Shock waves

If an aircraft moves faster than the speed of sound, it 'overtakes' its own wave fronts and shock waves are created. The overlapping spherical wave fronts reinforce one another to form a conical surface of very high pressure, propagating outwards. The pressure is much greater than in individual compressions and the intensity can reach 100 million W/m2 (corresponding to about 200 dB). The sonic boom is the audible sound created by this pressure wave, which is confined to the surface of a cone with the aircraft at its apex. This is called the Mach cone, after Ernst Mach.

Ernst Mach (1838-1916), an Austrian physicist, was the first to realise that if an object were to travel through air at a speed greater than the local speed of sound, a conical shock wave would be produced. The speed of aircraft is often designated in terms of Mach number. Mach 1 = speed of sound in air in the prevailing environmental conditions.

The cone created by the shock wave as an aircraft breaks the sound barrier (exceeds the speed of sound in the surrounding air) is called the Mach cone.

As the pressure wave created by a supersonic aircraft sweeps along the ground in the wake of the aircraft, the pressure variations can Shock wave at Mach 2. be large enough to cause

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Ernst Mach. Courtesy of Austrian Post.

significant damage. The Concorde passenger jet was only allowed to travel at supersonic speed over uninhabited areas and oceans.

The half angle of the Mach cone becomes smaller as the speed increases.x

Shock waves.

Shock waves.

X15 Jet. Courtesy of NASA. Vigorously paddling duck. Courtesy of

Piotr Pieranski.

X15 Jet. Courtesy of NASA. Vigorously paddling duck. Courtesy of

Piotr Pieranski.

7.7.6 Shock waves and light

Optical shock waves may be formed when high energy electrons and protons travel through materials at speeds in excess of the local speed of light. These waves are referred to as Cherenkov light.

Pavel Alekseyvich Cherenkov (1904-1990) was one of three Russian physicists who shared the 1958 Nobel Prize for Physics. In 1934, whilst a PhD student, Cherenkov discovered that there was a faint glow of blue light when high speed electrons passed through a transparent liquid. Igor Tamm and Ilya Frank, co-recipients of the Nobel Prize, interpreted his discovery.

Cherenkov light is emitted from the cores of nuclear reactors. A reactor may produce more than 1015 fissions per second whilst operating at full power. Each fission generates a huge amount of energy, mostly carried away by fission products.

These decays often produce high energy electrons travelling at speeds in excess of the speed of light in the water. This results in emission of characteristic intense blue light.

Astrophysical objects such as neutron stars and black holes emit high energy gamma radiation which interacts in the earth's upper atmosphere and generates electrons travelling faster than the local speed of light. Cherenkov light radiated by these electrons is detected by arrays of ground-based imaging telescopes, such as VERITAS in southern Arizona.

A historical interlude: The sound barrier

The speed of sound in air at sea level is about 350 ms-1, or 760 mph. At 40,000 ft, where the air is colder and more rarified, it is about 660 mph.

As planes became faster and faster, mysterious things seemed to happen when they reached speeds approaching Mach 1. Some pilots in World War I reported that controls froze in fighter planes when they were executing steep dives. There were even rumours, subsequently found to be false, that controls reversed at such speeds.

In the late 1940s, propeller aircraft began to be superseded by jets capable of flying speeds of over 600 mph, and it became crucial to find out if it was safe to fly as fast as, or faster than, the speed of sound. What happens if the plane overtakes its own sound waves? The interference pattern of the sound waves could of course be predicted, as for any waves created by a source travelling faster than the speed of the waves. They would form a 'Mach cone', like the bow wave of a ship travelling at a speed greater than the speed of waves

in water. Nobody could tell, however, if the plane would encounter new and extreme stress loads. Such stresses are normally measured in wind tunnels, but these could not go above 0.85 Mach.

There were speculations that a plane would encounter a 'sonic wall' which it could not penetrate and which indeed could destroy it. In 1946 Geoffrey de Havilland had tried to take one of his father's DH 108 jets through Mach 1. Tragically the plane disintegrated and he was killed. The mystery of the sound barrier had become another challenge for exploration.

In 1947, a new rocket plane, called the X-1, was being constructed in the US which it was believed might exceed Mach 1. The centre for testing this plane was established at Muroc Air Base in the desert in California. Test pilots had to have the courage to fly into the unknown, nerves of steel, and the ability to react instantly to any unforeseen problems as the speed of the plane came closer and closer to Mach 1.0. Chuck Yeager was the chief test pilot, whose skills in flying were legendary. As soon as he entered the cockpit, he was said to become part of the plane, irrespective of whether it was new to him or well known. Chuck had an outstanding record as a fighter pilot. He had been shot down over France, but was helped by the French underground to escape into Spain and return to England. Military rules prohibited his return to active combat, but exceptionally, by special permission from General Eisenhower, he returned to his squadron to continue with fighter missions. Chuck was a West Virginian, with a laid-back southern drawl, relaxed and confident, commanding the highest respect among his

Chuck Yeager. Courtesy of NASA.

flying colleagues. If anyone could guide the X-1 through the sound barrier, it was he.

The plan for the test flights and for the final record attempt was rather primitive compared to modern computerised techniques. In order to give it a flying start, the X-1 would be hooked under the bomb bay of a B29 bomber and lifted to an altitude of about 25,000 ft. At that point Yeager would climb down the ladder in the bomb bay and slide into the X-1 cockpit, lock the hatch, and wait to be dropped like a bomb. It literally was a bomb, filled to capacity with highly volatile fuel and liquid oxygen. As soon as he was dropped, and clear of the mother plane, Yeager would fire the rocket engines and shoot forwards from the mother ship. If the engines failed to ignite, the X-1, which was loaded with fuel, would not glide and would probably end up in a fatal spin.

On 14 October 1947, Yeager was scheduled to bring the plane up to Mach 0.97. Typically, in his nonchalant way, he had gone horse-riding with his wife, Glennis, two days before the test flight.

The horse stumbled and threw him, resulting in two broken ribs. Yeager could hardly raise his right hand, but kept this fact to himself, knowing full well that if it became known that he was thus handicapped, he would be grounded. He was determined to go ahead with the test, particularly since it might be his last flight for a while. He confided in his close friend Jack Ridley, the flight engineer who promised to keep mum and

The X-1 beside its mother plane. Courtesy of NASA.

to help him as far as possible. A minor but critical problem was that Yeager would have to pull hard with his right hand in order to lock the canopy above his head after he had lowered himself into the cockpit of the X-1. Jack's ingenuity provided a broken broom handle which Yeager could use as a lever with his left hand. With the problem of closing the cockpit solved, firing the rocket engine when in free fall did not seem to worry him as much.

As the critical moment arrived on the day of the test, and the plane began to fall freely from under the bay of the bomber, Yeager turned on the rocket engines in sequence. The machometer needle rose: 0.88, 0.92,..., 0.96 Mach. There was some buffeting, but the faster he flew, the smoother the ride. Since he still had some fuel left, he accelerated further. The needle went off the Mach scale, which was calibrated only up to Mach 1.0, and stayed off for about 20 seconds. The tracking station reported a distant rumble, the first sonic boom from an airplane. The sound barrier had been broken.

Yeager wrote later: 'The real barrier was not in the sky, but in our knowledge and experience of supersonic flight.' Modern jets can take off on their own power and reach speeds high above Mach 1.0.

Once the sound barrier had been broken, progress was fast in making faster and faster planes. Planes could take off under their own power and accelerate up to Mach 1.0 and beyond.

The F-104 Lockheed Starfighter was the first to reach Mach 1 in a climb. It could reach an altitude of over 100,000 ft and travel at

more than twice the speed of sound. On 10 December 1963, Chuck Yeager took the F-104 on a routine test flight. Travelling at a speed better than Mach 2 he fired the thrust rocket in its tail to bring it up to an altitude of 104,000 ft, when it went out of control and rap-

X-15 wreckage. Courtesy of NASA.

idly began to fall. The data recorder later showed that the plane had made 14 flat spins before it hit the desert floor.Yeager stayed with it for 13 of those spins before he ejected. As he wrote in his autobiography: 'I hated losing an expensive airplane, but I couldn't think of anything else to do.' He managed to eject successfully, although he was badly burned on his face by the rocket motor of his ejector seat.Yeager was not the only pilot to survive a crash of an X-series test plane. The picture shows the wreckage of an X-15 after Jack McKay had been forced to make an emergency landing at Mud Lake, Nevada.

Almost exactly 50 years after Yeager's pioneering flight, on 15 October 1997, the sound barrier was broken on land. A jet-powered car, driven by British Air Force pilot Andy Green, made two runs, in opposite directions across Black Rock Desert in Nevada. The car accelerated over a distance of 6 miles, then was timed over a measured mile, and then followed a further 6 miles to stop. The runs were timed at 759 and 766 mph, both faster than the speed of sound, which under the prevailing conditions was calculated at 748 mph. When the hard desert ground was later inspected, it was found to be pulverised by the Mach wave.

Andy Green's Thrust Super Sonic Car was 54 ft long and 12 ft wide. It was powered by a pair of Rolls-Royce jet engines from a Phantom fighter producing 50,000 pounds of thrust and 110,000 horsepower.

The aerodynamic design had to be faultless. If the front of the car were to lift by as little as half a degree, all the weight would come off the front wheels. The car would then nose up and flip over backwards. Alternatively, if the nose were to dip, it would somersault forwards. As one of the engineers remarked at the time: 'If the nose comes up, you're going flying. But equally, if it goes down, then you're going mining.'

Appendix 7.1 Derivation of Doppler frequency changes

Apparent frequency measured by a moving observer

In Figure 7.14, S is a point source of sound and O is an observer.

The observer runs a distance vo in one second, so he meets

Vo extra waves per second.

Figure 7.14 Doppler effect — moving observer and stationary source.

The source emits waves at a frequency f = .

The observer measures a frequency f ' =

If the observer 'runs away' from the source at the same speed, he measures a frequency:

As we inferred from common sense, the frequency is higher when the observer approaches the source and lower when he recedes from it.

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