An afterburner is an addition to an aircraft's engine. The afterburner consists of fuel nozzles that shoot fuel into the aircraft's already-hot exhaust. The fuel immediately combusts, raising the temperature of the exhaust even more. The gases expand due to the increased temperature to thrust the aircraft further. This is the most commonly used way for aircrafts to achieve supersonic speeds, because few engines alone can provide enough thrust to break the sound barrier. Engines that can sustain supersonic flight without afterburners are called supercruising engines.
This method of thrust is not without its problems. Afterburners work by combustion, which requires oxygen. The oxygen levels of the exhaust are lower than optimal because the exhaust is coming from the engine's combustion reactions which of course also use up oxygen. However, due to the inefficiency problems of the engine itself, there is still plenty of oxygen remaining. This is why afterburners are used at all. That they are easy to implement is yet another advantage.
As pointed out by Transitional Man in supercruise, the same amount of fuel that can sustain afterburning for a minute can sustain regular flight for an hour. Talk about inefficient! Because of this inefficiency, afterburners are not used throughout entire flights. There are two times afterburners are commonly used. The first is during takeoff, when a large acceleration is necessary. This is especially true on an aircraft carrier, which has a very limited runway. The other time is during a dogfight, when speed is one of the most important factors in survival.
Alternative methods of supersonic flight include supercruise, ramjets, and scramjets. Supercruise is simply the name attached to aircrafts capable of sustaining supersonic flight without afterburners. Ramjets work by harnessing the 'ram compression effect' which is the slowing of air as it passes through an aircraft's engine. This is really only effective at speeds greater than 600mph (mach .79) so the aircraft needs an alternative method of thrust to get it to that speed. They are also usually designed to work under very specific conditions. However, ramjets are efficient in that they require no moving parts. They are used primarily in missiles. Scramjets are the same as ramjets except that they do not slow the air to subsonic speeds. This allows scramjets to achieve much higher speeds, up to somewhere between mach 12 (9134mph) and mach 24 (18269mph)! Like ramjets, scramjets are not effective until a certain speed, which The Custodian in scramjet places at mach 3 (2284mph). Unfortunately (or fortunately), a lot of information regarding scramjets is classified.
Sources:
http://science.howstuffworks.com/question374.htm
http://en.wikipedia.org/wiki/Afterburner
http://en.wikipedia.org/wiki/Supercruise
http://en.wikipedia.org/wiki/Ramjet
Supercruise and Scramjet
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