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{{Image|Signal envelopes.png|right|150px|Top and bottom envelope functions for a modulated sine wave.}}
{{See also|Modulation}}
In [[physics]] and [[engineering]], the '''envelope function''' of a rapidly varying [[Signal (communications)|signal]] is a smooth curve outlining its extremes in amplitude.<ref name=Johnson/> The figure illustrates a sine wave varying between an upper and a lower envelope. The envelope function may be a function of time, or of space, or indeed of any variable.

==Example: Beating waves==
{{see also|Beat (acoustics)}}
{{Image|Modulated wave.png|right|150px|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:<ref name=Kinsman>

{{cite book |title=Wind Waves: Their Generation and Propagation on the Ocean Surface |author=Blair Kinsman |year=2002 |url=http://books.google.com/books?id=RlhZc4HAS5oC&pg=PA186 |pages=p. 186 |publisher=Courier Dover Publications |isbn=0486495116 |edition=Reprint of Prentice-Hall 1965}}

</ref>
:<math>F(x, \ t) = \sin \left[  2 \pi \left( \frac {x}{\lambda - \Delta \lambda } - ( f + \Delta f )t \right) \right] </math><math>+  \sin \left[  2 \pi \left( \frac {x}{\lambda + \Delta \lambda } - ( f - \Delta f )t \right) \right] </math>
:::<math>\approx 2\cos  \left[  2 \pi \left( \frac {x} {\lambda_{mod}} -  \Delta f \ t \right) \right] \ \sin \left[  2 \pi \left( \frac {x}{\lambda} -  f \ t \right) \right]  \ , </math>
which uses the trigonometric formula for the addition of two sine waves, and the approximation &Delta;&lambda;<<&lambda;:

:<math>\frac{1}{\lambda \pm \Delta \lambda}=\frac {1}{\lambda}\ \frac{1}{1\pm\Delta \lambda / \lambda }\approx \frac{1}{\lambda}\mp \frac {\Delta \lambda}{\lambda^2} .</math>

Here the ''modulation wavelength'' &lambda;<sub>mod</sub> is given by:<ref name=Kinsman/><ref name=Denny>

{{cite book |title=Air and Water: The Biology and Physics of Life's Media |author=Mark W. Denny |url=http://books.google.com/books?id=XjNS6v7q130C&pg=PA289 |pages=p. 289  |publisher=Princeton University Press 
|isbn=0691025185 |year=1993}}

</ref>
:<math> \lambda_{mod} =  \frac {\lambda^2}{\Delta \lambda}\ . </math>
The modulation wavelength is double that of the envelope itself because each half-wavelength of the modulating cosine wave governs both positive and negative values of the modulated sine wave. Likewise the ''beat frequency'' is that of the envelope, twice that of the modulating wave, or 2&Delta;''f''.<ref name=Tipler/>

If this wave is a sound wave, the ear hears the frequency associated with ''f'' and the amplitude of this sound varies with the beat frequency.<ref name=Tipler/>

===Phase and group velocity===
The argument of the sinusoids above apart from a factor 2&pi; are:
:<math>\xi_c =\left( \frac {x}{\lambda} -  f \ t \right)\ , </math>
:<math>\xi_e=\left( \frac {x} {\lambda_{mod}} -  \Delta f \ t \right) \ , </math>
with subscripts ''c'' and ''e'' referring to the ''carrier'' and the ''envelope''. The same amplitude ''F'' of the wave results for the same value of &xi;<sub>c</sub> and &xi;<sub>b</sub>, even though this value results for different choices of ''x'' and ''t''. This invariance means that one can trace these waveforms in space to find how a position of fixed amplitude propagates in time with a speed that keeps &xi; fixed; that is, for the carrier:
:<math>\left( \frac {x}{\lambda} -  f \ t \right) = \left( \frac {x+\Delta x}{\lambda} -  f (t + \Delta t) \right)\ , </math>
which determines for a constant amplitude the distance &Delta;''x'' is related to the time interval &Delta;''t'' by the so-called ''phase velocity'' ''v<sub>p</sub>''
:<math>v_p = \frac{\Delta x}{\Delta t} = \lambda f \ . </math>
On the other hand, the same considerations show the envelope propagates at the so-called ''group velocity'' ''v<sub>g</sub>'':<ref name=Eberly/>
:<math>v_g = \frac{\Delta x}{\Delta t} = \lambda_{mod}\Delta f =\lambda^2 \frac{\Delta f}{\Delta \lambda} \ . </math>
A more common expression for the group velocity is obtained by introducing the ''wavevector'' ''k'':
:<math>k=\frac{2\pi}{\lambda} \ . </math>
We notice that for small changes &Delta;&lambda;, the magnitude of the corresponding small change in wavevector:
:<math> \Delta k = \left|\frac{dk}{d\lambda}\right|\Delta \lambda = 2\pi \frac{\Delta \lambda}{\lambda^2} \ , </math>
so the group velocity can be rewritten as:
:<math> v_g= \frac {2\pi\Delta f}{\Delta k} \ . </math>
In all media, frequency and wavevector are related by a ''dispersion relation'', 2&pi;''f'' ≡ &omega; = &omega;(''k''), and the group velocity can be written:
:<math>v_g =\frac{d\omega (k)}{dk} \ . </math>
Here &omega; is the frequency in radians/s. In a medium such as [[classical vacuum]] the dispersion relation for electromagnetic waves is:
:<math>\omega = c_0 k </math>
where ''c''<sub>0</sub> is the [[speed of light]] in classical vacuum. For this case, the phase and group velocities both are ''c''<sub>0</sub>. In so-called ''dispersive media'' the dispersion relation can be a complicated function of wavevector, and the phase and group velocities are not the same. In the general case, the phase and group velocities may have different directions.<ref name=Cerveny/>

==Example: Envelope function approximation==
In [[condensed matter physics]] the wavefunction for a mobile charge carrier in a crystal can be expressed as a [[Bloch wave]]:
:<math>\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}).</math>

where ''n'' is the index for the band (for example, conduction or valence band) '''''r''''' is the spatial location of the particle, and '''''k''''' is its wavevector. The exponential is a sinusoidally varying function corresponding to a slowly varying envelope modulating the rapidly varying part of the wavefunction ''u'', describing the behavior of the wavefunction close to the cores  of the atoms of the lattice.

In determining the behavior of the carriers using [[quantum mechanics]], the ''envelope approximation'' usually is used in which the [[Schrödinger equation]] is simplified to refer only to the behavior of the envelope, and boundary conditions are applied to the envelope function directly, rather than to the complete wavefunction.<ref name= Schuller/> For example, the wavefunction of a carrier trapped near an impurity is governed by an envelope function ''F'' that becomes a superposition of Bloch envelope functions:
:<math>F(\mathbf r)=\frac{1}{\sqrt V} \sum_{\mathbf k } A(\mathbf k ) e^{i\mathbf{k\cdot r}} \ , </math>
where the coefficients ''A''('''''k''''') are found from the approximate Schrödinger equation, and ''V'' is the crystal volume.<ref name=impurity/>

==Example: Diffraction patterns==
{{Image|Double-slit diffraction pattern.png|right|200px|Diffraction pattern of a double slit has a single-slit envelope.}}
Diffraction patterns from multiple slits have envelopes determined by the single slit diffraction pattern. For a single slit the pattern is given by:<ref name=Harris/>
:<math>I_1=I_0 \sin^2\left(\frac {\pi d \sin \alpha}{\lambda}\right) / \left(\frac {\pi d \sin \alpha }{\lambda}\right)^2 \ , </math>
where &alpha; is the diffraction angle, ''d'' is the slit width, and &lambda; is the wavelength. For multiple slits, the pattern is <ref name=Harris/>
:<math>I_q = I_1 \sin^2 \left( \frac {q\pi g \sin \alpha} {\lambda} \right)  /  \sin^2  \left( \frac{ \pi g \sin \alpha}{\lambda}\right) \ , </math>
where ''q'' is the number of slits, and ''g'' is the grating constant. The first factor, the single-slit result, modulates the more rapidly varying second factor that depends upon the number of slits and their spacing.

==References== 
{{reflist|refs=

<ref name=Cerveny>

{{cite book |author=V. Cerveny, Vlastislav Červený |title=Seismic Ray Theory |url=http://books.google.com/books?id=Moj_V-8U0tEC&pg=PA35 |pages=p. 35 |chapter=§2.2.9 Relation between the phase and group velocity vectors |isbn=0521018226 |year=2005 |publisher=Cambridge University Press}}

</ref>

<ref name=Eberly>
{{cite book |title=Laser Physics |author=Peter W. Milonni, Joseph H. Eberly |url=http://books.google.com/books?id=f7g0Mx5RR3cC&pg=PA336 |pages=p. 336 |chapter=§8.3 Group velocity |isbn=0470387718 |year=2010 |publisher=John Wiley & Sons |edition=2nd ed}}
</ref>

<ref name=Harris>
{{cite book |title=Handbook of physics |url=http://books.google.com/books?id=c60mCxGRMR8C&pg=PA306 |pages=pp. 306 ''ff'' |editor=John W Harris, Walter Benenson, Horst Stöcker, Holger Lutz, editors|chapter=Intensity distribution  for diffraction by a slit ''and'' Intensity pattern for diffraction by a grating |isbn=0387952691 |year=2002 |publisher=Springer}}
</ref>

<ref name=impurity>
For example, see {{cite book |title=Solid State Physics |url=http://books.google.com/books?id=L5RrQbbvWn8C&pg=PA478 |pages=p. 478 |author=Giuseppe Grosso, Giuseppe Pastori Parravicini |isbn=012304460X |year=2000 |publisher=	Academic Press |edition=6th ed}}

</ref>

<ref name=Johnson>
{{cite book |title=Software Receiver Design: Build Your Own Digital Communication System in Five Easy Steps |url=http://books.google.com/books?id=LNea1qui1KcC&pg=PA417 |pages=p. 417 |chapter=Figure C.1: The envelope of a function outlines its extremes in a smooth manner |author=C. Richard Johnson, Jr, William A. Sethares, Andrew G. Klein |isbn=0521189446 |year=2011 |publisher=Cambridge University Press}}
</ref>

<ref name=Schuller>
{{cite book |title=Inelastic Light Scattering of Semiconductor Nanostructures: Fundamentals And Recent Advances |author=Christian Schüller |url=http://books.google.com/books?id=gOrVwg05Xk4C&pg=PA22 |pages=p. 22 |chapter=§2.4.1 Envelope function approximation (EFA) |isbn=3540365257 |year=2006 |publisher=Springer}}
</ref>

<ref name=Tipler >
{{cite book |url=http://books.google.com/books?id=BMVR37-8Jh0C&pg=PA538 |pages=p. 538 |isbn=142920124X |publisher=Macmillan |year=2008 |author=Paul Allen Tipler, Gene Mosca |edition=6th ed |title=Physics for Scientists and Engineers, Volume 1}}
</ref>
}}