mach number, $M_2$ , is slightly lower, and the flow is turned upward through the angle $\theta$ , where $\theta = \delta$ . The wave angle is denoted by $\beta$ . A streamline therefore bends upward as it crosses the oblique wave. As $\delta \to 0$ , the wave angle $\beta$ approaches the Mach angle, $\mu$ . The Mach number decreases across the oblique compression wave, while the pressure, density, and temperature all increase. For a finite turning angle, the compression waves coalesce to form an oblique shock wave.

The formation of compression and expansion waves from concave and convex corners.

Now consider an expansion wave. The geometry is similar, except that the wall turns downward, forming a convex corner. Again, assume a supersonic incoming flow. The flow turning angle is $\theta = \delta$ , but the flow now deflects downward. As $\delta \to 0$ , the expansion wave also approaches a Mach wave inclined at the Mach angle, $\mu$ . Across this wave, the Mach number increases, while the pressure, density, and temperature all decrease, opposite to the changes across a compression wave. Therefore, as $\delta \to 0$ , either type of wave approaches an infinitesimally weak Mach wave. For a finite expansion angle, the flow turns smoothly through a Prandtl–Meyer expansion fan composed of a continuous collection of infinitesimal expansion waves that “fan out” from the corner, a process described next.

Compression Waves & the θ-β-M Relation

When a compression surface turns a supersonic flow with a finite angle, such as a wedge, ramp, or airfoil section, an oblique shock wave forms at an angle $\beta$ to the incoming flow direction, as shown in Figure 8.

An oblique shock wave forms when a concave corner deflects a supersonic flow.

The deflection angle $\theta$ , shock angle $\beta$ , and upstream Mach number $M_1$ are related through the $\theta$ – $\beta$ – $M$ relation, which is a fundamental equation in compressible aerodynamics, i.e.,

(2) $\begin{equation*} \tan \theta = 2 \cot \beta \, \frac{M_1^2 \sin^2 \beta - 1}{M_1^2(\gamma + \cos 2\beta) + 2} \end{equation*}$

Equation 2 arises from applying the conservation laws of mass, momentum, and energy across an oblique shock wave, and it assumes steady, two-dimensional, inviscid flow with a perfect gas. The upstream and downstream flow quantities can be resolved into components normal and tangential to the shock, as shown in Figure 9. The normal component undergoes the same changes as flow through a normal shock, while the tangential velocity component remains unchanged.

Normal and tangential components of the upstream and downstream flow across an oblique shock wave.

The upstream and downstream velocity components normal and tangential to the shock satisfy

(3) $\begin{equation*} V_{t,2} = V_{t,1} \qquad \text{and} \qquad \frac{V_{n,2}}{V_{n,1}} = \frac{(\gamma - 1)M_1^2\sin^2\beta + 2} {(\gamma + 1)M_1^2\sin^2\beta} \end{equation*}$

The geometry in Figure 9 gives

(4) $\begin{equation*} \tan\beta = \frac{V_{n,1}}{V_{t,1}} \qquad \text{and} \qquad \tan(\beta-\theta) = \frac{V_{n,2}}{V_{t,2}} \end{equation*}$

and therefore

(5) $\begin{equation*} \frac{\tan(\beta-\theta)}{\tan\beta} = \frac{(\gamma - 1)M_1^2\sin^2\beta + 2} {(\gamma + 1)M_1^2\sin^2\beta} \end{equation*}$

Using the tangent-difference identity and rearranging gives Eq. 2, which provides the key analytical relationship needed to determine either the shock angle $\beta$ or the deflection angle $\theta$ given the upstream Mach number $M_1$ . In the hypersonic limit ( $M_1 \gg 1$ ), the shock angle becomes relatively insensitive to further increases in Mach number. For a slender wedge in hypersonic flow, the shock lies close to the surface, so $\beta$ may be only moderately greater than $\theta$ , producing the characteristic thin shock layer.

Notice that Eq. 2 is nonlinear and is typically solved graphically or numerically. A family of $\theta$ – $\beta$ curves for different Mach numbers provides quick visual insight, as shown in Figure 10, and is widely used in aerodynamics and propulsion. It will be apparent that for a given upstream Mach number, $M_1$ , a maximum deflection angle $\theta_{\max}$ exists beyond which an oblique shock cannot remain attached. When $\theta > \theta_{\max}$ , the shock becomes detached and forms a bow shock.

Relationship between the flow deflection angle $\theta$ , the shock wave angle $\beta$ , and the upstream Mach number $M_1$ for oblique shock waves.

It will be apparent from the plot that for each $\theta$ there are generally two solutions for $\beta$ : a weak shock, with a lower shock angle and less compression, and a strong shock, with a higher shock angle and greater compression. The downstream flow is usually supersonic for the weak-shock solution and may be subsonic for the strong-shock solution. In most external aerodynamic flows, the weak-shock solution is observed. The strong-shock solution can occur when imposed downstream conditions require it.^[6]

Prandtl-Meyer Expansion Theory

The behavior of supersonic flow undergoing expansion around a convex corner is governed by the Prandtl-Meyer theory. In contrast to the discontinuous nature of shock waves, the expansion process is continuous and isentropic, consisting of a fan of Mach waves that enables the flow to turn smoothly and accelerate, as shown in Figure 11. Notice that for any ray in the expansion fan, the Mach angle is $\mu$ .

A Prandtl-Meyer expansion fan is formed when a supersonic flow encounters a convex corner.

To quantify this expansion process, the Prandtl-Meyer function, $\nu(M)$ , gives the angle through which a flow must be turned to accelerate from Mach 1 to a given Mach number $M$ . Its development begins by considering an infinitesimal turning of the flow through the angle $d\theta$ across a Mach wave. The differential relation between the turning angle and Mach number is

(6) $\begin{equation*} d\theta = \frac{\sqrt{M^2-1}}{M\left(1+\dfrac{\gamma-1}{2}M^2\right)}\,dM \end{equation*}$

Integrating from the sonic condition, where $M$ = 1 and $\nu$ = 0, to an arbitrary supersonic Mach number, $M$ , gives

(7) $\begin{equation*} \nu(M) = \mathop{\Large\int}\limits_1^M \frac{\sqrt{\widetilde{M}^{\,2}-1}} {\widetilde{M}\left[1+\dfrac{\gamma-1}{2}\widetilde{M}^{\,2}\right]} \,d\widetilde{M} \end{equation*}$

which evaluates to

(8) $\begin{equation*} \nu(M) = \sqrt{\frac{\gamma + 1}{\gamma - 1}} \tan^{-1} \left( \sqrt{\frac{\gamma - 1}{\gamma + 1}(M^2 - 1)} \right) - \tan^{-1} \left( \sqrt{M^2 - 1} \right) \end{equation*}$

where $\gamma$ = 1.4 is the ratio of specific heats for air. The Prandtl-Meyer function is zero at Mach 1 and increases monotonically with increasing Mach number, as shown in Figure 12.

The Prandtl-Meyer function for different values of the ratio of specific heats.

Notice that $\nu(M)$ is an angle but not a physical angle in the flow. The total turning angle $\theta$ required to expand a flow from an initial Mach number $M_1$ to a higher Mach number $M_2$ is given by the difference in the values of the Prandtl-Meyer function, i.e.,

(9) $\begin{equation*} \theta = \nu(M_2) - \nu(M_1) \end{equation*}$

This equation allows the calculation of the downstream Mach number and turning angle for any given supersonic expansion. Because the process is isentropic, total pressure remains constant, and only the static properties, i.e., pressure, temperature, and density, decrease across the expansion fan as the Mach number increases.

If the upstream Mach number, $M_1$ , and the expansion angle, $\theta$ , are known, then the downstream Mach number, $M_2$ , is found in three steps. First, calculate the Prandtl-Meyer angle corresponding to the upstream Mach number, i.e.,

(10) $\begin{equation*} \nu_1 = \nu(M_1) \end{equation*}$

Second, add the expansion angle to obtain the downstream Prandtl-Meyer angle, i.e.,

(11) $\begin{equation*} \nu_2 = \nu_1 + \theta \end{equation*}$

or, equivalently,

(12) $\begin{equation*} \nu(M_2) = \nu(M_1) + \theta \end{equation*}$

Third, use the Prandtl-Meyer table or graph in Figure 12 to find the Mach number corresponding to $\nu_2$ , i.e.,

(13) $\begin{equation*} M_2 = \nu^{-1}(\nu_2) \end{equation*}$

where $\nu^{-1}$ denotes finding the Mach number corresponding to a specified value of the Prandtl-Meyer function. Because the equation for $\nu(M)$ cannot be rearranged explicitly to give $M$ , the value of $M_2$ must be obtained from a table or graph, or found numerically by trying values of $M_2$ until the calculated value of $\nu(M_2)$ agrees with $\nu_2$ .

If both $M_1$ and $M_2$ are known, then the turning angle is found directly from

(14) $\begin{equation*} \theta = \nu(M_2) - \nu(M_1) \end{equation*}$

The dashed lines in Figure 12 show the limiting value of the Prandtl-Meyer function as the Mach number approaches infinity, i.e.,

(15) $\begin{equation*} \nu_{\infty} = \frac{\pi}{2} \left( \sqrt{\frac{\gamma + 1}{\gamma - 1}} - 1 \right) \end{equation*}$

Prandtl-Meyer function table.
Mach, M	ν(M) (deg.)	Mach, M	ν(M) (deg.)	Mach, M	ν(M) (deg.)	Mach, M	ν(M) (deg.)
1.0	0.00	1.2	3.56	1.4	8.99	1.6	14.86
1.8	20.73	2.0	26.38	2.2	31.73	2.4	36.75
2.6	41.41	2.8	45.75	3.0	49.76	3.2	53.47
3.4	56.91	3.6	60.09	3.8	63.04	4.0	65.78
4.2	68.33	4.4	70.71	4.6	72.92	4.8	74.99
5.0	76.92	5.2	78.73	5.4	80.43	5.6	82.03
5.8	83.54	6.0	84.96	6.2	86.29	6.4	87.56
6.6	88.76	6.8	89.89	7.0	90.97	7.2	92.00
7.4	92.97	7.6	93.90	7.8	94.78	8.0	95.62
8.2	96.43	8.4	97.20	8.6	97.94	8.8	98.64
9.0	99.32	9.2	99.97	9.4	100.59	9.6	101.19

Prandtl-Meyer expansions play a crucial role in the design of nozzles for supersonic and hypersonic applications, where flows are expanded to achieve desired velocities. They also appear in external flows around corners and the trailing edges of supersonic airfoils, where expansion fans determine surface pressure distributions and lift.

Summary of Shock Flow Relations

Across a shock wave, the flow is no longer isentropic because entropy increases and irreversible losses occur. Therefore, the isentropic relations are no longer valid across the shock itself, and one must use the normal shock relations based on conservation of mass, momentum, and energy. For an oblique shock, these relations are applied to the normal component of the upstream Mach number, i.e.,

(16) $\begin{equation*} M_{n,1} = M_1 \sin \beta \end{equation*}$

The key ratios of pressure, temperature, and density across the shock are

(17) $\begin{equation*} \frac{p_2}{p_1} = 1 + \frac{2\gamma}{\gamma + 1} \left(M_{n,1}^2 - 1\right) \quad \text{and} \quad \frac{T_2}{T_1} = \frac{\left( 2\gamma M_{n,1}^2 - (\gamma - 1) \right) \left( (\gamma - 1) M_{n,1}^2 + 2 \right)}{(\gamma + 1)^2 M_{n,1}^2} \end{equation*}$

as well as

(18) $\begin{equation*} \frac{\varrho_2}{\varrho_1} = \frac{(\gamma + 1) M_{n,1}^2}{(\gamma - 1) M_{n,1}^2 + 2} \end{equation*}$

The downstream Mach number is then obtained from the post-shock normal Mach number, i.e.,

(19) $\begin{equation*} M_{n,2}^2 = \frac{1 + \dfrac{\gamma - 1}{2} M_{n,1}^2}{\gamma M_{n,1}^2 - \dfrac{\gamma - 1}{2}} \quad \text{and} \quad M_2 = \frac{M_{n,2}}{\sin(\beta - \theta)} \end{equation*}$

In contrast, an expansion fan involves smooth, continuous, isentropic turning of the flow. The flow remains isentropic through the expansion, so the pressure, temperature, and density ratios are

(20) $\begin{equation*} \frac{p_2}{p_1} = \left( \dfrac{1 + \dfrac{\gamma - 1}{2} M_1^2} {1 + \dfrac{\gamma - 1}{2} M_2^2} \right)^{\gamma/(\gamma - 1)} \qquad \text{and} \qquad \frac{T_2}{T_1} = \frac{1 + \dfrac{\gamma - 1}{2} M_1^2} {1 + \dfrac{\gamma - 1}{2} M_2^2} \end{equation*}$

and

(21) $\begin{equation*} \frac{\varrho_2}{\varrho_1} = \left( \dfrac{1 + \dfrac{\gamma - 1}{2} M_1^2} {1 + \dfrac{\gamma - 1}{2} M_2^2} \right)^{1/(\gamma - 1)} \end{equation*}$

These expressions enable a complete description of changes in flow properties across compression shocks and expansion waves in supersonic flow.

Why is γ =1.4 for air?

In compressible flow analysis, the ratio of specific heats, denoted by $\gamma$ , plays a critical role. For air at standard conditions, it is well known that $\gamma$ = 1.4. But where does this number come from? The ratio $\gamma$ (also called the adiabatic index) is defined as the ratio of the specific heat at constant pressure $c_p$ to the specific heat at constant volume ${c_v}$ , i.e.,

$\gamma = \frac{c_p}{c_v}$

These specific heats indicate the amount of energy per unit mass required to raise the temperature of a gas under different constraints: ${c_v}$ is at constant volume (no expansion), and $c_p$ is at constant pressure (allowing expansion).

Air is primarily a diatomic gas (about 78% nitrogen and 21% oxygen), and at moderate temperatures (below approximately 1,000 K), it behaves like an ideal diatomic gas. According to the equipartition theorem, each active quadratic degree of freedom contributes one-half of the appropriate gas constant to the constant-volume specific heat. For a diatomic gas at room temperature, there are three translational degrees of freedom and two rotational degrees of freedom. The vibrational modes are largely frozen at these temperatures. Therefore,

$c_v = \small \left( \frac{5}{2} \right) R$

Using the thermodynamic relationship

$c_p = c_v + R$

then $\gamma$ becomes

$\gamma = \frac{c_p}{c_v} = \frac{c_v + R }{c_v} = \small \frac{7/2}{5/2} = \frac{7}{5} = 1.4$

This value of $\gamma$ = 1.4 is essential in compressible flows because it appears in the speed of sound, i.e., $a = \sqrt{\gamma \, R \, T}$ as well as in isentropic flow relations, shock wave equations, etc. It is used to model air as a perfect gas in subsonic, supersonic, and many hypersonic flows. However, the assumption $\gamma$ = 1.4 becomes invalid under the following conditions:

High temperatures (typically above 1,000 K): vibrational modes become active, increasing ${c_v}$ and reducing $\gamma$ .
Real gas effects: dissociation, ionization, or chemically reacting flows.
Different gas types: monatomic gases, polyatomic gases, etc.

For example, monatomic gases such as helium have $\gamma = 5/3 \approx 1.67$ , while diatomic gases at elevated temperatures have $\gamma < 1.4$ . Therefore, $\gamma$ = 1.4 is used for air because it behaves like an ideal diatomic gas with five active degrees of freedom near standard atmospheric temperatures. This approximation is foundational to many compressible flow models, including the isentropic and shock relations previously discussed. However, this assumption must be revisited at high temperatures or in chemically reacting flows.

Supersonic Airfoil Characteristics

The flow physics of airfoils and wings at supersonic flight conditions are significantly different from the corresponding flow physics in subsonic flight, which is why the Mach number is crucial in classifying aerodynamic flows. In supersonic flight, the formation of shock waves and expansion waves significantly impacts the performance of the wing or airfoil. On the one hand, shock waves cause drag and increase the pressure on the wing’s surface. On the other hand, expansion waves arise when a supersonic flow is turned away from itself, reducing the pressure on the wing’s surface and creating lift. Therefore, the physics of these compressibility effects must be carefully considered when designing airfoils and wings for supersonic airplanes.

Supersonic Airfoil Shapes

Figure 13 shows that a supersonic airfoil or wing typically features a sharp leading edge and relatively flat upper and lower surfaces to minimize wave drag while producing the required lift. The airfoil’s maximum thickness is critical to its performance at supersonic speeds. The best supersonic airfoil designs are thin (i.e., low thickness-to-chord ratios) and mildly cambered, featuring specific thickness distributions and curvature to manage shock waves and regions of flow expansion. These designs focus on minimizing wave drag and optimizing the achievable lift-to-drag ratios. A diamond-wedge airfoil is a specific type of airfoil shape used in certain supersonic flight applications.

Flows about supersonic airfoils inevitably involve oblique shock waves and expansion fans.

Diamond-wedge airfoils, also called double-wedge airfoils, are often used for control surfaces on missiles and other supersonic vehicles. However, simple diamond-wedge sections are less suitable for aircraft that must also operate efficiently at low Mach numbers and high lift coefficients, such as during takeoff and landing. Operational supersonic aircraft may instead employ thin biconvex circular-arc or modified supersonic sections, supplemented by suitable planform shaping and high-lift devices for low-speed flight.

Flow Patterns & Pressure Distributions

Consider the flow about an airfoil in the form of a double-wedge or diamond shape experiencing a supersonic flow, as shown in Figure 14. Notice that in supersonic flow, pressure disturbances cannot propagate upstream of the airfoil; the upstream streamlines remain straight until they encounter the leading-edge shock waves. Oblique compression shock waves occur at the leading edge of the airfoil. The Mach number across the shock waves decreases (but remains supersonic), whereas the static pressure increases relative to the freestream value. At a positive angle of attack, the greater pressure increase occurs on the lower surface, thereby contributing significantly to lift.

Representative flow pattern and pressure distributions about a supersonic double-wedge (diamond) airfoil shape.

At the points of maximum airfoil thickness, expansion waves form, causing the Mach number to increase and the pressure to decrease after the expansion is complete. The upper surface of the airfoil now contributes to lift production. Compression waves form at the trailing edge as the flow turns back toward the freestream direction. These waves reduce the Mach number and raise the pressure toward the freestream value. This pressure distribution explains why the resultant lift on a symmetric thin airfoil in linearized supersonic flow acts at the mid-chord. Therefore, for nonzero lift, the center of pressure is located at the mid-chord, which is also the aerodynamic center.

Check Your Understanding #1 – Supersonic flow over a double wedge

A supersonic airflow at $M_\infty = 2.5$ , $p_\infty$ = 50,000 Pa, and $T_\infty$ = 250 K encounters a double wedge airfoil. The first wedge deflects the flow upward by $\theta_1 = 10^\circ$ (compression), and the second wedge deflects the flow downward by $\theta_2 = -7^\circ$ (expansion). Assume air is a perfect gas with $\gamma$ = 1.4. Determine:

The Mach number and pressure immediately after the oblique shock, i.e., $M_1$ and $p_1$ , respectively.
The Mach number and pressure after the expansion fan, i.e., $M_2$ and $p_2$ , respectively.

Show solution/hide solution.

We can use the $\theta$ – $\beta$ – $M$ relation for a compression wave to find the shock angle $\beta$ for $M_\infty = 2.5$ and $\theta = 10^\circ$ . From the graph or the equation, this gives the weak-shock solution $\beta = 31.9^\circ$ . Then we can compute the normal component of the Mach number using
$M_n = M_\infty \sin \beta = 2.5 \sin(31.9^\circ) = 1.32$

The normal shock relations give the downstream normal Mach number, i.e.,

$M_{n,2}^2 = \frac{1 + \dfrac{\gamma - 1}{2} M_n^2}{\gamma M_n^2 - \dfrac{\gamma - 1}{2}} = 0.776^2$

The post-shock Mach number is then

$M_1 = \frac{M_{n,2}}{\sin(\beta - \theta)} = \frac{0.776}{\sin(21.9^\circ)} = 2.09$

The pressure ratio across the shock is

$\frac{p_1}{p_\infty} = 1 + \frac{2\gamma}{\gamma + 1}(M_n^2 - 1) = 1.86$

so that

$p_1 = 1.86 \times 50,000 = 93,200 \text{ Pa}$
Using the Prandtl-Meyer expansion function, then
$\nu(M_1 = 2.09) = 28.7^\circ \quad \Rightarrow \quad \nu(M_2) = \nu(M_1) + 7^\circ = 35.7^\circ$

Inverting the function numerically or using tables, then $M_2 = 2.36$ . Using the isentropic relation for pressure gives

$\frac{p_2}{p_1} = \left( \dfrac{1 + \dfrac{\gamma - 1}{2} M_1^2}{1 + \dfrac{\gamma - 1}{2} M_2^2} \right)^{\gamma/(\gamma - 1)} = 0.654$

so that

$p_2 = 0.654 \times 93,200 = 60,900 \text{ Pa}$

Lift & Drag Characteristics

For a thin airfoil in incompressible flow (i.e., linearized thin airfoil theory below stall), the relationship between the lift coefficient, $C_l$ , and angle of attack, ${\alpha}$ , is given by

(22) $\begin{equation*} C_l = C_{l_{\alpha}} \, \alpha = 2 \pi \, \alpha \end{equation*}$

where the lift-curve slope, $C_{l_{\alpha}}$ , is equal to $2 \pi$ per radian of angle of attack. For linearized subsonic flow, the relationship changes to

(23) $\begin{equation*} C_l = \frac{2 \pi \, \alpha }{\sqrt{1 - M_{\infty}^2}} \end{equation*}$

where $\sqrt{1 - M_{\infty}^2}$ is known as the Prandtl-Glauert factor. Therefore, it is apparent from Eq. 23 that the lift-curve slope continuously increases in subsonic flow with increasing Mach number. However, notice that this result becomes singular as $M_{\infty}$ approaches one, which is one limitation of the linear theory.

For linearized supersonic flow, the relationship between the lift coefficient and angle of attack is given by

(24) $\begin{equation*} C_l = \frac{4 \, \alpha }{\sqrt{M_{\infty}^2 - 1}} \end{equation*}$

This result indicates that the supersonic lift-curve slope decreases with increasing Mach number. As with the corresponding subsonic result, the linearized expression becomes singular as $M_{\infty}$ approaches 1 and is not valid in the transonic regime.

The results in Eqs. 22 and 24 are considered classical solutions in airfoil theory and allow determination of the lift coefficients for airfoils at small angles of attack. These thin-airfoil approximations agree well with measurements of thin two-dimensional airfoils, as shown in Figure 15. For finite wings, three-dimensional effects must also be accounted for, although the two-dimensional results remain useful for interpreting local sectional behavior. However, for supersonic wings, the regions influenced by subsonic versus supersonic flow must still be considered.

Linearized subsonic and supersonic thin airfoil theory largely agrees with measurements, at least up to a Mach number of 3.0.

For symmetric thin airfoils, linearized supersonic theory shows that the lift coefficient at a given angle of attack is independent of the thickness distribution. This means that a flat plate, a symmetric diamond-wedge airfoil, and a symmetric biconvex airfoil have the same lift coefficient to first order. The drag, however, is different because the dominant source of drag is wave drag, which depends strongly on the details of the airfoil shape.

For a flat plate airfoil, the wave drag coefficient, $C_{d_{w}}$ , is

(25) $\begin{equation*} C_{d_{w}} = \frac{4 \alpha^2 }{\sqrt{M_{\infty}^2 - 1}} \end{equation*}$

and for a diamond-wedge airfoil with a half angle, $\epsilon$ , then

(26) $\begin{equation*} C_{d_{w}} = \frac{4 \left( \alpha^2 + \epsilon^2 \right) }{\sqrt{M_{\infty}^2 - 1}} \end{equation*}$

If the chord of the airfoil is ${c}$ , with a maximum thickness $t$ at mid-chord (i.e., at $c/2$ ), from the geometry of the wedge then

(27) $\begin{equation*} \tan \epsilon = \frac{t}{c} = \epsilon ~\mbox{~for small angles} \end{equation*}$

Therefore, Eq. 26 can be written as

(28) $\begin{equation*} C_{d_{w}} = \frac{4}{\sqrt{M_{\infty}^2 - 1}} \bigg( \alpha^2 + \left( \frac{t}{c} \right)^2 \bigg) \end{equation*}$

which shows that wave drag increases with the square of the airfoil thickness. The wave drag of a biconvex airfoil with small curvature, i.e., a large radius of curvature, is

(29) $\begin{equation*} C_{d_{w}} = \frac{4}{\sqrt{M_{\infty}^2 - 1}} \bigg( \alpha^2 + \frac{4}{3} \left( \frac{t}{c} \right)^2 \bigg) \end{equation*}$

In general, the wave drag of a supersonic airfoil can be expressed as

(30) $\begin{equation*} C_{d_{w}} = \frac{4}{\sqrt{M_{\infty}^2 - 1}} \bigg( \alpha^2 + k_t \left( \frac{t}{c} \right)^2 + k_c \left( \frac{f}{c} \right)^2 \bigg) \end{equation*}$

where $t/c$ is the maximum thickness-to-chord ratio and $f/c$ is the maximum camber-to-chord ratio. The values of the constants $k_t$ and $k_c$ depend on the exact airfoil shape.

Nevertheless, regardless of the shape, one of the most interesting and essential characteristics of an airfoil operating in supersonic flow is that its wave drag is proportional to the square of its thickness-to-chord ratio and the square of its camber. Therefore, it becomes clear why the airfoils used on the wings of supersonic airplanes must be very thin and mildly cambered compared to those used on subsonic airplanes.

Check Your Understanding #2 – Lift-to-drag ratio of a supersonic airfoil.

At what angle of attack is the maximum lift-to-drag ratio obtained for a diamond-wedge supersonic airfoil with a wedge half angle $\epsilon$ ?

Show solution/hide solution.

The lift coefficient for a diamond-wedge airfoil is given by

$C_l = \frac{4}{\sqrt{M_{\infty}^2 - 1}} \, \alpha$

and the wave drag coefficient by

$C_{d_{w}} = \frac{4 \left( \alpha^2 + \epsilon^2 \right) }{\sqrt{M_{\infty}^2 - 1}}$

Therefore, the lift-to-drag ratio is

$\frac{C_l}{C_{d_{w}}} = \frac{\alpha}{\alpha^2 + \epsilon^2}$

Differentiation with respect to ${\alpha}$ gives

$\frac{d (C_l/C_{d_{w}})}{d \alpha} = \frac{\epsilon^2 - \alpha^2}{(\alpha^2 + \epsilon^2)^2}$

which must be equal to zero for a maximum or minimum, i.e.,

$\frac{\epsilon^2 - \alpha^2}{(\alpha^2 + \epsilon^2)^2} = 0$

Therefore, the positive-lift maximum occurs at

$\alpha = \epsilon$

where both angles are expressed in radians. Substituting back gives the maximum lift-to-drag ratio as

$\left. \frac{C_l}{C_{d_w}} \right\rvert_{\rm max} = \frac{1}{2 \epsilon}~\mbox{~at $\alpha = \epsilon$}$

Pitching Moments

Another important characteristic of supersonic airfoils and wings is the aft movement of the aerodynamic center. For a thin subsonic airfoil, the aerodynamic center is close to the quarter-chord. For a thin airfoil in linearized supersonic flow, the aerodynamic center is close to the mid-chord. This result follows from the pressure distribution produced by compression and expansion waves, which places the incremental lift distribution farther aft than in subsonic flow.

The practical consequence is that a wing or airplane transitioning from subsonic to supersonic flight can experience a significant change in pitching moment. If the airplane’s center of gravity remains fixed while the aerodynamic center moves aft, then changes in lift act with a longer nose-down moment arm about the center of gravity. This effect has important implications for longitudinal trim and stability, especially during acceleration through the transonic regime and into supersonic flight.

As shown in Figure 16, this aft movement of the aerodynamic center can produce a sizeable nose-down pitching moment about the airplane’s center of gravity. This fundamental change in aerodynamics has severe implications for the longitudinal trim and stability of a supersonic airplane as it transitions between subsonic and supersonic flight conditions.^[7]

In supersonic flight, the aerodynamic center moves aft toward the mid-chord of the airfoil section, which can produce significant nose-down pitching moments.

Today, designers can balance these moments using aerodynamic forces, such as from a foreplane canard on some supersonic fighter airplanes, or by pumping a specific weight of fuel to the trim tanks further aft inside the wings to gain the proper location of the center of gravity, such as was done with the fuel system on Concorde.

Check Your Understanding #3 – Calculation of the lift-to-drag ratio of a supersonic airfoil.

A supersonic double-wedge airfoil with a thickness-to-chord ratio of 10% is operated at an angle of attack of 5.5 degrees at a Mach number of 2.5. For these conditions, calculate the lift and drag coefficients, the lift-to-drag ratio, and the moment coefficient about the leading edge.

Show solution/hide solution.

In this case, the values given lead to

$\sqrt{M_{\infty}^2 - 1} = \sqrt{2.5^2 - 1} = 2.291$

and

$\alpha = 5.5 \times \frac{\pi}{180} = 0.096~\mbox{rads}$

The lift coefficient is

$C_l = \frac{4 \, \alpha }{\sqrt{M_{\infty}^2 - 1}} = \frac{ 4 \times 0.096}{2.291} = 0.168$

The wave drag coefficient is

$C_{d_{w}} = \frac{4}{\sqrt{M_{\infty}^2 - 1}} \bigg( \alpha^2 + \left( \frac{t}{c} \right)^2 \bigg) = \frac{4.0}{2.291} \left( 0.096^2 + 0.1^2 \right) = 0.0336$

Therefore, the lift-to-drag ratio is

$\frac{C_l}{C_{d_{w}} } = \frac{0.168}{0.0336} = 5.0$

The pitching moment coefficient about the leading edge will be

$C_{m_{\rm le}} = -\frac{x_{\rm cp}}{c} \, C_l = -0.5 \times 0.168 = -0.084$

The minus sign denotes a nose-down moment.

Supersonic Finite Wings

Interest in the theory and practice of supersonic flight continued at various research centers such as the NACA, with wind tunnel tests confirming the development of the high drag on an airplane in supersonic flight, which had been known as the “sound barrier.” Although Mach 1 is not a barrier per se, these conditions result in significant changes in the flow physics of the airplanes that can affect their flight performance, stability, and control characteristics.

After the work of Betz and Busemann before WWII, Robert T. Jones at NACA Langley developed an aerodynamic theory for supersonic delta wings, his work “Wing Plan Forms for High-Speed Flight” being published as NACA TR-863 in 1947. His approach combined the theory of thin supersonic airfoils with highly swept delta wings. This wing design served as the basis for the first generation of supersonic delta-wing airplanes.

Subsequently, several different supersonic airplane prototypes were built and flown by American and British manufacturers, including the Convair XF-92 and the Fairey Delta 2, the latter setting a new World air speed record on 10 March 1956, achieving Mach 1.73. However, such airplanes did not always have good handling qualities and required high landing speeds. Consequently, there were many mishaps with the first generation of supersonic airplanes, including stall/spin events and other departures from controlled flight, often resulting in fatalities. In this regard, the Lockheed F-104 Starfighter became one of the most notorious among other jet fighter airplanes.

Mach Angle

As an airplane approaches Mach 1, pressure disturbances generated by different parts of the airplane strengthen and may coalesce into shock waves. Once the airplane exceeds Mach 1, disturbances are confined within Mach cones that extend downstream from their sources. In three dimensions, the envelope of infinitesimal pressure disturbances forms a Mach cone that extends downstream from each disturbance source, as shown schematically in Figure 17.

As an airplane approaches the speed of sound, pressure waves merge ahead of it, and at supersonic speeds, a Mach cone forms at the nose.

Figure 18 shows schlieren visualization images of the supersonic flow about a model of a fighter airplane in a wind tunnel. Notice the build-up in the number and intensity of the almost normal shock waves as Mach 1 is approached. In supersonic flight, the Mach cone becomes increasingly swept back with increasing flight Mach number, $M_{\infty}$ . The angle $\mu$ is used to denote the Mach angle, as given by

(31) $\begin{equation*} \mu = \sin^{-1} \left( \frac{1}{M_{\infty}} \right) \end{equation*}$

which is the half-apex angle of the Mach cone.

Motiv: Slirbilder av vindtunnelmodell av flygplan J 35 i skala 1:30. Tagna i vindtunnel S4 på Flygtekniska försöksanstalten, FFA. Ur Flygvapenmuseums bildarkiv. — Schlieren flow visualization images show the formation of shock waves as a fighter airplane model in a wind tunnel as the flow transitions from transonic to supersonic.

Therefore, the Mach angle is 90 $^{\circ}$ at $M_{\infty} = 1$ , and its value decreases quickly beyond Mach 1, as shown in Figure 19. Notice that even by a Mach number of 2, the Mach cone is already swept back 60 $^{\circ}$ to a half apex angle of $\mu = 30^{\circ}$ . At hypersonic speeds, the Mach angle becomes very small, so sweeping the wing behind the Mach cone is generally impractical. Therefore, a blunt or bluff body shape is often a better design solution for keeping shock waves away from surfaces.

Variation in the Mach angle with Mach number, showing that the Mach angle decreases rapidly beyond Mach 1.

In Figure 20, the signature of the shock waves and Mach cone can also be seen through the natural condensation of water vapor in the air in the lower-pressure and slightly lower temperature regions produced over the airplane. In this case, the visible condensation forms in regions where the flow expands, causing the static pressure and temperature to decrease. A compression shock raises the static pressure and temperature and does not produce this condensation. This effect occurs only with humid air conditions, and the lighting must also be suitable to capture such an image.

Natural condensation of water vapor reveals regions of reduced pressure around an airplane flying near the speed of sound.

The photograph in Figure 21 is a fascinating schlieren image of the plethora of Mach (shock) waves generated by an actual airplane during supersonic flight. The optical method for visualizing this flow is called Airborne Background Oriented Schlieren (ABOS), the light source is the sun, and the image is taken from a chase airplane. Notice the particularly strong (i.e., darker-looking) Mach waves at the nose and tail of the airplane, which are those responsible for the sonic booms heard on the ground. Also apparent in this image is the turbulence from the engine exhaust.

A rather dramatic image of the Mach waves generated by an airplane in supersonic flight as obtained using an “Airborne Background Oriented Schlieren” system. In this case, the Mach angle indicates that the aircraft is flying at approximately Mach 1.4. (NASA image.)

Check Your Understanding #4 – Flight Mach number of an airplane from the shock wave angle

The half-angle of the Mach cone generated by a supersonic jet fighter is 25 degrees. What is the flight Mach number of the airplane? What is the airplane’s true airspeed if the local air temperature is -20^oC?

Show solution/hide solution.

The half apex angle of the Mach cone angle, $\mu$ , is given by

${}\mu = \sin^{-1} \left( \frac{1}{M_{\infty}} \right)$

Therefore,

$M_{\infty} = \frac{1}{\sin \mu} = \frac{1}{\sin 25^{\circ}} = 2.36$

The absolute temperature, $T_{\infty}$ , is -20 + 273.15 = 253.15 K, so the speed of sound is

$a_{\infty} = \sqrt{\gamma \, R \, T_{\infty} } = \sqrt{ 1.4 \times 287.057 \times 253.15} = 318.96~\mbox{m/s}$

Therefore, the true airspeed of the jet airplane is

$V_{\infty} = a_{\infty} M_{\infty} = 318.96 \times 2.36 = 752.75~\mbox{m/s}$

Wing Sweep

The Mach angle is essential in supersonic wing design because pressure disturbances in a supersonic flow are confined to the cone region as determined by the Mach angle. Unlike a subsonic flow, there is no upstream influence outside the Mach cone in a supersonic flow; pressure disturbances are confined to the Mach cone and the region downstream within it. The consequence of this fact is important because, if the wing’s leading edge is swept back behind the Mach cone, the wing experiences relatively lower drag (i.e., lower wave drag), as shown in the schlieren images in Figure 22. Otherwise, if the shock wave reaches the wing, the shock’s effects will disrupt the boundary-layer flow, increasing drag and potentially causing flow separation.

Schlieren flow visualization images made in the wind tunnel show that sweeping the wings behind the Mach wave eliminates adverse interactions between the shock waves and the wings. (NACA image.)

Consider a thin wing of infinite span, as shown in Figure 23. The wing’s leading edge is swept back through an angle ${\Lambda}$ measured from the spanwise direction, so the component of the freestream Mach number normal to the leading edge is $M_{\infty}\cos\Lambda$ . The normal component, $M_{\infty} \cos \Lambda$ , affects the wing’s chordwise pressures and aerodynamic loading; the spanwise component $M_{\infty} \sin \Lambda$ is assumed to have no effect on the flow developments over the wing. This result is called the independence principle of sweep, or simply the independence principle, after Adolf Busemann. Notice also that the “effective” airfoil shape affected by the flow has a longer chord and a lower thickness-to-chord ratio, thereby increasing the critical Mach number.

The independence principle states only the Mach number normal to the leading edge affects the aerodynamics.

Consider Figure 24, which shows a two-dimensional section of a swept wing in supersonic flow. If the component $M_{\infty} \cos \Lambda$ is less than unity, then the leading edge is subsonic in the sense that disturbances can propagate along the leading edge. The relevant condition is

(32) $\begin{equation*} M_{\infty}\cos\Lambda < 1 \end{equation*}$

or

(33) $\begin{equation*} \cos \Lambda < \frac{1}{M_{\infty}} \end{equation*}$

which is equivalent to

(34) $\begin{equation*} \frac{\pi}{2}-\Lambda < \mu \end{equation*}$

where $\mu$ is the Mach angle. In this case, the leading-edge shock is avoided or greatly weakened, and the leading-edge contribution to wave drag is reduced. However, a finite-thickness finite wing may still have wave drag associated with its thickness, volume distribution, lift, and other compressibility effects.

A swept wing with subsonic and supersonic leading edges. Limiting the buildup of wave drag on the wing requires that the leading edge be swept behind the Mach cone.

If the component $M_{\infty} \cos \Lambda$ is greater than unity, then the chordwise pressures and wing loadings will be the same as those on an unswept wing in a supersonic freestream of Mach number $M_{\infty} \cos \Lambda$ . In this case, point Q is outside the Mach cone from all points such as P, and the wing is said to have a supersonic leading edge, i.e.,

(35) $\begin{equation*} \cos \Lambda > \frac{1}{M_{\infty}} \end{equation*}$

which means

(36) $\begin{equation*} \sin \left( \frac{\pi}{2} - \Lambda \right) > \sin \mu \end{equation*}$

so $\pi/2 - \Lambda > \mu$ . Therefore, it becomes clear that a wing flying with a supersonic leading edge will create a larger wave drag.

Therefore, a desirable design goal for many supersonic wings is to employ sufficient sweep to obtain a subsonic leading edge at the design Mach number. In this case, the leading edge lies inside the Mach cone, so the flow can adjust more gradually around the leading edge and the leading-edge shock contribution to wave drag is reduced. These preceding arguments apply most directly to an idealized wing of infinite span, or to portions of a finite-span wing where the flow can be considered approximately two-dimensional, i.e., away from the wing tips.

Using wing sweepback to reduce (wave) drag is most effective for larger aspect ratio wings because using too much sweep also reduces the effective aspect ratio of the wing. The disadvantage of using sweepback is that the reduced freestream velocity component normal to the wing’s leading edge lowers the lift-curve slope and generally increases the airplane’s stall airspeed. The lower effective aspect ratio results in higher induced drag on the wing during subsonic operation, which is undesirable. Nevertheless, the wing must fly at lower airspeeds and higher angles of attack during takeoff and landing.

Figure 25, derived from wind-tunnel measurements, summarizes how wing sweep angle profoundly affects drag in the high subsonic, transonic, and supersonic regimes. As is now apparent, this effect arises because swept-back wings reduce the wave drag and prevent the shock waves from interfering with the flow over the wings, which can create flow separation and a further increase in drag. Notice that sweep delays the rise of the transonic drag and reduces the rate at which the drag increases in the transonic regime. However, although swept wings can delay the increase in drag from compressibility effects, they can also introduce other aerodynamic problems, higher structural weight, and aeroelastic concerns. Therefore, in practice, airplane designers tend to use as little wing sweep as possible to achieve aerodynamic goals, with 20-30 degrees of sweep typical of modern commercial jet airliners.

Sweepback on a wing provides for a very significant reduction in its drag.

The effects of sweep angle are also summarized in Figure 26, which shows the lift-to-drag ratio of a swept wing versus an unswept wing as a function of flight Mach number. In light of the preceding analysis, it becomes clear why some sweepback of a wing can minimize transonic and supersonic drag. On the one hand, as an airplane with an unswept wing approaches Mach 1, its drag increases rapidly, and its lift-to-drag ratio plummets immediately. On the other hand, it can be seen that the highly swept delta wing has a much lower lift-to-drag ratio at lower Mach numbers compared to the unswept wing, but it has a much more acceptable lift-to-drag ratio in supersonic flight.

The variation in the lift-to-drag ratio versus Mach number in the supersonic regime of unswept and swept wings.

Therefore, there must be trades and design compromises when selecting the sweep angle of an airplane wing. The moderately swept-back cantilever wing with high-lift devices is optimal for flying at Mach 0.8, while the delta wing is best for flight at Mach 2. Delta wings, in particular, are well-suited for continuous supersonic flight. A highly swept wing with a completely subsonic leading edge will perform very well at supersonic speeds and low lift coefficient, but at the cost of poorer low-speed performance, i.e., higher induced drag because of its low aspect ratio wing. A swept wing produces less lift than an equivalent unswept wing, resulting in a higher stall speed and also less maneuver capability because of the lower stall margin. This is why some airplanes, including the B-1 bomber, the F-14, and Tornado fighters, use variable-sweep wings to merge the benefits of unswept and swept wings over the full Mach number envelope of the airplane. However, this aerodynamic advantage comes at the cost of structural complexity and weight, among other drawbacks.

Supersonic Delta Wings

For sustained supersonic flight, it is known that a delta or triangular wing is close to the optimal type of supersonic wing planform, first theoretically defined by R. T. Jones. It has since been confirmed by wind tunnel and flight test measurements. The advantage of a delta wing is that it can have a larger sweep angle and a greater wing area than a wing just swept back. However, other wing planforms can also provide similar aerodynamic benefits, such as low wave drag, in supersonic flight.

For a delta wing, the flow is conical from the apex of the wing at O, as shown in Figure 27. The pressure is constant along any radius vector from O. Two cases must be considered:

Subsonic leading edge.
Supersonic leading edge.

With the subsonic leading edge then $\cos \Lambda < 1/M_{\infty}$ . In this case, as shown in the figure, there is a low leading edge pressure (suction) over AB and CD, but the average pressures are less than those for a two-dimensional wing with no sweep. With a supersonic leading edge, then $\pi/2 - \Lambda > \mu$ . It is found that the average pressures across AB and CD equal that for a two-dimensional wing with no sweep.

A delta wing with subsonic and supersonic leading edges.

Lift & Drag of Supersonic Delta Wings

The key factors affecting the lift on a supersonic wing are its angle of attack, ${\alpha}$ , operating lift coefficient, $C_L$ , the freestream (flight) Mach number, $M_{\infty}$ , and the wing’s leading edge sweep angle, ${\Lambda}$ . For a delta wing, the wave drag is strongly influenced by the leading edge sweep angle, ${\Lambda}$ . When the leading edge is subsonic, the effective sweep angle is greater than the Mach angle, which reduces the strength of shock waves and, consequently, the wave drag. A supersonic leading edge produces stronger shock waves on the wings, increasing drag and affecting lift characteristics. In contrast, a subsonic leading edge allows smoother airflow and less pronounced shock effects, resulting in different lift and drag characteristics as functions of angle of attack and Mach number.

Supersonic Leading Edge

For a thin delta wing with a supersonic leading edge, the normal Mach number to the leading edge exceeds unity, i.e.,

(37) $\begin{equation*} M_{\infty}\cos\Lambda > 1 \end{equation*}$

where $\Lambda$ is the leading-edge sweep angle measured from the spanwise direction. In this case, the leading edge behaves more like a supersonic edge, and local pressure changes are governed by attached shocks and expansion waves. For portions of the wing that behave approximately as two-dimensional sections, the lift coefficient may be estimated using the linearized supersonic result

(38) $\begin{equation*} C_L \approx \frac{4 \alpha}{\sqrt{M_{\infty}^2 - 1}} \end{equation*}$

where $\alpha$ is in radians. This approximation should be interpreted as a sectional or local estimate, not as a universal finite-delta-wing lift formula.

The corresponding drag on a delta wing comprises three main components. The contributions differ based on whether the leading edge is supersonic or subsonic. For a delta wing with a supersonic leading edge, the sources of drag are:

Wave drag is the dominant source of drag because of the creation of relatively strong shock waves.
Induced drag, or “drag due to lift,” arises from the effects of the wing tip vortices.
Skin friction or boundary layer shear stress drag.

The wave drag coefficient, $C_{D_{w}}$ , can be approximated by

(39) $\begin{equation*} C_{D_{w}} \approx \frac{C_L^2}{4} \sqrt{M_{\infty}^2 - 1} \end{equation*}$

Corrections to the wave drag can be applied for wing thickness in the same manner as two-dimensional airfoils.

The induced drag for a delta wing can be more complex to characterize than that of a high aspect ratio wing. For a thin delta wing, the induced drag coefficient, ${C_{D_{i}}}$ , is often expressed as

(40) $\begin{equation*} C_{D_{i}} = \frac{C_L^2}{\pi \, AR \, e} \end{equation*}$

where $AR$ is the wing’s aspect ratio, and ${e}$ is Oswald’s efficiency factor. As airspeed, ${V_{\infty}}$ , and flight Mach number increase, the lift coefficient for a given airplane weight, $W$ , will decrease with the inverse square of the airspeed, i.e.,

(41) $\begin{equation*} C_L = \dfrac{2W}{\varrho_{\infty} V_{\infty}^2 S} \end{equation*}$

where $S$ is the reference wing area and $\varrho_{\infty}$ is the ambient air density. Therefore, the induced drag becomes an increasingly smaller fraction of the total drag. For supersonic flight, the induced drag is typically relatively low because of the low lift coefficients on the wing.

The boundary layer or viscous shear drag, $D_v$ , on a supersonic wing can be approximated by considering the skin friction coefficient, $C_f$ , and the wetted area, $S_w$ , of the wing, i.e.,

(42) $\begin{equation*} D_v = C_f \, q_{\infty} \, S_w \end{equation*}$

so that $C_{D_{v}}$ is given by

(43) $\begin{equation*} C_{D_{v}} = \dfrac{D_v}{ q_{\infty} \, S} \end{equation*}$

where $S$ is the standard wing reference area. Here, $C_f$ is influenced by the Mach number and Reynolds number. For a turbulent boundary layer, $C_f$ can be estimated using empirical correlations such as

(44) $\begin{equation*} C_f \approx \frac{0.074}{Re^{~0.2}} \end{equation*}$

where the Reynolds number is based on the wing chord. For supersonic flows, modifications to account for compressibility effects are needed, such as using the van Driest transformation, i.e.,

(45) $\begin{equation*} C_f \approx \frac{0.074}{Re^{~0.2}} \left( \frac{1}{1 + 0.15 \, M_{\infty}^2} \right)^{0.58} \end{equation*}$

The skin-friction coefficient on a supersonic wing may be reduced by compressibility effects relative to an incompressible estimate at the same Reynolds number. Still, the overall drag is influenced by higher dynamic pressure and complex interactions between shock waves and boundary layers.

The total drag coefficient, $C_D$ , is then the sum of the wave drag, $C_{D_{w}}$ , induced drag, ${C_{D_{i}}}$ , and skin friction drag, $C_{D_{v}}$ , i.e.,

(46) $\begin{equation*} C_D = C_{D_{v}} + C_{D_{i}} + C_{D_{w}} = C_{D_{v}} + \frac{C_L^2}{\pi \, AR \, e} + \frac{C_L^2}{4} \sqrt{M_{\infty}^2 - 1} \end{equation*}$

Subsonic Leading Edge

A subsonic leading edge is an ideal operating condition for a supersonic wing, i.e., the leading edge of the wing is swept back behind the Mach cone. For a leading-edge sweep angle $\Lambda$ measured from the spanwise direction, the condition for a subsonic leading edge is

(47) $\begin{equation*} M_{\infty}\cos\Lambda < 1 \end{equation*}$

As shown in Figure 28, the geometric goal for a supersonic airplane at a given design flight Mach number is to sweep the leading edge back behind the Mach cone just enough, with some margin, to ensure it always has a subsonic leading edge. The design should avoid excessive sweep to maintain the wing’s lifting area and aspect ratio. In this regard, the supersonic planforms shown in Figure 28 appear approximately correct.^[8] The idea of using a larger sweep angle near the root of the wing is to account for the slightly higher Mach numbers caused by wing/fuselage interference.

A design feature of a supersonic delta is to keep the leading edge swept back behind the Mach cone to improve the aerodynamics and reduce drag. For comparison, the planforms are normalized by wing span.

For a delta wing with a subsonic leading edge, the lift coefficient at small angles of attack may be written as

(48) $\begin{equation*} C_L = C_{L_{\alpha}} \, \alpha \end{equation*}$

where the lift-curve slope, $C_{L_{\alpha}}$ , depends on the wing aspect ratio, leading-edge sweep angle, and freestream Mach number. It is not generally equal to $2\pi$ for a finite delta wing.

For a delta wing with a subsonic leading edge, the leading-edge shock contribution to wave drag is reduced because the normal Mach number to the leading edge is less than unity. The remaining drag still includes skin-friction drag, drag due to lift, and wave drag associated with thickness, volume distribution, and lift. Therefore, for conceptual design it is better to write the total drag schematically as

(49) $\begin{equation*} C_D = C_{D_f} + C_{D_i} + C_{D_w} \end{equation*}$

where $C_{D_f}$ is the viscous skin-friction contribution, $C_{D_i}$ is the drag due to lift, and $C_{D_w}$ is the remaining wave-drag contribution. The wave-drag term depends on Mach number, sweep, thickness distribution, volume distribution, and lift, and is normally estimated using linearized supersonic theory, slender-body/area-rule concepts, CFD, or wind-tunnel data rather than by a single universal expression.

Supersonic Boundary Layer Characteristics

The viscous shear drag considerations mentioned earlier apply generally to supersonic wings but require further specification depending on whether the leading edge is subsonic or supersonic. When the leading edge is subsonic, disturbances can propagate along the leading edge, and the local flow behaves more like subsonic flow. However, the boundary layer is not necessarily attached. The boundary layer begins forming smoothly from the stagnation point, and the primary concern is the development of the boundary layer along the wing’s chord. In this case, the flow over the leading edge does not generate strong shock waves, and the skin-friction drag behaves similarly to that in high-speed subsonic flows, with compressibility effects accounted for.

For a supersonic leading edge, the flow encounters an attached shock wave at the leading edge. This introduces a strong shock-boundary-layer interaction that can significantly affect viscous shear drag. The boundary layer becomes more prone to thickening and separation because of the adverse pressure gradients induced by the shock wave. In addition, the shock can trigger an early transition to turbulence immediately downstream of the leading edge, thereby further increasing skin-friction drag.

Delta Wings in the Transonic Regime

All wings, including delta wings, exhibit unique aerodynamic characteristics because shock waves form when transitioning through the transonic regime. This regime is typically defined for a supersonic-capable wing over a flight Mach number range of approximately 0.9 to 1.1, as shown in Figure 29. These shock waves create wave drag and can also cause flow separation near the wing’s trailing edge, further increasing drag and reducing lift. The lift generated by delta wings in the transonic regime is influenced by the angle of attack and the strength and positions of the shock waves. It is undesirable to continuously operate a supersonic wing (or aircraft) in this transonic regime, not just because of the high drag but also because of the buffeting caused by shock wave boundary layer interactions and trailing edge flow separation.

The drag on a wing, including a highly swept delta wing, will increase markedly when transitioning through the transonic flight regime.

The increase in drag and the commensurate increase in thrust required for flight led to this transonic flight regime being called the “sound barrier.” While there is no intrinsic aerodynamic barrier to flight at supersonic speeds, the substantial increase in thrust required to pass through the transonic regime often requires afterburning on supersonic aircraft that lack sufficient dry thrust. After that, the drag coefficient (but not the drag!) diminishes somewhat, and supersonic flight can continue without the afterburner. Outside the transonic regime, linearized theory predicts that the lift coefficient varies as $C_L \propto 2\pi \alpha/\sqrt{1-M_{\infty}^2}$ in subsonic flow and as $C_L \propto 4\alpha/\sqrt{M_{\infty}^2-1}$ in supersonic flow.

While cruising at Mach 2, the Concorde could operate without afterburners because the engines and their air inlets were optimized for sustained supersonic flight. However, using the afterburners was essential to overcome the increased drag in the transonic regime. Furthermore, the aerodynamic center shifts aft from near the quarter-chord in subsonic flow toward the mid-chord in fully supersonic flow. The accompanying changes in pressure distribution and pitching moment through the transonic regime affect the airplane’s stability, control, and trim. These effects are often mitigated on supersonic airplanes by rebalancing the fuel weight, as was done on the Concorde, or by using an “all-flying” foreplane canard.

The transonic region is generally considered a transitional flight regime because of its inherently low aerodynamic performance and potential stability and control issues. Supersonic aircraft are designed to traverse this regime quickly during acceleration from subsonic to supersonic speeds or deceleration, thereby minimizing the time spent in this complex and unfavorable aerodynamic condition.

Other Variations of the Delta Wing

Many variants of the delta-wing design have since emerged in the evolution of more efficient supersonic aircraft, as shown in Figure 30. The tailless delta wing is the classic design, although experience has shown it could be improved in terms of flight handling qualities. To this end, tailed delta wings have improved the airplane’s pitch control and overall handling qualities, especially when transitioning to and from supersonic flight. In addition, compound and ogival delta wings have been found to exhibit superior low-speed and high-angle-of-attack characteristics while retaining the advantages of supersonic flight.

Examples of some variations of the basic delta wing planform.

The closely coupled canard delta configuration is widely used because it improves overall flight characteristics and handling qualities in both supersonic and subsonic regimes. The design of supersonic fighter airplanes, including the Eurofighter Typhoon, has become standard, as shown in Figure 31. This closely coupled configuration has a smaller “all-flying” delta foreplane, or canard, located in front of and above the main delta wing. This canard-delta configuration modifies the airflow over the wing at high angles of attack, keeping it more attached and yielding the aircraft lower landing speeds and improved handling qualities. The all-flying delta-shaped foreplane also provides significant pitching-moment authority for control, trim, and maneuverability.

The Eurofighter Typhoon has a closely-coupled canard delta wing.

Several other issues have emerged when using pure delta-wing airplanes, including:

1. Controlling the significant changes in the airplane’s pitching moments relative to its center of gravity during transitions to and from supersonic flight is difficult with normal use of the wing flight controls.

2. Concerns about the airplane’s low-speed flight characteristics, including high angles of attack, vortex breakdown, and susceptibility to a behavior known as wing rock. Therefore, early delta-wing aircraft had high landing speeds, which posed numerous operational challenges.

3. Because relatively thin wings were required for aerodynamic reasons, it was challenging to achieve the required structural strength and stiffness while ensuring the wing was free of aeroelastic twisting and flutter.

With a delta wing or one of its variants, the supersonic wave drag may be minimized further by employing Whitcomb’s Area Rule along the length of a slender fuselage. This rule can also be used to improve the spanwise lift distribution over the wing as it passes the fuselage; Dietrich Küchemann had originally hypothesized these ideas during WWII. The subtle use of the Area Rule is often noted when examining the fuselage shapes of supersonic fighter airplanes, which may have carefully contoured fuselage shapes to give smooth variations in the cross-sectional shape of the airplane; see Figure 32. This rule is tantamount to stating that the rate of change of cross-sectional area with respect to distance measured from the nose should be as gradual as possible. A significant constraint in this aerodynamic shaping process for a fighter is engine integration because the fuselage must contain the engine, the afterburner, and all the engine-associated systems.

Graph comparing an aircraft that is capable of supersonic flight and one that is incapable. — The principle of the “area rule” is to maintain a smooth variation in the net cross-sectional area of the airplanes to reduce the compressibility effects and wave drag. Wind tunnel and flight testing have both confirmed the benefits.

Variable Sweep Supersonic Wing Designs

To widen an aircraft’s flight envelope, designers strive to integrate the benefits of different aircraft shapes that perform well in various flight regimes. For example, combining the benefits of efficient subsonic and supersonic flight is desirable. As shown in Figure 33, these benefits can be achieved using a variable-sweep, or “swing-wing,” design. However, it is not always a feasible engineering solution, partly because of the extra mechanical complexity and the significant weight penalty.

Military variable sweep or “swing-wing” airplanes are examples designed to take advantage of the aerodynamic benefits of both subsonic and supersonic flight.

There can be several missions where a supersonic capability combined with good subsonic efficiency is a highly desirable characteristic. For example, a mission may require an efficient subsonic cruise over long distances to conserve fuel, followed by a shorter supersonic flight to the target, and then a subsonic cruise back to base. The variable-sweep, or “swing-wing,” design can better maintain trim as the center of lift on the wing shifts from subsonic to supersonic flight and back again, while the center of gravity also moves. On the F-14, F-111, and B-1 supersonic bombers, the wings are swept forward for takeoff and landing, thereby significantly reducing stall speed and improving low-speed handling qualities.

The lift-to-drag ratio of the F-111 airplane for both subsonic and supersonic flight is shown in Figure 34. Notice the improvement in the lift-to-drag ratio as the wings are swept forward, which is an expected behavior based on the corresponding increase in wing aspect ratio. A lift-to-drag ratio of 4-5 in supersonic flight is typical for any swept delta-wing airplane.

The lift-to-drag ratio of the General Dynamics F-111 as a function of wing sweep angle at subsonic and supersonic flight Mach numbers.

One disadvantage of a swing-wing airplane is the added structural weight and mechanical complexity associated with the sweep pivot joint and its actuator mechanisms. Additionally, variable-geometry wings are more difficult to design with low radar cross-sections, posing a significant challenge for military applications.

Aircraft such as fighter aircraft require wing pylons to carry bombs, air-launched weapons, and long-range fuel tanks. Therefore, another disadvantage of a variable-sweep wing is that pylon stations on the wing’s outer (swing) part are only possible if they can be mechanically designed to be reoriented to point into the relative flow. Pivoting pylons have been used on some variable-geometry aircraft. While they incur a weight penalty, they can significantly enhance the aircraft’s capabilities by enabling it to carry weapons or external fuel tanks.

Pivoting wing pylon systems for external stores and weapons carriage have been used on the General Dynamics F-111 and the Panavia Tornado to keep them facing into the flow as the wing sweeps backward and forward. Mechanically, this can be done using a form of pantograph, with one end fixed to the fuselage and the other to the pylon. The F-111’s outermost pair of pylons did not swivel and were used only when the wing was fully swept forward, limiting the aircraft to subsonic flight with fuel tanks, such as for long-range ferry flights.

Engines for Supersonic Flight

A turbojet engine is suitable for use in supersonic flight. The thrust produced by a turbojet engine can remain relatively constant or increase over part of the supersonic flight regime, depending on the inlet pressure recovery, engine operating limits, altitude, and nozzle performance. This behavior appears favorable for supersonic propulsion, i.e., the faster the aircraft flies, the more air it captures, and the more thrust the engine can potentially generate. However, this trend holds only within a limited Mach-number range. In reality, several aerodynamic and thermodynamic constraints emerge at higher Mach numbers, complicating and ultimately limiting thrust production.

One critical factor is the need to decelerate the incoming supersonic air to subsonic speeds before it enters the engine’s compressor. Compressors are not designed to handle supersonic inflow; doing so would cause shock formation, severe flow separation, pressure losses, and potential compressor stall. To avoid this, the inlet system must include a series of shock waves and compression surfaces to slow and compress the flow in a controlled manner. While this process recovers some of the kinetic energy as pressure, it also introduces total pressure losses and thermal loading.

Additionally, as the Mach number increases, stagnation temperatures rise sharply, pushing materials and cooling systems to their thermal limits. Furthermore, drag increases with Mach number, meaning that the net propulsive efficiency may decrease even as engine thrust increases. Therefore, while turbojets can operate effectively into the low supersonic regime, their performance becomes increasingly constrained at higher Mach numbers, motivating the development of specialized configurations such as ramjets and scramjets for high-speed flight.

Inlet Conditions

As the freestream Mach number exceeds unity, the incoming air must be decelerated to subsonic speeds before entering the compressor stage of a turbojet engine, owing to the aerodynamic and mechanical limitations of axial compressors. In supersonic flight, this requirement imposes stringent demands on the inlet system, which must be carefully designed to control the location and strength of shock waves and minimize the likelihood of strong shock-wave/boundary-layer interactions. The objective is to use a suitable inlet to reduce flow speed to subsonic conditions at the compressor face while maintaining as much total pressure as possible, i.e., by minimizing all sources of loss.

For a supersonic-capable turbojet, this goal is typically achieved by using a fixed- or variable-geometry duct equipped with conical spikes, ramps, doors, or baffles that control flow within the inlet upstream of the engine. For example, on the SR-71, the engines were equipped with a variable-geometry spike inlet and bypass system. As the aircraft accelerated, the conical spike was gradually retracted to control the engine’s inlet conditions. Newer concepts such as the diverterless supersonic inlet (DSI) aim to simplify the design by eliminating diverters. While DSIs offer advantages in reduced weight, mechanical complexity, and radar signature, they are typically optimized for a narrow Mach range. They must be carefully matched to the engine and flight profile to ensure stable operation across the envelope.

Figure 35 shows an example of the system used on the Rolls-Royce/Snecma Olympus 593 turbojet engines that were used to power the Concorde SST. Its variable-geometry inlet system employed a movable intake ramp and spill doors to manage shock-wave structures and airflow across its flight envelope. This arrangement enabled precise control of the positions of intake shock waves, thereby maintaining high pressure recovery and stable engine operation at supersonic cruise. At takeoff and during subsonic acceleration, the intake ramps were fully retracted (lifted) to maximize intake area and deliver the maximum possible mass flow to the engines. The auxiliary inlet doors were fully open, allowing additional air into the engine intake, while the downstream door admitted bypass air for engine cooling.

Intake ramp and nozzle configurations for Concorde’s Rolls-Royce/Snecma Olympus 593 turbojet engine.

A turbojet engine cannot realize its maximum performance potential in supersonic cruise without effective pressure recovery. Therefore, as the Concorde accelerated, the intake ramps were moved downward to keep the engine inlet conditions subsonic. At cruise, near Mach 2.0, the intake ramps were fully deployed to generate external oblique shock waves that decelerated the flow to about Mach 1.5. A terminal shock wave was formed at the throat of the diffuser, bringing the flow to subsonic conditions. The increasing area of the diffuser further slowed the flow and raised the static pressure before it reached the engine intake. Spill doors vented any excess air during throttle transients to maintain the stability of the intake flow. Ramp angles and spill door positions were pre-programmed based on Mach number using a digital control system to hold the terminal shock near the throat. Therefore, the engine received a continuous, steady, subsonic, high-pressure airflow, ensuring stable, efficient cruise performance. On Concorde, it is noteworthy that the close-to-ideal conditions produced by the variable-geometry inlets contributed to over 60% of the total propulsive force at Mach 2.

Thrust Production

The net thrust from a turbojet engine is governed by a momentum balance across its control volume. For steady flow with negligible pressure thrust (e.g., an ideally expanded nozzle), the thrust is given by

(50) $\begin{equation*} F = \overbigdot{m}_e V_e - \overbigdot{m}_{\infty} V_{\infty} \end{equation*}$

where $\overbigdot{m}_e$ and $V_e$ are the exhaust mass flow rate and jet velocity, respectively, and $\overbigdot{m}_{\infty}$ and ${V_{\infty}}$ refer to the freestream conditions. If the fuel mass flow is negligible relative to the incoming air, so that $\overbigdot{m}_e \approx \overbigdot{m}_{\infty} = \overbigdot{m}$ , this latter result simplifies to

(51) $\begin{equation*} { F = \overbigdot{m} (V_e - V_{\infty}) } \end{equation*}$

Notice that supersonic engines use variable-geometry convergent-divergent nozzles to ensure ideal engine flow expansion. For example, Concorde employed a nozzle that was continuously adjusted during flight. By matching the nozzle exit pressure to the ambient pressure, especially at higher altitudes, the system minimized thrust losses and improved the engine’s overall propulsive efficiency.

The engine mass flow rate is approximately

(52) $\begin{equation*} \overbigdot{m} = \varrho_{\infty} \, A_{\text{in}} \, V_{\infty} = \left( \frac{p_{\infty}}{R \, T_{\infty}} \right) A_{\text{in}} \, M_{\infty} \, a_{\infty} \end{equation*}$

assuming ideal gas behavior and a fixed inlet area, $A_{\text{in}}$ . As Mach number increases, both $\overbigdot{m}$ and ${V_{\infty}}$ increase, so turbojet thrust generally increases with flight speed over much of the supersonic operating range. However, the increase is not strictly linear because net thrust is also affected by ram drag, inlet pressure recovery, stagnation-temperature limits, nozzle performance, and the reduction in air density at higher altitude, as illustrated in Figure 36.

Representative variations in thrust produced by an advanced turbojet engine as functions of flight Mach number and operational altitude.

The ability to produce a high exhaust jet velocity $V_e$ with a turbojet depends on the stagnation conditions available at the engine intake. In supersonic flight, the inlet must decelerate and compress the freestream flow. As previously explained, this goal is achieved using shock waves and duct geometry, which increase pressure and temperature but also incur losses. These losses can be represented by the inlet pressure recovery ratio $\pi_r$ , defined as the ratio of stagnation pressure at the engine intake, $p_{t,\text{engine}}$ , to the freestream stagnation pressure, $p_{t,\infty}$ , i.e.,

(53) $\begin{equation*} \pi_r = \frac{p_{t,\text{engine}}}{p_{t,\infty}} \end{equation*}$

The resulting pressure loss will limit the maximum pressure available to support combustion in the engine core, thereby limiting the exhaust velocity, $V_e$ . Assuming isentropic expansion through the nozzle from the turbine exit to the nozzle exit pressure, the exhaust velocity can be estimated from the energy equation as

(54) $\begin{equation*} V_e = \sqrt{ 2 \, c_p \, T_{t5} \left[ 1- \left( \frac{p_e}{p_{t5}} \right)^{(\gamma-1)/\gamma} \right] } \end{equation*}$

where $c_p$ is the specific heat at constant pressure, $T_{t5}$ and $p_{t5}$ are the stagnation temperature and pressure at the turbine exit, respectively, and $p_e$ is the nozzle exit pressure. At higher Mach numbers, $\pi_r$ decreases in value because of increasing shock wave and boundary layer thickening losses. Meanwhile, the total temperature of the freestream increases according to

(55) $\begin{equation*} T_{t, \infty} = T_\infty \left( 1 + \frac{\gamma - 1}{2} M_\infty^2 \right) \end{equation*}$

which raises the inlet temperature. However, because of turbine material constraints, this increase limits the allowable temperature rise during combustion. Therefore, although the total incoming energy increases, the usable thermal drop across the turbine and nozzle constrains engine operation.

Therefore, it becomes clear that intake design is critical for delivering mass flow and pressure to the engine and producing thrust. On Concorde, the variable-geometry inlets achieved a pressure recovery ratio of approximately $\pi_r \approx 0.94$ at Mach 2. This enabled the engine to operate more efficiently and deliver higher net thrust; the inlet alone contributed over 60% of the total propulsive force at cruise.

TSFC Characteristics

Efficient pressure recovery is, therefore, essential to realizing the performance potential of supersonic turbojet propulsion. The highest inlet pressure recovery is also critical because inlet losses dominate the TSFC in supersonic cruise. Representative variations of the TSFC of a turbojet engine in supersonic flow are shown in Figure 37. The thrust-specific fuel consumption (TSFC) is defined as

(56) $\begin{equation*} \text{TSFC} = \frac{\overbigdot{m}_f}{F} \end{equation*}$

Notice that the TSFC of a turbojet engine generally gets worse (i.e., increases) with increasing flight Mach number into the supersonic regime. Although intake pressure increases with Mach number and thrust, the fuel flow rate increases more rapidly, thereby reducing propulsive efficiency and increasing TSFC.

Representative variations in TSFC for an advanced turbojet engine as functions of flight Mach number and operational altitude.

While most supersonic aircraft use afterburners to boost thrust, particularly in the transonic regime, this comes at the cost of a sharp increase in thrust-specific fuel consumption (TSFC). The afterburner works by injecting additional fuel downstream of the turbine to reheat the exhaust gases before expansion in the nozzle, thereby increasing jet velocity and augmenting thrust. However, this process is thermodynamically inefficient, and the fuel flow rate rises rapidly, making sustained supersonic cruise with afterburner operation extremely fuel-inefficient. The Tupolev Tu-144, for example, required continuous afterburner use to maintain supersonic flight, unlike Concorde, whose engines could produce sufficient thrust without afterburners during cruise.

To improve overall performance and reduce TSFC, a supersonic airplane can use low-bypass turbofans instead of pure turbojets. These engines incorporate a bypass stream that increases the total mass flow through the propulsion system without giving a proportional increase in core temperature. The bypass stream produces thrust at a lower exhaust velocity, improving the engine’s overall propulsive efficiency, which is defined as

(57) $\begin{equation*} \eta_p = \frac{2}{1 + \dfrac{V_e}{V_{\infty}}} \end{equation*}$

This ratio becomes more favorable when $V_e$ is closer to ${V_{\infty}}$ , as in the case of a turbofan. Although the bypass ratio must remain low to avoid aerodynamic drag from the engine installation at supersonic speeds, even a modest bypass stream can provide significant benefits by reducing TSFC and improving overall thrust performance.

Design & Integration Challenges

Maintaining efficient supersonic cruise, especially over long ranges, is one of the most formidable challenges in aircraft design. It requires more than simply generating high thrust; achieving this performance depends on minimizing wave drag, which increases rapidly with Mach number and is highly sensitive to even minor geometric deviations. Consequently, the airframe must be precisely shaped, with long, slender fuselages, smoothly contoured surfaces, and thin wings with short spans and highly swept leading edges to reduce both drag and structural weight.

In addition, the propulsion system cannot be treated in isolation. Inlet geometry, nozzle configuration, and thermal management must all be carefully integrated with the airframe to ensure stable engine operation, good pressure recovery, and efficient thrust across the flight envelope. Thrust-specific fuel consumption (TSFC) also becomes a critical factor, as high fuel burn at supersonic speeds can severely limit range unless the propulsion system is optimized for cruise efficiency. Sustained supersonic cruise is not merely an extension of subsonic design practices. It is a fundamentally different and more exacting design problem that requires close coordination across structures, aerodynamics, propulsion, and systems.

Supersonic Transport (SST) Airplanes

There was considerable interest in developing supersonic transport (SST) aircraft during the 1960s, with significant technical efforts led by the British and French aircraft industries. The Anglo-French Concorde first flew in 1969 and entered operational service in 1976, becoming the first successful SST in sustained commercial service. The Concorde used a tailless ogival-shaped “slender-delta” wing, carefully designed and integrated with the propulsion system to give good lift-to-drag in supersonic flight but also acceptable low-speed flight characteristics for takeoff and landing; see Figure 38. However, a big issue for the Concorde was the intensity of its “sonic boom,” which limited its routes mostly over the North Atlantic between New York and London.

The Anglo-French *Concorde* and the Russian Tu-144 were SSTs designed to cruise near Mach 2 and carry about 120 passengers. Only the *Concorde* proved technically and operationally successful.

The Russians developed the Tupolev Tu-144, which resembled the Concorde but was distinctive in its use of retractable foreplane canards. Both the Concorde and Tu-144 were designed for cruising at just over Mach 2 at 60,000 feet, which was the highest Mach number that could be obtained without excessive kinetic heating of the airframe. Consequently, the Concorde and the Tu-144 had to use special high-temperature aluminum alloys in their construction and various other design features to allow the airframe to expand during flight. Conventional aircraft use various aluminum alloys, but these materials tend to soften and lose strength at the temperatures generated by kinetic heating at Mach 2 and above.

Twenty Concorde aircraft were built, including six development aircraft and 14 production aircraft, and the airplane was never economically viable for broader airline service except over the North Atlantic. Nevertheless, it was a very popular airplane with both the passengers and the flight crews alike. British Airways and Air France flew the Concorde for over 25 years until it was retired in 2003, three years after the fatal Air France Flight 4590 crash and amid rising maintenance costs and declining demand.^[9] With a take-off speed of about 220 knots (250 mph) and a cruising speed of 1,350 mph, which was just a little more than twice the speed of sound, a typical London to New York crossing was accomplished in 3 and 1/2 hours, compared to 8 hours for a subsonic flight. The Concorde holds many aviation records, including the fastest transatlantic airliner flight (New York JFK to London Heathrow) in 2 hours and 53 minutes. However, the Tu-144 was unsuccessful as a commercial airplane. After two crashes, including one on a delivery flight to Aeroflot, it was permanently retired after only 55 passenger flights.

In the 1970s, Boeing planned to develop an SST, initially using a variable-geometry, or “swing-wing,” design, the final evolution of the concept shown in Figure 39. The idea of the swing-wing design has already been discussed. For civil aircraft, this design could enable efficient subsonic flight over land and within terminal areas while retaining the advantages of supersonic cruise.

Lockheed and Boeing proposed SST designs in the early 1970s, although neither went beyond the conceptual design stage.

However, despite the potential aerodynamic benefits of a swing-wing concept, Boeing’s detailed engineering analyses proved too challenging from an airframe weight perspective. Boeing’s final 2707 SST concept was a more conventional supersonic delta-shaped wing. The Boeing design was also much larger than the Concorde or the Tu-144 and was intended to cruise at Mach 3 with more passengers. However, the combination of technical problems, high development costs, concerns about the intensity of sonic booms, and the lack of a passenger market for airlines eventually led to its cancellation.

Slender Delta Concept

Most delta-wing aircraft do not inherently exhibit favorable aerodynamic characteristics at low speeds and high angles of attack, such as during takeoff and landing. Takeoff and landing speeds are high for such airplanes, requiring long runways. Concorde was specially designed to exploit vortex lift during low-speed, high-angle-of-attack flight. The airplane had no high-lift devices, so to reduce landing speeds, the wing was designed to promote a phenomenon known as vortex lift.

In the early 1950s, Weber and Küchemann at the RAE (Royal Aircraft Establishment) in the U.K. published a series of papers on a new type of delta-wing planform known as the “slender delta” concept. If designed appropriately, delta wings can produce robust, stable vortex flows on their upper surfaces at low airspeeds and high angles of attack, as shown in the schematic in Figure 40. The presence of these vortices creates very low-pressure zones, significantly increasing the lift on the wing. Many modern supersonic aircraft with highly swept or delta wings are designed to exploit the benefits of vortex lift at low airspeeds.

Vortex lift from a slender delta wing at high angles of attack allows supersonic airplanes to have much lower takeoff and landing airspeeds.

Küchemann is most famous for his book The Aerodynamic Design of Aircraft, which many consider the classic text on modern aerodynamic design. After Küchemann died in 1976, Weber completed the book in 1978. In addition, Weber developed methods to predict the drag on slender delta-winged airplanes during supersonic flight.

Weber and Küchemann determined that lift from leading-edge vortex flows could be maximized by increasing the wing’s slenderness and slightly drooping the leading-edge camber. This approach would retain good supersonic performance while also allowing for high maximum lift at low airspeeds, as shown in Figure 41, resulting in much lower takeoff and landing speeds. The unfortunate byproduct is also a significant increase in drag, which requires considerable thrust and power to overcome, even when landing.

The generation of “vortex lift” at low airspeeds and high angles of attack is also accompanied by significant increases in drag.

However, Concorde had to be flown at an extremely high angle of attack to achieve this condition, which was a distinctive (if not rather alarming) flight attitude during takeoffs and landings. Indeed, the angle of attack during takeoff and landing was so high that the aircraft was designed with a “drooped-nose” feature to allow pilots to see over the nose. Concorde was also distinctive because of its exceptionally tall landing gear, which was needed to enable the airplane to rotate to the required angle of attack for takeoff without the tail striking the runway.

Polhamus developed a practical model^[10] to account for the additional lift generated by leading-edge vortices on sharp-edged delta wings at high angles of attack. He proposed an analogy with the leading-edge suction effect observed in attached potential flow over rounded edges. Although the flow over delta wings is separated, the vortices serve as a surrogate for the suction peak, thereby maintaining an attached-like lift behavior in a time-averaged sense. The total lift coefficient is modeled as a combination of potential-flow lift and vortex lift, expressed as

(58) $\begin{equation*} C_L = K_p \, \sin \alpha \, \cos^2 \alpha + K_v \, \cos \alpha \, \sin^2 \alpha \end{equation*}$

where $\alpha$ is the angle of attack, $K_p$ is the potential flow lift coefficient (geometry-dependent), and $K_v$ is the vortex lift coefficient (empirically determined). This model provides a useful semi-empirical prediction of lift for slender wings with sharp leading edges and is especially applicable to delta-wing aircraft operating at high angles of attack. Polhamus later extended the model^[11] to estimate induced drag from vortex lift.

Kinetic Heating

A significant challenge for supersonic and hypersonic flight vehicles is the intense aerodynamic heating of the airframe caused by convective heat transfer from the hot boundary-layer flow to the vehicle surface. This thermal environment posed a significant design constraint for the Concorde. Convective heating refers to the transfer of thermal energy from the air to the vehicle surface, where the kinetic energy of the incoming flow is partially converted into heat as the air decelerates and compresses within the boundary layer. In this region, temperature gradients develop between the hotter, compressed air near the surface and the cooler structure beneath, resulting in a continuous heat influx into the airframe.

At high flight speeds, the convective heat flux to an aircraft surface increases approximately with the cube of the airspeed. For stagnation-region heating, a commonly used approximate scaling is

(59) $\begin{equation*} \overbigdot{q}'' \ \propto \ \sqrt{\frac{\varrho_{\infty}}{R_n}} \, V_{\infty}^3 \end{equation*}$

where $R_n$ is the local nose radius. The flow velocity is related to the flight Mach number by $V_{\infty} = M_{\infty}a_{\infty}$ , and the local ambient speed of sound is

(60) $\begin{equation*} a_{\infty} = \sqrt{\gamma \, R \, T_{\infty}} \end{equation*}$

If ambient density, ambient temperature, nose radius, gas properties, and the applicable heating correlation are all held constant, then $V_{\infty}\propto M_{\infty}$ and the approximate scaling becomes

(61) $\begin{equation*} \overbigdot{q}'' \ \propto \ M_{\infty}^3 \end{equation*}$

Under these restricted assumptions, doubling the Mach number would increase the predicted convective heat flux by a factor of eight. This is not a general flight-to-flight scaling because density, temperature, altitude, vehicle geometry, boundary-layer state, and high-temperature gas effects may also change substantially with Mach number.

The consequence of this heating is that the air temperature adjacent to the aircraft’s skin rises significantly above the ambient static temperature. This thermal energy is then transferred into the structure by convection. An estimate of the surface temperature under these conditions can be obtained using the adiabatic wall temperature, which is the equilibrium surface temperature corresponding to zero net convective heat transfer at the wall. Assuming an adiabatic wall, this temperature is estimated from the recovery temperature

(62) $\begin{equation*} T_{\text{wall}} \approx T_{aw} = T_\infty \left(1 + r \, \frac{\gamma - 1}{2} M_{\infty}^2 \right) \end{equation*}$

where $r$ is the recovery factor. For a turbulent boundary layer, $r \approx Pr^{1/3}$ , which is approximately 0.9 for air, so

(63) $\begin{equation*} T_{aw} \approx T_\infty \left(1 + 0.18 \, M_{\infty}^2 \right) \end{equation*}$

where $T_{\infty}$ is the ambient static temperature, and $\gamma = 1.4$ for air. This expression provides an estimate of the adiabatic wall temperature in supersonic flow and shows that it increases with the square of the flight Mach number under the stated assumptions. The levels of aerodynamic kinetic heating impose strict thermal requirements on materials and structural design for both supersonic and hypersonic vehicles.

The graphic in Figure 42 illustrates the variation in skin temperature experienced by Concorde during cruise at Mach 2.1 and an altitude of approximately 60,000 feet. At this altitude, the ambient temperature is about -69 $^{\circ}$ F (-56 $^{\circ}$ C). In contrast, the surface temperature of the aircraft reached approximately 266 $^{\circ}$ F (130 $^{\circ}$ C) at the nose and about 199 $^{\circ}$ F (93 $^{\circ}$ C) at the tail. The resulting thermal expansion, compounded by the sustained high-speed flight, caused the fuselage to lengthen by nearly 18 cm (about 7 inches). The airframe incorporated special design features to accommodate this expansion without introducing excessive structural stresses, including expansion joints and flexible mounting structures at critical locations inside the wings and along the fuselage.

Temperature contours on the *Concorde* at its cruise Mach number of 2.1 required the development of special high-temperature aluminum alloys.

Traditional aluminum alloys cannot tolerate the elevated temperatures encountered in sustained supersonic flight, so specialized materials were required to construct the Concorde. To address this, high-temperature aluminum alloys were employed, most notably Hiduminium RR58, which is a light alloy with significantly greater thermal resistance than conventional aluminum. Hiduminium RR58 was used extensively in the aircraft’s primary structure. Additionally, high-temperature steel and stainless steel honeycombs were used in areas exposed to higher thermal and mechanical loads. In contrast, insulated resin-bonded glass fiber composites were applied in the nose region, where surface temperatures were highest.

Sonic Booms

A significant challenge facing supersonic transport (SST) aircraft and other flight vehicles operating at supersonic or hypersonic speeds is the generation of “sonic booms.” These booms arise from a complex system of shock waves and expansion fans that form around the vehicle, with dominant contributions from a bow shock at the nose and a recompression shock at the tail. Shock waves are abrupt pressure disturbances that propagate through the atmosphere while weakening because of geometric spreading, atmospheric absorption, and other propagation effects. To an observer on the ground below the overflying aircraft, these disturbances arrive as a rapid pressure change in the form of an “N-wave,” usually perceived as an overpressure followed quickly by an underpressure, producing the characteristic “boom-boom” noise, as illustrated in Figure 43.

A ground observer often describes a sonic boom as sounding like two sharp cracks, similar to pistol shots, occurring in rapid succession. This phenomenon is not only disruptive but also startling. In the case of Concorde, the sonic boom overpressure produced sound levels exceeding 135 dB, which was strong enough to rattle windows, even many miles away from the flight path. Despite these concerns, research continues into sonic boom mitigation, including reshaping the aircraft’s nose, wings, and fuselage to control the formation and propagation of shock waves and to reduce ground-level noise. However, it remains unclear whether this sonic boom problem can be sufficiently mitigated for any future SST to meet noise regulations.

Geometry of Sonic Booms

These shock waves, or “sonic booms,” do not reach the ground uniformly. Instead, they form curved traces whose geometry is governed by the intersection of the Mach cone with the Earth’s surface, as illustrated in Figure 44. As the aircraft exceeds the speed of sound, it continuously generates spherical pressure waves that trail behind it. The shock waves that form the boundary of two Mach cones are effectively carried along with the aircraft, extending indefinitely behind it. Recall that the half-angle of this cone, known as the Mach angle, is given by

(64) $\begin{equation*} \mu = \sin^{-1} \left( \frac{1}{M_{\infty}} \right) \end{equation*}$

where $M_{\infty}$ is the flight Mach number and $\mu$ is the angle between the Mach cone and the flight path.

The bow shock and tail recompression shock form separate wave surfaces that intersect the ground in two distinct hyperbolic patterns.

To determine how a Mach cone intersects the ground, consider an aircraft flying supersonically at a constant altitude ${h}$ . At a given instant, let the aircraft be located at $x = x_a$ , $y = 0$ , and $z = h$ , with the flight path parallel to the $x$ -axis. The Mach cone extends downstream from the aircraft with a half-apex angle $\mu$ , where

(65) $\begin{equation*} \sin \mu = \frac{1}{M_{\infty}} \end{equation*}$

A point $(x,y,z)$ on the cone satisfies

(66) $\begin{equation*} y^2 + (z-h)^2 = (x_a-x)^2 \tan^2\mu \end{equation*}$

where $x < x_a$ corresponds to points behind the airplane.

On the ground plane, where $z=0$ , the intersection becomes

(67) $\begin{equation*} y^2+h^2 = (x_a-x)^2\tan^2\mu \end{equation*}$

or

(68) $\begin{equation*} \frac{(x_a-x)^2}{h^2\cot^2\mu} - \frac{y^2}{h^2} = 1 \end{equation*}$

This equation describes one branch of a hyperbola on the ground behind the aircraft. Its vertex lies directly beneath a point located a horizontal distance

(69) $\begin{equation*} x_a-x_v = h\cot\mu = h\sqrt{M_{\infty}^2-1} \end{equation*}$

behind the airplane.

The bow shock and tail recompression shock each form separate wave surfaces that intersect the ground in distinct hyperbolic patterns. These wavefronts correspond to the perceived sonic boom of an overflying airplane, as the overpressure and underpressure zones sweep over a ground observer. The time separation between the arrival of the bow shock and the tail recompression shock produces the well-known “N-wave” signature. The pressure rises abruptly at the bow shock, decreases gradually through the waveform, drops abruptly to an underpressure at the tail shock, and then returns gradually to the ambient pressure. Understanding this wavefront geometry and its extent is crucial for analyzing the impact zone or sonic carpet^[12] of sonic booms but also for designing aircraft profiles that can reduce the sharpness or intensity of these wavefronts, thereby mitigating their effects on populated areas.

Sonic Boom Mitigation?

Sonic booms are so intrinsic to supersonic flight that they have been studied by NASA and other organizations for many decades, with the research documented in an excellent review article . The development of supersonic transport (SST) airplanes, such as Concorde, and various military flight programs have prompted numerous publications on sonic booms and the factors that affect them. The principles of sonic-boom mitigation are well established, and shaped-boom concepts have been investigated in flight, but their suitability for routine commercial operations and their acceptability to communities remain to be demonstrated at the required scale. In this context, sonic boom mitigation refers to aerodynamic design methods and/or flight operation techniques that reduce the intensity and audibility of sonic booms generated by supersonic aircraft. These strategies can include reshaping the airframe to alter the near-field pressure signature, thereby reducing the strength and abruptness of the pressure wave that reaches the ground. Such efforts are essential to enabling future civil supersonic aircraft to operate without generating objectionable sonic-boom footprints.

With renewed interest in SSTs, in 2017, NASA issued a draft request for proposals for developing a Quiet Supersonic Transport (QueSST) low-boom flight demonstrator. This program called for the design and testing of a new X-plane to support the development of future-generation SST concepts. The contract was awarded to Lockheed Martin, and its Skunk Works X-59 supersonic demonstrator has been designed with a long, carefully shaped nose, as shown in the artist’s rendering in Figure 45. The X-59 is designed to conduct its principal supersonic tests at approximately 55,000 feet and Mach 1.4. The X-59 completed its first flight on October 28, 2025, marking the beginning of its flight-test program. Subsequent flights are intended to expand the envelope gradually before the aircraft is used to collect community-response data for low-boom supersonic flight.

The NASA/Lockheed Martin Skunk Works X-59 demonstrator is designed to conduct its principal supersonic tests at approximately 55,000 feet and Mach 1.4. Its shape is designed to generate a softer sonic boom.

Following initial flight tests, NASA intends to conduct community overflight studies across selected U.S. cities to assess public perception of the aircraft’s (predicted) reduced sonic boom signature, referred to as a “sonic thump,” as further explained in Figure 46 based on NASA research. Still, unless the reductions in sonic boom intensity are significant, this demonstration will unlikely change the public’s opinion about sonic booms after the psychoacoustic supersonic flight tests conducted over London and Oklahoma City in the 1960s, which were performed at similar flight Mach numbers. Nevertheless, these flight test campaigns may ultimately inform FAA regulatory decisions and contribute to ICAO’s development of international sonic boom standards, potentially paving the way for future commercial supersonic flight over land.

The X-59 is designed to test the theory that the N-wave sonic boom can be changed to an S-wave sonic thump by suitable airframe shape modifications. (NASA image.)

Sonic Boom Intensity

The X-59 is a research demonstrator rather than a commercial transport, and its Mach 1.4 design condition is lower than the cruise Mach numbers proposed for many SST concepts. Therefore, the scaling of its low-boom results to larger, heavier, and faster commercial aircraft must be considered. The scaling behavior differs depending on whether the overpressure is measured nearer to the aircraft or at the ground. Close to the aircraft, particularly at the nose, the pressure rise across the bow shock is governed by standard compressible-flow relationships. In the near field, the pressure disturbance generated by a fixed aircraft shape scales with the freestream dynamic pressure. In simplified form,

(70) $\begin{equation*} \Delta p \ \propto \ q_{\infty} C_p = \frac{1}{2} \, \varrho_{\infty}\, V_{\infty}^2 \, C_p \end{equation*}$

where ${C_p}$ depends on the local surface slope, aircraft geometry, and flight Mach number. Because ${C_p}$ also varies with Mach number, the pressure disturbance does not generally increase in direct proportion to $M_{\infty}^2$ . Its subsequent propagation to the ground further changes both the amplitude and shape of the pressure signature.

As the pressure signature propagates from the aircraft toward the ground, nonlinear steepening, geometric spreading, atmospheric stratification, absorption, and molecular relaxation alter its amplitude and shape. The resulting ground-level peak overpressure cannot be represented by a universal scaling proportional to $W M_{\infty}^3/h^2$ . Such an expression is dimensionally incomplete and greatly exaggerates the dependence on aircraft weight and Mach number. For geometrically similar conventional aircraft producing fully developed far-field N-waves, simplified sonic-boom theory gives an approximate scaling of the form

(71) $\begin{equation*} \Delta p_{\max} \ \propto \ K_S K_p \frac{\sqrt{W}} {h_e^{3/4} l^{1/4}} \left(M_{\infty}^2-1\right)^{1/8} \end{equation*}$

where $W$ is the aircraft weight, $l$ is a characteristic aircraft length, $h_e$ is an effective propagation altitude that accounts for atmospheric conditions, and $K_S$ and $K_p$ represent configuration and propagation factors. The exact numerical coefficient depends on the system of units and the particular simplified prediction method used.

This result shows that the peak overpressure increases approximately with the square root of aircraft weight and decreases with increasing aircraft length and propagation altitude. The direct Mach-number dependence is comparatively weak. For otherwise similar conditions, increasing the Mach number from $M_1$ to $M_2$ gives approximately

(72) $\begin{equation*} \frac{\Delta p_2}{\Delta p_1} \approx \left( \frac{M_2^2-1}{M_1^2-1} \right)^{1/8} \end{equation*}$

For example, increasing the Mach number from 2 to 6 gives

(73) $\begin{equation*} \frac{\Delta p_6}{\Delta p_2} \approx \left( \frac{6^2-1}{2^2-1} \right)^{1/8} = 1.36 \end{equation*}$

which corresponds to an increase in pressure level of

(74) $\begin{equation*} \Delta L_p = 20\log_{10}(1.36) \approx 2.7~\text{dB} \end{equation*}$

Aircraft geometry remains critical because the complete pressure signature depends on the longitudinal distributions of volume and lift, commonly represented through an equivalent-area distribution and its associated Whitham $F$ -function. Modern sonic-boom calculations propagate this near-field signature through the real atmosphere rather than applying a single universal power law in Mach number.

The sound experienced by an observer on the ground depends logarithmically on the pressure amplitude. A logarithmic pressure level based on a specified boom-pressure amplitude may be defined as

(75) $\begin{equation*} L_p = 20 \log_{10} \left( \frac{\Delta p}{p_0} \right) \end{equation*}$

where $p_0 = 20 \, \mu\text{Pa}$ is the reference acoustic pressure. The difference in pressure level between two boom overpressures is

(76) $\begin{equation*} \Delta L_p = 20 \log_{10} \left( \frac{\Delta p_2}{\Delta p_1} \right) \end{equation*}$

For example, a 36% increase in overpressure corresponds to

(77) $\begin{equation*} \Delta L_p = 20 \log_{10}(1.36) \approx 2.7~\text{dB} \end{equation*}$

Even when the peak overpressure is moderate, the rapid pressure rise of a sonic boom can make it sound abrupt and startling, similar to two sharp cracks in rapid succession.

While the sound pressure level (SPL) quantifies the amplitude or loudness of a sonic boom, it does not capture the whole character of the auditory experience. Sonic booms are impulsive in nature, characterized by sudden changes in pressure over very short time scales. The rate at which this pressure rises, expressed as the time derivative of pressure, ${dp/dt}$ , plays a crucial role in how startling or intrusive the sound is perceived to be.

A sharp N-wave, characterized by a nearly vertical shock front, causes a rapid increase in pressure, resulting in a more abrupt and jarring noise event for an observer on the ground. In contrast, a more gradual rise in pressure produces a softer, less disruptive boom. Human perception of noise is susceptible to transients; rapid onsets elicit a much stronger physiological and psychological response than slowly varying sounds of the same amplitude, similar to the difference between the sound of a helicopter and an airplane. Consequently, a slow-rising boom may resemble distant thunder, whereas a fast-rising boom is perceived more like gunshots. This distinction explains why sonic booms with similar SPLs can differ dramatically in their perceived severity.

In regulatory and psychoacoustic contexts, the impact of a sonic boom can be represented using metrics such as Perceived Loudness (PLdB) and Sound Exposure Level (SEL). These metrics account for waveform duration, frequency content, and human sensitivity to sharp pressure transitions. The PLdB metric weights the acoustic pressure spectrum to match the human hearing curve. It tends to increase slightly more rapidly than SPL because of its sensitivity to rapid increases in pressure. The SEL, commonly used in environmental noise assessment, integrates the acoustic energy of the event over its duration, i.e.,

(78) $\begin{equation*} \text{SEL} = 10 \log_{10} \left( \frac{1}{t_0} \int_0^T \left( \frac{p(t)}{p_0} \right)^2 \, dt \right) \end{equation*}$

where $T$ is the duration of the event and $t_0$ is the reference time (typically 1 second). Both PLdB and SEL will be necessary to evaluate the acceptability of SSTs for overland flight. Inevitably, boom mitigation strategies must be evaluated by their ability to reduce the measured values of these metrics to tolerable levels for the general public.

Why is civil supersonic flight over land normally restricted?

Public opposition to civil supersonic flight in the U.S. began with a controversial FAA/USAF test program over Oklahoma City in 1961–1962. Various types of military jets generated sonic booms over the city multiple times daily for several months, resulting in tens of thousands of complaints and as many damage claims. The backlash was a public relations disaster and severely undermined support for the FAA’s supersonic transport initiative, which contributed to Congress’s cancellation of funding for the Boeing 2707 SST in 1971. Soon after, the FAA adopted restrictions on civil supersonic flight. Under the current provisions of 14 CFR § 91.817, civil aircraft may not operate in the United States above Mach 1 unless the operator complies with the conditions of an authorization issued under § 91.818. Flights entering or leaving the United States must also comply with limitations intended to prevent sonic booms from reaching the surface within the United States unless such an authorization has been issued.

Cross-Sectional Area Shaping

The Achilles’ heel of any new SST designed for Mach 1.7 to 2.5 flight is the “sonic boom,” and its design will have to incorporate some form of boom mitigation strategies if overland flight of the SST is ever to be acceptable. While the hyperbolic ground pattern and resulting boom carpet of sonic boom wavefronts cannot be eliminated because they are a geometric consequence of the Mach cone intersecting the ground, there is certainly evidence that the noise effects can be mitigated.

There is ample evidence to show that by carefully shaping the aircraft to spread pressure disturbances gradually along the fuselage length, the resulting ground-level sonic boom waveform can be made flatter and less intense, transitioning from the sharp “N-wave” boom to a milder “S-wave” boom. This is the essence of the X-59’s “low-boom” design and, if validated, may represent a key step in demonstrating that this concept works. In designing a “low boom” SST, two fundamental aerodynamic results in high-speed aerodynamics, i.e., Whitham’s supersonic sonic boom theory and Whitcomb’s transonic area rule, can be invoked. While these results appear in distinct contexts, they are closely related through a shared geometric principle in that both identify the second derivative of the aircraft’s cross-sectional area distribution along its length as a primary contributor to compressibility effects and wave drag in high-speed flow.

In linearized supersonic flow theory, the pressure disturbances generated by a slender body depend strongly on how rapidly its cross-sectional area changes along its length. In simplified form,

(79) $\begin{equation*} \Delta p(x) \ \propto \ \frac{d^2 A(x)}{d x^2} \end{equation*}$

where $A(x)$ is the aircraft cross-sectional area distribution. Large and abrupt changes in area curvature produce stronger compression and expansion waves, while gradual changes produce weaker and more distributed pressure disturbances. The Whitham $F$ -function provides the more complete mathematical relationship between the aircraft shape and the resulting pressure signature. For a lifting aircraft, the relevant distribution is an equivalent area that includes both volume and lift contributions. The inverse-design challenge is to determine an aircraft shape and lift distribution that produce an acceptably weak ground-level pressure signature.

The principles, at least, can be readily explained. For legacy supersonic designs such as Concorde, the cross-sectional area increases rapidly near the wing leading edge and decreases sharply at the trailing edge, as shown in Figure 47. These transitions produce large values of $\dfrac{d^2A}{dx^2}$ , resulting in concentrated compressive and expansive pressures. The corresponding ground overpressure signature is a strong, nearly symmetric “N-wave” with a high peak and a rapid rise time, which an observer perceives as a loud, abrupt boom.

Comparison of sonic boom shaping for a “low-boom” SST design versus a legacy design.

In contrast, a “low boom” design, as predicted for the X-59 demonstrator, employs a long forebody and smoothly blended changes in cross-sectional area and volume. The cross-sectional area increases gradually, peaks further aft, and exhibits gentler curvature throughout, lowering the magnitude of $\dfrac{d^2A}{dx^2}$ , and so producing a ground signature composed of lower-amplitude pressure disturbances. Notice that the more “S-wave” shaped boom reduces both the peak overpressure and the rate of pressure rise, thereby minimizing acoustic intensity and startling effects to an observer on the ground.

Atmospheric Conditions

Atmospheric conditions can also influence the strength and reach of sonic booms. Temperature gradients and wind shear can refract shock waves upward or laterally, potentially reducing their intensity at the surface. Concepts such as “boomless cruise” aim to take advantage of these effects by operating in regions and altitudes where shock waves are refracted away from the ground. However, most of the U.S. and much of the world do not exhibit the persistent atmospheric profiles required to deflect sonic booms upward with any predictable reliability. In the troposphere, temperature generally decreases with altitude, causing the effective speed of sound to decrease with altitude and refracting sonic-boom rays upward. This effect limits the lateral extent of the primary boom carpet and can create shadow regions beyond the sonic cutoff. Even under favorable conditions, such as temperature inversion, slight variations in humidity, turbulence, or terrain can still scatter acoustic energy back toward the ground. The lateral spread of a sonic boom often extends 50 miles or more, making it likely that people far from the flight path will still hear it. Therefore, while atmospheric refraction may alter the footprint of the sonic boom, it does not provide a reliable or practical method for mitigating it.

From an operational and regulatory perspective, relying on atmospheric refraction to mitigate sonic booms introduces substantial uncertainty. Exploiting such effects would require high-resolution, real-time knowledge of upper-atmosphere temperature, humidity, and wind profiles along the entire flight path, data that are difficult to obtain and that vary rapidly. Even minor deviations in these conditions can shift the location and intensity of the refracted shock waves, making it impossible to guarantee a consistently quiet ground footprint. Moreover, any certification framework developed by the FAA, EASA, or other authorities will need to account for applicable ICAO standards and will require predictable and repeatable noise signatures for overland supersonic operations. Consequently, practical boom mitigation efforts will remain focused on deterministic strategies, such as airframe shaping and other low-boom design principles, regardless of atmospheric variability.

Check Your Understanding #5 – Sonic-boom Mach-number scaling

Consider two geometrically similar supersonic aircraft of equal weight and length flying at the same effective altitude. Aircraft 1 flies at $M_1=2$ , while Aircraft 2 flies at $M_2=6$ . Using the approximate far-field N-wave scaling

$\Delta p_{\max} \ \propto \ \left(M_{\infty}^2-1\right)^{1/8}$

determine:

The ratio of the peak ground-level overpressures, $\Delta p_2/\Delta p_1$ .
The corresponding difference in pressure level, $\Delta L_p$ .
The physical significance of the result.

Show solution/hide solution.

The overpressure ratio is
$\frac{\Delta p_2}{\Delta p_1} = \left( \frac{M_2^2-1}{M_1^2-1} \right)^{1/8}$

Substituting $M_1=2$ and $M_2=6$ gives

$\frac{\Delta p_2}{\Delta p_1} = \left( \frac{6^2-1}{2^2-1} \right)^{1/8} = \left( \frac{35}{3} \right)^{1/8} = 1.36$
The corresponding change in pressure level is
$\Delta L_p = 20\log_{10} \left( \frac{\Delta p_2}{\Delta p_1} \right)$

so that

$\Delta L_p = 20\log_{10}(1.36) = 2.7~\text{dB}$
The direct dependence of far-field N-wave peak overpressure on Mach number is relatively weak. Increasing the Mach number from 2 to 6 increases the predicted peak overpressure by about 36%, corresponding to approximately 2.7 dB. Aircraft weight, length, lift distribution, volume distribution, altitude, and atmospheric propagation can have equally important or stronger effects on the ground signature.

Any Future for SSTs?

The Anglo-French Concorde remains the only successful SST to date. Although many other SST designs have been studied, whether any future concept will prove economically viable for scheduled airline service remains to be seen. The inherently lower lift-to-drag ratios of aircraft operating at supersonic speeds, even when the wing and airframe are highly aerodynamically and structurally optimized, mean SSTs will always consume significantly more fuel than subsonic designs. Consequently, the maximum achievable range of a supersonic airliner will always be less than that of a subsonic aircraft. Like the Concorde, transatlantic routes for a Mach 2 capable SST are undoubtedly technically and economically feasible, with an optimally designed SST. However, longer routes, such as from the U.S. to Asia or Australia, would require refueling stops, likely eroding any speed advantage, particularly for a Mach 1.5 to 1.7 capable SST. Additional routing detours to avoid sonic booms over land may further reduce the practical benefit of higher cruise speeds unless the SST can sustain Mach 2 or greater.

Costs

The resulting seat-mile costs for an SST will undoubtedly be higher than those of subsonic airplanes, perhaps three to five times more, despite the advantage to passengers of shorter flight times. Therefore, to be commercially viable, airlines would need to charge significantly higher fares for an SST, limiting the market to passengers willing to pay a premium. As a reference, Concorde tickets in the 1990s typically cost US$10,000–12,000 round trip (equivalent to US$18,000–20,000 or more today). While reduced travel times may appeal to some passengers, particularly for the novelty of supersonic flight, passengers have historically chosen travel based on price. Therefore, the economics of an SST will remain difficult to justify for both airlines and manufacturers. Maintenance and repair costs are also likely to be significantly higher for an SST as a consequence of the thermal and mechanical stresses imposed on the airframe and propulsion systems during high-speed cruise, which can increase downtime and reduce airplane availability.

Certification

Certification remains a significant hurdle for any proposed SST and will likely entail a lengthy, uncertain process. Concorde underwent more than 4,000 hours of certification testing over seven years, plus 1,000 hours of route-proving under airline operating conditions. A more fundamental obstacle today is that civil supersonic flight over land remains tightly restricted in both the U.S. and Europe because of the disruptive effects of sonic booms. Regulatory frameworks such as FAA Part 36 (Noise Standards) and ICAO Annex 16, Volume I, mandate noise certification procedures but provide no standards for acceptable sonic boom loudness. Although efforts are underway to define a low-boom certification metric, no enforceable regulatory pathway exists. Unless these restrictions are revised, demonstrating low-boom capability will not be enough because a viable SST must ultimately comply with yet-to-be-defined regulatory limits on boom intensity and community impact. Until then, overland routes served by any future SSTs will remain restricted, limiting flexibility and reducing the aircraft’s commercial appeal. Additional operational limitations may arise at airports because of noise abatement requirements or runway length constraints.^[13]

Optimization or Optimism?

Pursuing a new SST can be seen as either an ambitious exercise in optimization or a stubborn reinvention of a concept that history has already judged. Future SST players face immense technical, certification, and economic challenges, as well as substantial public skepticism. None of the major civil airframe manufacturers is pursuing supersonic flight. In contrast, smaller supersonic aircraft such as supersonic business jets (SSBJs) have received somewhat more recent attention. Several companies have proposed SSBJ concepts capable of transporting 10-80 passengers. Typically, these aircraft would be about twice the size of a conventional business jet but still much smaller than the Concorde. Yet the technical and certification issues remain, and the economics remain speculative.

One startup, Boom Supersonic, claims to have a Mach 1.7 capable SST called Overture in development and expects it to enter service with airlines by 2029. Beyond the fundamental issues associated with the sonic boom, a major technical hurdle remains: the absence of a suitable propulsion system. No engine manufacturers currently produce quiet, fuel-efficient engines optimized for supersonic cruise, even as low as Mach 1.7, that meet civil certification requirements. Developing and certifying a new supersonic-capable engine for civil use could take a decade or more, given the absence of existing engine types that meet both supersonic performance and current noise and emissions regulations. Nevertheless, several groups argue that integrating technological innovation, market demand, and environmental adaptation could enable a new era of sustainable supersonic travel. However, in light of the well-known challenges and other potential issues that may yet emerge, this remains highly speculative. For now, it is reasonable to conclude that fare-paying passengers are unlikely to fly supersonic again anytime soon.

Summary & Closure

Humans have long aspired to fly faster, a pursuit that has driven the development of aerodynamically refined airframes and advanced propulsion systems for supersonic flight. Despite decades of progress, the realization of practical, routine supersonic transportation remains a formidable engineering challenge. The principal technical requirements include the need to minimize wave drag and shock interactions through precise airframe shaping, as well as the development of propulsion systems capable of maintaining high efficiency at supersonic speeds. Continued advances in aerodynamics, propulsion, and high-temperature materials are essential to enabling future breakthroughs in high-speed flight.

Beyond technical barriers, the economics of supersonic transport are equally complex. The research, design, testing, and certification of supersonic aircraft demand significant financial investment, which must be recovered through sales and operations. Manufacturing costs are substantially higher than those for conventional subsonic airplanes because of the need for specialized materials and production techniques. In operation, supersonic aircraft consume more fuel per passenger mile or passenger kilometer and are likely to incur higher maintenance costs. Consequently, airlines must impose premium pricing to maintain profitability. The commercial viability of any future supersonic transport will ultimately depend not only on technological maturity but also on market demand and its ability to compete with established subsonic alternatives.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Do some background research to determine what efforts are being explored to reduce the intensity of the “sonic boom” produced by supersonic airplanes.
Research some engineering issues associated with kinetic heating on a supersonic airplane that cruises between Mach 2 and 3. What kinds of construction materials might be required?
What kinds of supersonic passenger airplanes, i.e., SSTs, might there be in the future? Explain why their success ultimately depends on the availability of an engine suitable for supersonic flight.
Study the image below, which is a schlieren flow visualization image of what is produced when a bullet is fired from a gun. Can you explain what is happening here and the origin of all the waves?

Consider the Mach angle as a function of the Mach number. What happens to the Mach angle at hypersonic Mach numbers, and why is this important?

Other Useful Online Resources

For additional resources on supersonic flight, follow up on some of these online resources:

A film by Shell Oil on transonic flight.
Airplanes approaching the sound barrier – another film by Shell Oil.
High-speed flight: Part 1, and Parts 2 & 3.
A great video on the Miles M52 supersonic demonstrator – 1,000 mph in the 1940s.
The Miles M52 versus the Bell X-1 – an interesting video.
Video about the World’s fastest subsonic airliner: The Convair 990A Coronado.
The Insane Engineering of the X-15 – see the video here.
The Lockheed A-12 – speed matters!
The SR-71 Blackbird – the king of speed!
Project canceled: BAC TSR2 – The British Cold War strike and reconnaissance aircraft that was canceled.
Great video about the English Electric Lightning –The British supersonic fighter and interceptor aircraft.
Video about the World’s fastest bomber: The Mach 3 capable XB-70 Valkyrie.
Death of a Valkyrie: The 1966 XB-70 Midair Collision
Kelly Johnson and the Skunk Works. The genius who changed aviation.

The author gratefully acknowledges his teacher, Professor Bryan Edward Richards. After graduating in 1960, Professor Richards joined the British Aircraft Corporation (BAC), where he contributed to the design of Concorde and its supersonic aerodynamics. He earned his Ph.D. in 1966, with a focus on hypersonic film cooling. He later became a professor at the Von Kármán Institute in Belgium, where he led research in hypersonic aerodynamics until 1979. He then served as the Mechan Chair of Aeronautics at the University of Glasgow, where he taught and conducted research in high-speed aerodynamics and computational fluid dynamics until his retirement in 2003. ↵
Some of the most notable were in Germany. See: Peter Wegener, "The Peenemünde Wind Tunnels – A Memoir," Yale University Press, 1996. ↵
Ackeret, J., "Luftkräfte auf Flügeln, die mit größerer als Schallgeschwindigkeit bewegt werden" in der Zeitschrift für Flugtechnik und Motorluftschiffahrt. ↵
One solution was using dive recovery flaps or airbrakes designed to disrupt the airflow over the wings and tail surfaces, reducing the nose-down pitching tendency during high-speed dives. ↵
The theory for this was developed by Theodor Meyer in his influential 1908 doctoral dissertation under Ludwig Prandtl, which also contained an early systematic treatment of oblique shock waves. ↵
A related application of this relation involves determining the conditions under which the post-shock Mach number $M_2$ drops to unity, known as the sonic line. This delineates the boundary between attached and detached shock behavior. ↵
This effect is related to the pitch-down tendency often called "Mach tuck," although Mach tuck in transonic flight also involves shock-induced changes in the wing pressure distribution, shock-induced separation, and changes in tail effectiveness. ↵
An exception may be the Boom Overture design, yet unflown, which has more average wing sweep than needed for a Mach number of 1.7. ↵
The British Airways Concorde fleet made 50,000 flights and collectively flew more than 2.5 million passengers. ↵
NASA Technical Note D-3767: "A Concept of the Vortex Lift of Sharp-Edge Delta Wings Based on a Leading-Edge-Suction Analogy." ↵
NASA Technical Note D-4739: "Application of the Leading-Edge-Suction Analogy of Vortex Lift to the Drag Due to Lift of Sharp-Edge Delta Wings." ↵
Which may extend out 50 or more miles. ↵
It is fair to say that any SST will not have short-field takeoff capabilities. ↵

License

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2026 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.

Introduction[1]

History

Fundamentals of Supersonic Flow

Basic Flow Physics

Mach Waves & Shock Waves

Compression Waves & the θ-β-M Relation

Prandtl-Meyer Expansion Theory

Summary of Shock Flow Relations

Supersonic Airfoil Characteristics

Supersonic Airfoil Shapes

Flow Patterns & Pressure Distributions

Lift & Drag Characteristics

Pitching Moments

Supersonic Finite Wings

Mach Angle

Wing Sweep

Supersonic Delta Wings

Lift & Drag of Supersonic Delta Wings

Supersonic Leading Edge

Subsonic Leading Edge

Supersonic Boundary Layer Characteristics

Delta Wings in the Transonic Regime

Other Variations of the Delta Wing

Variable Sweep Supersonic Wing Designs

Engines for Supersonic Flight

Inlet Conditions

Thrust Production

TSFC Characteristics

Design & Integration Challenges

Supersonic Transport (SST) Airplanes

Slender Delta Concept

Kinetic Heating

Sonic Booms

Geometry of Sonic Booms

Sonic Boom Mitigation?

Sonic Boom Intensity

Cross-Sectional Area Shaping

Atmospheric Conditions

Any Future for SSTs?

Costs

Certification

Optimization or Optimism?

Summary & Closure

License

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Introduction^[1]