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In optics, evanescent waves are formed when sinusoidal waves are (internally) reflected off an interface at an angle greater than the critical angle so that total internal reflection occurs.
"Evanescent" means "tending to vanish", which is appropriate because the intensity of evanescent waves decays exponentially (rather than sinusoidally) with the distance from the interface at which they are formed.
Source: http://www.andrew.cmu.edu/user/dcprieve/Evanescent%20waves.htm
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