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"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

Mathematically, evanescent waves are characterized by having an imaginary value of the wavenumber, k.