Top and bottom envelope functions for a modulated sine wave.
In physics and engineering, the envelope function of an oscillating signal is a smooth curve outlining its extremes.^{[1]} 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
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:^{[2]}

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 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] \ ,
which uses the trigonometric formula for the addition of two sine waves, and the approximation Δλ<<λ:

\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} .
Here the modulation wavelength λ_{mod} is given by:^{[2]}^{[3]}

\lambda_{mod} = \frac {\lambda^2}{\Delta \lambda}\ .
The modulation wavelength is double that of the envelope itself because each halfwavelength 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 2Δf.^{[4]}
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.^{[4]}
Phase and group velocity
The argument of the sinusoids above apart from a factor 2π are:

\xi_C =\left( \frac {x}{\lambda}  f \ t \right)\ ,

\xi_E=\left( \frac {x} {\lambda_{mod}}  \Delta f \ t \right) \ ,
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 ξ_{C} and ξ_{E}, and a particular value results for many different but properly related choices of x and t. This invariance means that one can trace these waveforms in space to find the speed of a position of fixed amplitude as it propagates in time; for the argument of the carrier wave to stay the same, the condition is:

\left( \frac {x}{\lambda}  f \ t \right) = \left( \frac {x+\Delta x}{\lambda}  f (t + \Delta t) \right)\ ,
which shows to keep a constant amplitude the distance Δx is related to the time interval Δt by the socalled phase velocity v_{p}

v_p = \frac{\Delta x}{\Delta t} = \lambda f \ .
On the other hand, the same considerations show the envelope propagates at the socalled group velocity v_{g}:^{[5]}

v_g = \frac{\Delta x}{\Delta t} = \lambda_{mod}\Delta f =\lambda^2 \frac{\Delta f}{\Delta \lambda} \ .
A more common expression for the group velocity is obtained by introducing the wavevector k:

k=\frac{2\pi}{\lambda} \ .
We notice that for small changes Δλ, the magnitude of the corresponding small change in wavevector, say Δk, is:

\Delta k = \left\frac{dk}{d\lambda}\right\Delta \lambda = 2\pi \frac{\Delta \lambda}{\lambda^2} \ ,
so the group velocity can be rewritten as:

v_g= \frac {2\pi\Delta f}{\Delta k} =\frac {\Delta \omega}{\Delta k}\ ,
where ω is the frequency in radians/s: ω = 2πf. In all media, frequency and wavevector are related by a dispersion relation, ω = ω(k), and the group velocity can be written:

v_g =\frac{d\omega (k)}{dk} \ .
Dispersion relation ω=ω(k) for some waves corresponding to lattice vibrations in GaAs.^{[6]}
In a medium such as classical vacuum the dispersion relation for electromagnetic waves is:

\omega = c_0 k
where _{0}c is the speed of light in classical vacuum. For this case, the phase and group velocities both are c_{0}.
In socalled dispersive media the dispersion relation can be a complicated function of wavevector, and the phase and group velocities are not the same. For example, for several types of waves exhibited by atomic vibrations (phonons) in GaAs, the dispersion relations are shown in the figure for various directions of wavevector k. In the general case, the phase and group velocities may have different directions.^{[7]}
Example: Envelope function approximation
Electron probabilities in lowest two quantum states of a 160Ǻ GaAs quantum well in a GaAsGaAlAs
heterostructure as calculated from envelope functions.
^{[8]}
In condensed matter physics an energy eigenfunction for a mobile charge carrier in a crystal can be expressed as a Bloch wave:

\psi_{n\mathbf{k}}(\mathbf{r})=e^{i\mathbf{k}\cdot\mathbf{r}}u_{n\mathbf{k}}(\mathbf{r}) \ ,
where n is the index for the band (for example, conduction or valence band) r is a spatial location, and k is a wavevector. The exponential is a sinusoidally varying function corresponding to a slowly varying envelope modulating the rapidly varying part of the wavefunction u_{n,k} describing the behavior of the wavefunction close to the cores of the atoms of the lattice. The envelope is restricted to kvalues within a range limited by the Brillouin zone of the crystal, and that limits how rapidly it can vary with location r.
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.^{[9]} For example, the wavefunction of a carrier trapped near an impurity is governed by an envelope function F that governs a superposition of Bloch functions:

\psi( \mathbf r )= \sum_{\mathbf k } F( \mathbf k ) e^{i\mathbf{k\cdot r}}u_{\mathbf {k}}(\mathbf r ) \ ,
where the Fourier components of the envelope F(k) are found from the approximate Schrödinger equation.^{[10]} In some applications, the periodic part u_{k} is replaced by its value near the band edge, say k=k_{0}, and then:^{[9]}

\psi( \mathbf r )\approx \left( \sum_{\mathbf k } F( \mathbf k ) e^{i\mathbf{k\cdot r}}\right)u_{\mathbf {k=k_0}}(\mathbf r ) = F( \mathbf r )u_{\mathbf {k=k_0}}(\mathbf r ) \ .
Example: Diffraction patterns
Diffraction pattern of a double slit has a singleslit envelope.
Diffraction patterns from multiple slits have envelopes determined by the single slit diffraction pattern. For a single slit the pattern is given by:^{[11]}

I_1=I_0 \sin^2\left(\frac {\pi d \sin \alpha}{\lambda}\right) / \left(\frac {\pi d \sin \alpha }{\lambda}\right)^2 \ ,
where α is the diffraction angle, d is the slit width, and λ is the wavelength. For multiple slits, the pattern is ^{[11]}

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) \ ,
where q is the number of slits, and g is the grating constant. The first factor, the singleslit result I_{1}, modulates the more rapidly varying second factor that depends upon the number of slits and their spacing.
See also
References

^ C. Richard Johnson, Jr, William A. Sethares, Andrew G. Klein (2011). "Figure C.1: The envelope of a function outlines its extremes in a smooth manner". Software Receiver Design: Build Your Own Digital Communication System in Five Easy Steps. Cambridge University Press. p. 417.

^ ^{}a ^{b} Blair Kinsman (2002). Wind Waves: Their Generation and Propagation on the Ocean Surface (Reprint of PrenticeHall 1965 ed.). Courier Dover Publications. p. 186.

^ Mark W. Denny (1993). Air and Water: The Biology and Physics of Life's Media. Princeton University Press. p. 289.

^ ^{}a ^{b} Paul Allen Tipler, Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1 (6th ed.). Macmillan. p. 538.

^ Peter W. Milonni, Joseph H. Eberly (2010). "§8.3 Group velocity". Laser Physics (2nd ed.). John Wiley & Sons. p. 336.

^ Peter Y. Yu, Manuel Cardona (2010). "Fig. 3.2: Phonon dispersion curves in GaAs along highsymmetry axes". Fundamentals of Semiconductors: Physics and Materials Properties (4th ed.). Springer. p. 111.

^ V. Cerveny, Vlastislav Červený (2005). "§2.2.9 Relation between the phase and group velocity vectors". Seismic Ray Theory. Cambridge University Press. p. 35.

^ G Bastard, JA Brum, R Ferreira (1991). "Figure 10 in Electronic States in Semiconductor Heterostructures". In Henry Ehrenreich, David Turnbull, eds. Solid state physics: Semiconductor Heterostructures and Nanostructures. p. 259.

^ ^{}a ^{b} Christian Schüller (2006). "§2.4.1 Envelope function approximation (EFA)". Inelastic Light Scattering of Semiconductor Nanostructures: Fundamentals And Recent Advances. Springer. p. 22.

^ For example, see Marco Fanciulli (2009). "§1.1 Envelope function approximation". Electron Spin Resonance and Related Phenomena in LowDimensional Structures. Springer. pp. 224 ff.

^ ^{}a ^{b} Kordt Griepenkerl (2002). "Intensity distribution for diffraction by a slit and Intensity pattern for diffraction by a grating". In John W Harris, Walter Benenson, Horst Stöcker, Holger Lutz. Handbook of physics. Springer. pp. 306 ff.
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