Envelope function: Difference between revisions
imported>John R. Brews (beating waves) |
imported>John R. Brews (→Phase and group velocity: elaborate on different velocities) |
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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 | In all media, frequency and wavevector are related by a ''dispersion relation'', 2π''f'' ≡ ω = ω(''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 ω is the frequency in radians/s. | Here ω 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== |
Revision as of 12:19, 8 April 2012
- 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)
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
- ↑ 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.
- ↑ 2.0 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.
- ↑ Mark W. Denny (1993). Air and Water: The Biology and Physics of Life's Media. Princeton University Press, p. 289. ISBN 0691025185.
- ↑ Paul Allen Tipler, Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1, 6th ed. Macmillan, p. 538. ISBN 142920124X.
- ↑ Peter W. Milonni, Joseph H. Eberly (2010). “§8.3 Group velocity”, Laser Physics, 2nd ed. John Wiley & Sons, p. 336. ISBN 0470387718.