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Envelope (waves)

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Envelope (waves)
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(File:Signal envelopes.png|right|thumb|Top and bottom envelope functions for a modulated sine wave.)In physics and engineering, the envelope of an oscillating signal is a smooth curve outlining its extremes. The envelope thus generalizes the concept of a constant amplitude. The figure illustrates a modulated sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.

Example: beating waves

{{see also|Beat (acoustics)}}(File:Modulated wave.png|right|thumb|A modulated wave resulting from adding two sine waves of nearly identical wavelength and frequency.)A common situation resulting in an envelope function in both space x and time t is the superposition of two waves of almost the same wavelength and frequency:BOOK, Wind Waves: Their Generation and Propagation on the Ocean Surface, Blair Kinsman, 2002,weblink 186, Courier Dover Publications, 0486495116, Reprint of Prentice-Hall 1965,
F(x, t) = sin left[ 2 pi left( frac {x}{lambda - Delta lambda } - ( f + Delta f )t right) right] + sin left[ 2 pi left( frac {x}{lambda + Delta lambda } - ( f - Delta f )t right) right]
approx 2cos left[ 2 pi left( frac {x} {lambda_{rm mod}} - Delta f t right) right] sin left[ 2 pi left( frac {x}{lambda} - f t right) right] ,
which uses the trigonometric formula for the addition of two sine waves, and the approximation Δλ

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