Envelope function: Difference between revisions

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so the group velocity can be rewritten as:
so the group velocity can be rewritten as:
:<math> v_g= \frac {2\pi\Delta f}{\Delta k} \ . </math>
:<math> v_g= \frac {2\pi\Delta f}{\Delta k} \ . </math>
In so-called ''dispersive media'' the frequency and wavevector are related by a ''dispersion relation'', 2&pi;''f'' ≡ &omega; = &omega;(''k''), and the group velocity can be written:
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>
:<math>v_g =\frac{d\omega (k)}{dk} \ . </math>
Here &omega; is the frequency in radians/s.
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.


==References==  
==References==  

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© Image: John R. Brews
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 is a smooth curve outlining its extremes in amplitude.[1] 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: Beat frequency

See also: Beat (acoustics)
(PD) Image: John R. Brews
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]

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

Here the modulation wavelength λmod is given by:[2][3]

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.

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 Δf.[4]

Phase and group velocity

The argument of the sinusoids above apart from a factor 2π are:

with subscripts c and e referring to the carrier and the envelope. The same values for these functions occur when these arguments increase by unity, whether that increase is a result of an increase in distance x or an increase in time t. That is, the same amplitude results for ξc and for ξc + 1, and likewise for ξb. This invariance means that one can trace these waveforms in space to find a position of fixed amplitude propagates in time with a speed that keeps ξ fixed; that is, for the carrier:

which determines for a constant amplitude the distance Δx is related to the time interval Δt by the so-called phase velocity vp

On the other hand, the same considerations show the envelope propagates at the so-called group velocity vg:[5]

A more common expression for the group velocity is obtained by introducing the wavevector k:

We notice that for small changes Δλ, the magnitude of the corresponding small change in wavevector:

so the group velocity can be rewritten as:

In all media, frequency and wavevector are related by a dispersion relation, 2πf ≡ ω = ω(k), and the group velocity can be written:

Here ω is the frequency in radians/s. In a medium such as classical vacuum the dispersion relation for electromagnetic waves is:

where c0 is the speed of light in classical vacuum. For this case, the phase and group velocities both are c0. 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.

References

  1. 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. ISBN 0521189446. 
  2. 2.0 2.1 Blair Kinsman (2002). Wind Waves: Their Generation and Propagation on the Ocean Surface, Reprint of Prentice-Hall 1965. Courier Dover Publications, p. 186. ISBN 0486495116. 
  3. Mark W. Denny (1993). Air and Water: The Biology and Physics of Life's Media. Princeton University Press, p. 289. ISBN 0691025185. 
  4. Paul Allen Tipler, Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1, 6th ed. Macmillan, p. 538. ISBN 142920124X. 
  5. Peter W. Milonni, Joseph H. Eberly (2010). “§8.3 Group velocity”, Laser Physics, 2nd ed. John Wiley & Sons, p. 336. ISBN 0470387718.