Lesson 5 · 40 min

Jets, Sweep and the Sound Barrier

In 1945 the fastest aircraft flew at about 750 km/h. Fifteen years later fighters flew at twice the speed of sound and airliners cruised at 900 km/h. Two ideas did it: the jet engine, whose thrust does not fall away with speed, and the swept wing, which lets an aircraft fly close to the speed of sound while its wing feels a much slower flow.

Learning objectives

Explain why propellers and thick straight wings limited speed, and what the jet engine changed.
Compute the speed of sound and the Mach number in the standard atmosphere.
Apply simple sweep theory, \(M_n = M\cos\Lambda\), and relate wing sweep to design Mach number.
Describe the area rule, delta and variable-sweep wings, and the lessons of the de Havilland Comet.

The jet engine

Frank Whittle in Britain patented a turbojet in 1930; Hans von Ohain in Germany developed one independently, and it flew first, in the Heinkel He 178 on 27 August 1939. By the end of the war the Messerschmitt Me 262 and the Gloster Meteor were in service.

The difference that mattered to designers is how thrust behaves with speed. A piston engine delivers roughly constant power, so the propeller's thrust \(T = \eta_p P/V\) falls as speed rises, and the propeller itself loses efficiency as its tips approach the speed of sound. A turbojet delivers roughly constant thrust, so the useful power \(TV\) grows with speed.

Example 5.1 — The power of a jet

The North American F-86E had one J47 turbojet of \(5200\ \text{lb}\) (\(23.1\ \text{kN}\)) of thrust and reached \(679\ \text{mph}\) (\(303.5\ \text{m/s}\)) at sea level. Find its thrust-to-weight ratio at \(\Wo = 6739\ \text{kg}\), and the thrust power \(TV\) at top speed. Compare with the P-51D's \(1111\ \text{kW}\) engine.

Show solution

\[ \frac{T}{W} = \frac{23\,130}{6739(9.81)} = 0.350, \qquad TV = 23\,130(303.5) = 7.02\times 10^6\ \text{W} = 7.0\ \text{MW} \]

At top speed the F-86's single engine does about six times the useful work of the Mustang's (whose thrust power is \(\eta_p P\), well under 1111 kW). No piston engine and propeller of 1947 could approach that.

Compressibility and the critical Mach number

Air accelerates over the upper surface of a wing. When the free stream reaches the critical Mach number \(M_{\text{crit}}\), the local flow somewhere on the wing reaches the speed of sound; a little faster, shock waves form, the boundary layer separates behind them, and the drag rises steeply (the drag-divergence Mach number). Pilots of fast propeller fighters diving from altitude met this as buffeting, loss of elevator effectiveness and a nose-down tuck. For the thick straight wings of the 1940s the drag rise began at about \(M = 0.7\).

Speed of sound and Mach number

\[ a = \sqrt{\gamma R T}, \qquad M = \frac{V}{a}, \qquad \gamma = 1.4,\ R = 287\ \text{J/(kg·K)} \]

In the standard atmosphere \(T = 288.15 - 0.0065h\) (K, \(h\) in m) up to 11 km, and \(216.65\ \text{K}\) from 11 to 20 km. At sea level \(a = 340\ \text{m/s}\); above 11 km, \(a = 295\ \text{m/s}\) (\(1062\ \text{km/h}\)).

Thinner airfoils raise \(M_{\text{crit}}\), but a thin wing is heavy (Lesson 3 again: a wing needs depth to carry bending). The breakthrough was geometric.

The swept wing

Adolf Busemann proposed sweeping the wing at the Volta conference in Rome in 1935, and German research during the war developed the idea; Robert T. Jones at NACA reached the same conclusion independently in 1945. The reasoning is simple sweep theory: on a long swept wing, only the component of the flow perpendicular to the leading edge sets the pressures on the airfoil. The spanwise component just slides along the wing.

Simple sweep theory

\[ M_n = M\cos\Lambda \]

A wing swept by \(\Lambda\) behaves, for compressibility, like a straight wing at the normal Mach number \(M_n\). In this idealization the drag rise is delayed to \(M_{dd}/\cos\Lambda\); real wings, with roots and tips, gain somewhat less.

Figure 5.1 Simple sweep theory. The free-stream Mach number (magenta) splits into a component normal to the swept leading edge (blue), which the airfoil feels, and a spanwise component (gray) that slides along the wing. Increase the sweep until the normal Mach number falls below the airfoil's drag-divergence Mach number: the wing turns from red (shock waves, steep drag rise) to green.

The first swept-wing jets flew in 1947. The North American F-86 fighter (October) and the Boeing B-47 bomber (December) both had 35° of sweep. The B-47 also had a thin, flexible, high-aspect-ratio wing and engines hung in pods on pylons below it. Loftin estimates its \(\LDmax\) at about 20, higher than any propeller aircraft. That layout, a swept wing with podded engines, became the Boeing 707 and nearly every jet airliner since.

Example 5.2 — The 707's wing in cruise

The Boeing 707-320B cruises at \(M = 0.83\) at \(35\,000\ \text{ft}\) (\(10\,668\ \text{m}\)) with a wing swept \(35^\circ\). Find its true airspeed and the normal Mach number its wing feels.

Show solution

\[ T = 288.15 - 0.0065(10\,668) = 218.8\ \text{K}, \qquad a = \sqrt{1.4(287)(218.8)} = 296.5\ \text{m/s} \] \[ V = Ma = 0.83(296.5) = 246.1\ \text{m/s} = 886\ \text{km/h}, \qquad M_n = 0.83\cos 35^\circ = 0.680 \]

Loftin lists \(550\ \text{mph}\) (\(885\ \text{km/h}\)) for economical cruise. The wing's airfoils work at about \(M = 0.68\), where a thick-ish section is still free of strong shocks.

Figure 5.2 Wing sweep against design Mach number for the jets in Loftin's tables: cruise Mach number for transports (orange) and maximum Mach number for fighters (magenta). The left curve keeps the normal Mach number at your chosen value; the right curve sweeps the leading edge behind the Mach cone, \(\Lambda = 90^\circ - \arcsin(1/M)\), which supersonic designs try to do. The straight-winged F-104 is the famous exception: its wing was very thin instead.

Through the sound barrier

On 14 October 1947 Chuck Yeager flew the rocket-powered Bell X-1 faster than sound. Practical supersonic aircraft needed two more ideas:

The area rule (Richard Whitcomb, NACA, 1952): near Mach 1 the wave drag depends on how the aircraft's total cross-sectional area varies along its length, so the fuselage should be narrowed where the wing is. The Convair F-102 could not exceed Mach 1 until its fuselage was "waisted" in 1954.
Wing planforms for supersonic flight: highly swept or delta wings (the F-106, 60° leading-edge sweep, reached Mach 2.3; Concorde flew in 1969), very thin straight wings (the F-104), and variable sweep (the F-111 of 1964 and the F-14), which spreads the wing for takeoff and cruise and sweeps it back for high speed.

The first jet airliners and the Comet

The de Havilland Comet flew in July 1949 and entered service in 1952, years ahead of any rival: four turbojets buried in the wing roots, a modest 20° of sweep, a pressurized cabin at 35 000 ft. In 1954 two Comets broke up in flight. A full-scale fuselage tested in a water tank at Farnborough, pressurized over and over, failed by metal fatigue: a crack grew from the corner of a cut-out in the skin. The investigation changed structural design everywhere: fatigue testing of complete airframes, fail-safe structures with multiple load paths, and later damage tolerance, the requirement that cracks can be found and stopped before they become critical.

By then Boeing had flown its private-venture 367-80 (the "Dash 80", July 1954), the prototype of the 707. The 707 and the Douglas DC-8 (1958) set the pattern for long-range jet transport: 30–35° sweep, engines in pods under the wing, \(\LDmax\) of 18 to 19, cruise at Mach 0.8 or more.

Check your understanding

Key takeaways

Propellers deliver roughly constant power and lose efficiency near Mach 1 at the tips; a jet delivers roughly constant thrust, so its useful power grows with speed.
\(a = \sqrt{\gamma R T}\), \(M = V/a\). Thick straight wings meet steep drag rise near \(M = 0.7\).
Simple sweep theory: \(M_n = M\cos\Lambda\). Sweep of 30–35° lets transports cruise at \(M \approx 0.8\) to 0.85.
The area rule, delta, thin and variable-sweep wings opened supersonic flight, at a cost in subsonic \(\LDmax\).
The Comet's fatigue failures created fail-safe and damage-tolerant design. Next, Lesson 6: how turbofans, better wings and new materials made the jet transport efficient.