Envelope (waves)

In beating waves

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:

${\displaystyle \begin{array}{cc}F(x,t)& =\sin \left[2\pi \left(\frac{x}{\lambda -\mathrm{\Delta }\lambda }-(f+\mathrm{\Delta }f)t\right)\right]+\sin \left[2\pi \left(\frac{x}{\lambda +\mathrm{\Delta }\lambda }-(f-\mathrm{\Delta }f)t\right)\right]\\ & \approx 2\cos \left[2\pi \left(\frac{x}{{\lambda }_{\mathrm{m}\mathrm{o}\mathrm{d}}}-\mathrm{\Delta }ft\right)\right]\sin \left[2\pi \left(\frac{x}{\lambda }-ft\right)\right]\end{array}}$

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

${\displaystyle \frac{1}{\lambda \pm \mathrm{\Delta }\lambda }=\frac{1}{\lambda }\frac{1}{1\pm \mathrm{\Delta }\lambda /\lambda }\approx \frac{1}{\lambda }\mp \frac{\mathrm{\Delta }\lambda }{{\lambda }^2}.}$

Here the modulation wavelength λ_mod is given by:

${\displaystyle {\lambda }_{\mathrm{m}\mathrm{o}\mathrm{d}}=\frac{{\lambda }^2}{\mathrm{\Delta }\lambda }.}$

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 2Δf.

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.

Phase and group velocity

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

${\displaystyle {\xi }_C=\left(\frac{x}{\lambda }-ft\right),}$

${\displaystyle {\xi }_E=\left(\frac{x}{{\lambda }_{\mathrm{m}\mathrm{o}\mathrm{d}}}-\mathrm{\Delta }ft\right),}$

with subscripts C and E referring to the carrier and the envelope. The same amplitude F of the wave results from the same values of ξ_C and ξ_E, each of which may itself return to the same value over 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:

${\displaystyle \left(\frac{x}{\lambda }-ft\right)=\left(\frac{x+\mathrm{\Delta }x}{\lambda }-f(t+\mathrm{\Delta }t)\right),}$

which shows to keep a constant amplitude the distance Δx is related to the time interval Δt by the so-called phase velocity v_p

${\displaystyle v_{\mathrm{p}}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}=\lambda f.}$

On the other hand, the same considerations show the envelope propagates at the so-called group velocity v_g:

${\displaystyle v_{\mathrm{g}}=\frac{\mathrm{\Delta }x}{\mathrm{\Delta }t}={\lambda }_{\mathrm{m}\mathrm{o}\mathrm{d}}\mathrm{\Delta }f={\lambda }^2\frac{\mathrm{\Delta }f}{\mathrm{\Delta }\lambda }.}$

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

${\displaystyle k=\frac{2\pi }{\lambda }.}$

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

${\displaystyle \mathrm{\Delta }k=\left|\frac{dk}{d\lambda }\right|\mathrm{\Delta }\lambda =2\pi \frac{\mathrm{\Delta }\lambda }{{\lambda }^2},}$

so the group velocity can be rewritten as:

${\displaystyle v_{\mathrm{g}}=\frac{2\pi \mathrm{\Delta }f}{\mathrm{\Delta }k}=\frac{\mathrm{\Delta }\omega }{\mathrm{\Delta }k},}$

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:

${\displaystyle v_{\mathrm{g}}=\frac{d\omega (k)}{dk}.}$

In a medium such as classical vacuum the dispersion relation for electromagnetic waves is:

${\displaystyle \omega =c_{0}k}$

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. 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.