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 into an instantaneous amplitude. The figure illustrates a modulated sine wave varying between an upper envelope and a lower envelope. The envelope function may be a function of time, space, angle, or indeed of any variable.
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:[1]
\begin{align} F(x, t)&=\sin\left[2\pi\left(
x | |
λ-\Deltaλ |
-(f+\Deltaf)t\right)\right]+\sin\left[2\pi\left(
x | |
λ+\Deltaλ |
-(f-\Deltaf)t\right)\right]\\[6pt] & ≈ 2\cos\left[2\pi\left(
x | |
λ\rm |
-\Deltaf t\right)\right] \sin\left[2\pi\left(
x | |
λ |
-f t\right)\right] \end{align}
which uses the trigonometric formula for the addition of two sine waves, and the approximation Δλ ≪ λ:
1 | = | |
λ\pm\Deltaλ |
1 | ||
λ |
1 | |
1\pm\Deltaλ/λ |
≈
1 | |
λ |
\mp
\Deltaλ | |
λ2 |
.
Here the modulation wavelength λmod is given by:[1] [2]
λ\rm=
λ2 | |
\Deltaλ |
.
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.
The argument of the sinusoids above apart from a factor 2 are:
\xiC=\left(
x | |
λ |
-f t\right) ,
\xiE=\left(
x | |
λ\rm |
-\Deltaf t\right) ,
\left(
x | |
λ |
-f t\right)=\left(
x+\Deltax | |
λ |
-f(t+\Deltat)\right) ,
v\rm=
\Deltax | |
\Deltat |
=λf .
v\rm=
\Deltax | |
\Deltat |
=λ\rm\Deltaf=λ2
\Deltaf | |
\Deltaλ |
.
A more common expression for the group velocity is obtained by introducing the wavevector k:
k= | 2\pi |
λ |
.
\Deltak=\left|
dk | |
dλ |
\right|\Deltaλ=2\pi
\Deltaλ | |
λ2 |
,
v\rm=
2\pi\Deltaf | = | |
\Deltak |
\Delta\omega | |
\Deltak |
,
where ω is the frequency in radians/s: ω = 2f. In all media, frequency and wavevector are related by a dispersion relation, ω = ω(k), and the group velocity can be written:
v\rm=
d\omega(k) | |
dk |
.
\omega=c0k
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. 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.
See also: k·p perturbation theory.
In condensed matter physics an energy eigenfunction for a mobile charge carrier in a crystal can be expressed as a Bloch wave:
\psink(r)=eik ⋅ runk(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 un,k describing the behavior of the wavefunction close to the cores of the atoms of the lattice. The envelope is restricted to k-values 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. 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(r)=\sumkF(k)eik ⋅ ruk(r) ,
\psi(r) ≈ \left(\sumkF(k)eik ⋅ r
\right)u | |
k=k0 |
(r)=F(
r)u | |
k=k0 |
(r) .
Diffraction patterns from multiple slits have envelopes determined by the single slit diffraction pattern. For a single slit the pattern is given by:
I1=I0
| ||||
\sin |
\right)/\left(
\pid\sin\alpha | |
λ |
\right)2 ,
Iq=I1\sin2\left(
q\pig\sin\alpha | |
λ |
\right)/\sin2\left(
\pig\sin\alpha | |
λ |
\right) ,
An envelope detector is a circuit that attempts to extract the envelope from an analog signal.
In digital signal processing, the envelope may be estimated employing the Hilbert transform or a moving RMS amplitude.[3]