Stealth technology is any technology that makes an aircraft more difficult to detect. The type of detection that has received the most attention is electromagnetic, or detection by radar. Nonetheless, there are many forms of stealth that also need to be considered in making an aircraft "low-observable."

• Radar cross section (RCS): Radar, short for radio detection and ranging, is an instument that sends out radio beams and then picks up any reflected energy from aircraft, ships, or other objects to determine the location and speed of these items. The range at which radar can detect an object is related to the power transmitted by the radar set, the fidelity of the radar antenna, the wavelength of the radar signal, and the radar cross section of the vehicle. RCS, a measure of the radar waves scattered in a given direction, is the only variable an aircraft designer has any control over. At first, designers tried to reduce RCS by placing various radar-absobant coatings on the aircraft exterior. However, these coatings typically only produced minor reductions in RCS since many parts of the aircraft cannot be covered (cockpit windows and engines, for example). Designers soon realized that the only way to make substantial redcutions in RCS was by carefully designing each portion of the aircraft to scatter radar waves away from their source. Some obvious solutions led to moderate reductions. For example, 90 degree corners and flat perpendicular surfaces were found to produce high radar returns, so designers rounded corners and used chined noses as well as canted vertical tails to reduce radar signatures. The Lockheed SR-71 Blackbird is an excellent example of these early efforts. However, these modifications were only made when they did not interfere drastically with the overall performance of the design, and though the final product might have been more stealthy, it was still detectable. The revolution in radar stealth came in the 1970s when computers were powerful enough to solve the Maxwell electromagnetic equations for reasonably complicated shapes. These equations determine how radar waves are reflected and scattered, and by developing a program capable of predicting the RCS of an entire aircraft from different angles, designers were able to drastically reduce the RCS. The major limitation of this early program was that it could only analyze flat panels, so the F-117 and its Have Blue prototypes were composed of a number of faceted panels. The massive improvements in computer technology over the past two decades have allowed the same basic method to be applied to smooth, contoured surfaces. These improved codes have been instrumental in designing aircraft like the B-2 and F-22. In addition to the overall shaping of the aircraft, a number of other aircraft components can produce high radar returns. These include:
  
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• engines: The fan blades at the front of a jet engine and the turbine blades on the back are very radar reflective. Most stealth aircraft place the engine ducts above the wing or fuselage to help block the engine interiors from radar sources below the aircraft. The F-117 also makes use of special screens on the engine inlets that block radar waves from reaching these surfaces. However, these are difficult to design because of their adverse impact on engine performance and thus have been abandobed in later designs. More recent stealth aircraft use S-ducted inlets that bend off center to hide the blades from being seen. The F-117 also makes use of a special "platypus" nozzle that effectively hides the turbine blades from any radar source behind the aircraft.
  
• cockpit and other windows: The interior of the cockpit is full of sharp corners and reflective metal objects. Even the pilot's helmet is radar reflective. The F-117 uses flat window panels and radar-absorbing treatments on the cockpit windows and on the windows housing the bomb laser-guidance systems that block radar waves from entering these areas.
  
• sharp perpendicular edges: Any kind of edge perpendicular to radar waves causes them to to be diffracted and reflected. In particular, the edges of landing gear doors and other access doors as well as the trailing edges of the wings produce strong radar returns. This effect can be minimized by sweeping the edges so they are not perpendicular to the radar waves. Thus, the edges of doors on the F-117 and other stealth aircraft are covered with small saw-tooths, or diamond-shaped edges that dissipate the radar energy in many directions.
  
• Infrared signature: Along with radar, the second major detection system used today is heat sensors, such as night-vision goggles. Most short-range missiles like Sidewinder home-in on heat sources, like engines. The key to reducing the infrared signature is to cool the exhaust air rapidly using long nozzle ducts or mixing the exhaust with cooler air. The F-117 platypus nozzle, one of the more difficult items to construct, does both of these things. The Stealth Fighter also uses a high-bypass turbofan engine that mixes the hot jet exhaust with cooler bypass air. The B-2 and YF-23 also route the hot exhaust through long troughs coated with heat-absorbing material that not only help cool the air but block the hot gases from being seen from below.
  
• Aural signature: Aircraft are noisy, and all the fancy RCS-reducing methods in the world mean nothing when someone on the ground hears your plane roaring by. Since the F-117 is a high-altitude bomber, this issue isn't so critical, but high-bypass turbofan engines do tend to be much quieter than turbojets. Most stealth aircraft are also subsonic so they do not create sonic booms.
  
• Visibility: The ultimate level of stealth is to make the aircraft invisible to the human eye. Obviously, nothing developed to date has reached this extreme, although experiments have been and are being performed to find methods of doing so. The most common and least sophisticated approach is through the use of camoflauge paint schemes. Light glinting off of canopies can also be reduced using flat panels or special coatings. More sophisticated approaches revolve around using lights or mirrors to bend beams of light around the aircraft making it more difficult to spot, but little is known about this kind of research. In addition, care must be taken to avoid contrails, the white trails of condensed water vapor that can be seen in engine exhausts. Cooling the exhaust goes a long way towards achieving this goal, and the B-2 also uses special fuel additives that help eliminate contrails.
  

I think you are asking two separate, if related, questions. The more general question about what makes a stealth plane different from a non-stealth plane covers a wide variety of technologies. The issue of making a plane quieter is more specific and isn't necessarily confined to stealth aircraft alone. It's more of an issue of the kinds of engines used on a plane.

Older military and commercial jets tend to use jet engines that are called turbojets or low-bypass turbofans. These engines suck in a small amount of air and accelerate it out the back at high speed to produce thrust. This process tends to be very noisy. For this reason, stealth planes (as well as newer commercial aircraft) use engines that are called high-bypass turbofans. These engines take in a much larger quantity of air but accelerate it less. However, because the mass of air is so big, you can actually get more thrust than with the older kinds of engines. The added benefit is that these newer engines create a large cushion of slower-moving air around the noisy part of the engine, and this cushion blocks much of the noise so the engine is significantly quieter. Commercial aircraft are also using these new quieter engines to meet government noise regulations.
Another technique used to make a stealth plane quieter is to place the engines on top of the plane so that their noise is blocked from reaching the ground by the body of the aircraft itself. You can see this method used on both the B-2 stealth bomber and F-117 stealth fighter.
answer by Jeff Scott

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