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Envelope (waves)
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<h2 id="in-beating-waves">In beating waves</h2> <p>A common situation resulting in an envelope function in both space <i>x</i> and time <i>t</i> is the superposition of two waves of almost the same wavelength and frequency:<a class="footnote-ref" id="fnref:2" href="#fn:2"><sup>2</sup></a> </p> F ( x , t ) = sin [ 2 π ( x λ − Δ λ − ( f + Δ f ) t ) ] + sin [ 2 π ( x λ + Δ λ − ( f − Δ f ) t ) ] ≈ 2 cos [ 2 π ( x λ m o d − Δ f t ) ] sin [ 2 π ( x λ − f t ) ] {\displaystyle {\begin{aligned}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]\\[6pt]&\approx 2\cos \left[2\pi \left({\frac {x}{\lambda _{\rm {mod}}}}-\Delta f\ t\right)\right]\ \sin \left[2\pi \left({\frac {x}{\lambda }}-f\ t\right)\right]\end{aligned}}} <p>which uses the trigonometric formula for the <a href="/facts/Trigonometric_identity/Xw9Jx8Vz">addition of two sine waves</a>, and the approximation Δ<i>λ</i> ≪ <i>λ</i>: </p> 1 λ ± Δ λ = 1 λ 1 1 ± Δ λ / λ ≈ 1 λ ∓ Δ λ λ 2 . {\displaystyle {\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}}}.} <p>Here the <i>modulation wavelength</i> <i>λ</i>mod is given by:<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> λ m o d = λ 2 Δ λ . {\displaystyle \lambda _{\rm {mod}}={\frac {\lambda ^{2}}{\Delta \lambda }}\ .} <p>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 <i><a href="/facts/Beat_frequency/zKhaKnMy">beat frequency</a></i> is that of the envelope, twice that of the modulating wave, or 2Δ<i>f</i>.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> </p><p>If this wave is a sound wave, the ear hears the frequency associated with <i>f</i> and the amplitude of this sound varies with the beat frequency.<a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a> </p> <h3>Phase and group velocity</h3> <p class="note">See also: <a href="/facts/Wave/fU0HH6St">Wave § Phase velocity and group velocity</a></p> <p>The argument of the sinusoids above apart from a factor 2π are: </p> ξ C = ( x λ − f t ) , {\displaystyle \xi _{C}=\left({\frac {x}{\lambda }}-f\ t\right)\ ,} ξ E = ( x λ m o d − Δ f t ) , {\displaystyle \xi _{E}=\left({\frac {x}{\lambda _{\rm {mod}}}}-\Delta f\ t\right)\ ,} <p>with subscripts <i>C</i> and <i>E</i> referring to the <i>carrier</i> and the <i>envelope</i>. The same amplitude <i>F</i> of the wave results from the same values of ξC and ξE, each of which may itself return to the same value over different but properly related choices of <i>x</i> and <i>t</i>. 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 <a href="/facts/Carrier_wave/ae98pBlY">carrier wave</a> to stay the same, the condition is: </p> ( x λ − f t ) = ( x + Δ x λ − f ( t + Δ t ) ) , {\displaystyle \left({\frac {x}{\lambda }}-f\ t\right)=\left({\frac {x+\Delta x}{\lambda }}-f(t+\Delta t)\right)\ ,} <p>which shows to keep a constant amplitude the distance Δ<i>x</i> is related to the time interval Δ<i>t</i> by the so-called <i><a href="/facts/Phase_velocity/m43q3X33">phase velocity</a></i> <i>v</i>p </p> v p = Δ x Δ t = λ f . {\displaystyle v_{\rm {p}}={\frac {\Delta x}{\Delta t}}=\lambda f\ .} <p>On the other hand, the same considerations show the envelope propagates at the so-called <i><a href="/facts/Group_velocity/HPIMFwaV">group velocity</a></i> <i>v</i>g:<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a> </p> v g = Δ x Δ t = λ m o d Δ f = λ 2 Δ f Δ λ . {\displaystyle v_{\rm {g}}={\frac {\Delta x}{\Delta t}}=\lambda _{\rm {mod}}\Delta f=\lambda ^{2}{\frac {\Delta f}{\Delta \lambda }}\ .} <p>A more common expression for the group velocity is obtained by introducing the <i>wavevector</i> <i>k</i>: </p> k = 2 π λ . {\displaystyle k={\frac {2\pi }{\lambda }}\ .} <p>We notice that for small changes Δ<i>λ</i>, the magnitude of the corresponding small change in wavevector, say Δ<i>k</i>, is: </p> Δ k = | d k d λ | Δ λ = 2 π Δ λ λ 2 , {\displaystyle \Delta k=\left|{\frac {dk}{d\lambda }}\right|\Delta \lambda =2\pi {\frac {\Delta \lambda }{\lambda ^{2}}}\ ,} <p>so the group velocity can be rewritten as: </p> v g = 2 π Δ f Δ k = Δ ω Δ k , {\displaystyle v_{\rm {g}}={\frac {2\pi \Delta f}{\Delta k}}={\frac {\Delta \omega }{\Delta k}}\ ,} <p>where <i>ω</i> is the frequency in radians/s: <i>ω</i> = 2π<i>f</i>. In all media, frequency and wavevector are related by a <i><a href="/facts/Dispersion_relation/oo7RM5eg">dispersion relation</a></i>, <i>ω</i> = <i>ω</i>(<i>k</i>), and the group velocity can be written: </p> v g = d ω ( k ) d k . {\displaystyle v_{\rm {g}}={\frac {d\omega (k)}{dk}}\ .} <p>In a medium such as <a href="/facts/Classical_vacuum/BTk3goyV">classical vacuum</a> the dispersion relation for electromagnetic waves is: </p> ω = c 0 k {\displaystyle \omega =c_{0}k} <p>where <i>c</i>0 is the <a href="/facts/Speed_of_light/sKSQwdNx">speed of light</a> in classical vacuum. For this case, the phase and group velocities both are <i>c</i>0. </p><p>In so-called <i><a href="/facts/Dispersion_(optics)/cUdHnYXz">dispersive media</a></i> the <a href="/facts/Dispersion_relation/oo7RM5eg">dispersion relation</a> 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 (<a href="/facts/Phonons/tflWW6vw">phonons</a>) in <a href="/facts/GaAs/BakLFjpp">GaAs</a>, the dispersion relations are shown in the figure for <a href="/facts/Brillouin_zone/V8unhAAl">various directions</a> of wavevector k. In the general case, the phase and group velocities may have different directions.<a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a> </p> <h2 id="in-function-approximation">In function approximation</h2> <p class="note">See also: <a href="/facts/K%25C2%25B7p_perturbation_theory/4nJWLPC4">k·p perturbation theory</a></p> <p>In <a href="/facts/Condensed_matter_physics/EBVI5nmL">condensed matter physics</a> an energy <a href="/facts/Eigenfunction/62EJS0XB">eigenfunction</a> for a mobile <a href="/facts/Charge_carrier/wpjjql4x">charge carrier</a> in a crystal can be expressed as a <a href="/facts/Bloch_wave/oxauccIU">Bloch wave</a>: </p> ψ n k ( r ) = e i k ⋅ r u n k ( r ) , {\displaystyle \psi _{n\mathbf {k} }(\mathbf {r} )=e^{i\mathbf {k} \cdot \mathbf {r} }u_{n\mathbf {k} }(\mathbf {r} )\ ,} <p>where <i>n</i> is the index for the band (for example, conduction or valence band) r is a spatial location, and k is a <a href="/facts/Wavevector/Jk8wUEft">wavevector</a>. The exponential is a sinusoidally varying function corresponding to a slowly varying envelope modulating the rapidly varying part of the <a href="/facts/Wave_function/KgM5ykjY">wave function</a> <i>u</i><i>n</i>,k describing the behavior of the wave function close to the cores of the atoms of the lattice. The envelope is restricted to k-values within a range limited by the <a href="/facts/Brillouin_zone/V8unhAAl">Brillouin zone</a> of the crystal, and that limits how rapidly it can vary with location r. </p><p>In determining the behavior of the carriers using <a href="/facts/Quantum_mechanics/BRc3Mzgr">quantum mechanics</a>, the <i>envelope approximation</i> usually is used in which the <a href="/facts/Schr%25C3%25B6dinger_equation/3csgSIWF">Schrödinger equation</a> 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 wave function.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a> For example, the wave function of a carrier trapped near an impurity is governed by an envelope function <i>F</i> that governs a superposition of Bloch functions: </p> ψ ( r ) = ∑ k F ( k ) e i k ⋅ r u k ( r ) , {\displaystyle \psi (\mathbf {r} )=\sum _{\mathbf {k} }F(\mathbf {k} )e^{i\mathbf {k\cdot r} }u_{\mathbf {k} }(\mathbf {r} )\ ,} <p>where the Fourier components of the envelope <i>F</i>(k) are found from the approximate Schrödinger equation.<a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a> In some applications, the periodic part <i>u</i>k is replaced by its value near the band edge, say k=k0, and then:<a class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a> </p> ψ ( r ) ≈ ( ∑ k F ( k ) e i k ⋅ r ) u k = k 0 ( r ) = F ( r ) u k = k 0 ( r ) . {\displaystyle \psi (\mathbf {r} )\approx \left(\sum _{\mathbf {k} }F(\mathbf {k} )e^{i\mathbf {k\cdot r} }\right)u_{\mathbf {k} =\mathbf {k} _{0}}(\mathbf {r} )=F(\mathbf {r} )u_{\mathbf {k} =\mathbf {k} _{0}}(\mathbf {r} )\ .} <h2 id="in-diffraction-patterns">In diffraction patterns</h2> <p><a href="/facts/Diffraction_pattern/Mk80xXHT">Diffraction patterns</a> from multiple slits have envelopes determined by the single slit diffraction pattern. For a single slit the pattern is given by:<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a> </p> I 1 = I 0 sin 2 ( π d sin α λ ) / ( π d sin α λ ) 2 , {\displaystyle I_{1}=I_{0}\sin ^{2}\left({\frac {\pi d\sin \alpha }{\lambda }}\right)/\left({\frac {\pi d\sin \alpha }{\lambda }}\right)^{2}\ ,} <p>where α is the diffraction angle, <i>d</i> is the slit width, and λ is the wavelength. For multiple slits, the pattern is <a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a> </p> I q = I 1 sin 2 ( q π g sin α λ ) / sin 2 ( π g sin α λ ) , {\displaystyle 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)\ ,} <p>where <i>q</i> is the number of slits, and <i>g</i> is the grating constant. The first factor, the single-slit result <i>I1</i>, modulates the more rapidly varying second factor that depends upon the number of slits and their spacing. </p> <h2 id="estimation">Estimation</h2> <p>An <a href="/facts/Envelope_detector/0sAQ7gmp">envelope detector</a> is a <a href="/facts/Electric_circuit/zt94kTW0">circuit</a> that attempts to extract the envelope from an <a href="/facts/Analog_signal/xDbb4uB5">analog signal</a>. </p><p>In <a href="/facts/Digital_signal_processing/PEG7azr8">digital signal processing</a>, the envelope may be estimated employing the <a href="/facts/Hilbert_transform/RoQaC8MS">Hilbert transform</a> or a <a href="/facts/Moving_average/tWY74500">moving</a> <a href="/facts/RMS_amplitude/aEckuXJM">RMS amplitude</a>.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a> </p> <h2 id="see-also">See also</h2> <ul><li><a href="/facts/Analytic_signal/8CB4Cyul">Analytic signal § Complex envelope/baseband</a></li> <li><a href="/facts/Empirical_mode_decomposition/4wd3JY4E">Empirical mode decomposition</a></li> <li><a href="/facts/Envelope_(mathematics)/jr5DkCGD">Envelope (mathematics)</a></li> <li><a href="/facts/Envelope_tracking/LLRHdFTb">Envelope tracking</a></li> <li><a href="/facts/Instantaneous_phase/fJ17zuan">Instantaneous phase</a></li> <li><a href="/facts/Modulation/aAXPjINv">Modulation</a></li> <li><a href="/facts/Mathematics_of_oscillation/JHMgNIhG">Mathematics of oscillation</a></li> <li><a href="/facts/Peak_envelope_power/2j6eS46U">Peak envelope power</a></li> <li><a href="/facts/Spectral_envelope/RJL6ajoo">Spectral envelope</a></li></ul> <p><i>This article incorporates material from the <a href="/facts/Citizendium/r8kIw6Hf">Citizendium</a> article "Envelope function", which is licensed under the Creative Commons Attribution-ShareAlike 3.0 Unported License but not under the GFDL.</i> </p> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>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 978-0521189446. <a href="978-0521189446" target="_blank">978-0521189446</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Blair Kinsman (2002). Wind Waves: Their Generation and Propagation on the Ocean Surface (Reprint of Prentice-Hall 1965 ed.). Courier Dover Publications. p. 186. ISBN 0486495116. <a href="0486495116" target="_blank">0486495116</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>Blair Kinsman (2002). Wind Waves: Their Generation and Propagation on the Ocean Surface (Reprint of Prentice-Hall 1965 ed.). Courier Dover Publications. p. 186. ISBN 0486495116. <a href="0486495116" target="_blank">0486495116</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>Mark W. Denny (1993). Air and Water: The Biology and Physics of Life's Media. Princeton University Press. pp. 289. ISBN 0691025185. <a href="0691025185" target="_blank">0691025185</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>Paul Allen Tipler; Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1 (6th ed.). Macmillan. p. 538. ISBN 978-1429201247. <a href="978-1429201247" target="_blank">978-1429201247</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> <li id="fn:6"><p>Paul Allen Tipler; Gene Mosca (2008). Physics for Scientists and Engineers, Volume 1 (6th ed.). Macmillan. p. 538. ISBN 978-1429201247. <a href="978-1429201247" target="_blank">978-1429201247</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li> <li id="fn:7"><p>Peter W. Milonni; Joseph H. Eberly (2010). "§8.3 Group velocity". Laser Physics (2nd ed.). John Wiley & Sons. p. 336. ISBN 978-0470387719. <a href="978-0470387719" target="_blank">978-0470387719</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li> <li id="fn:8"><p>V. Cerveny; Vlastislav Červený (2005). "§2.2.9 Relation between the phase and group velocity vectors". Seismic Ray Theory. Cambridge University Press. p. 35. ISBN 0521018226. <a href="0521018226" target="_blank">0521018226</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li> <li id="fn:9"><p>Christian Schüller (2006). "§2.4.1 Envelope function approximation (EFA)". Inelastic Light Scattering of Semiconductor Nanostructures: Fundamentals And Recent Advances. Springer. p. 22. ISBN 3540365257. <a href="3540365257" target="_blank">3540365257</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li> <li id="fn:10"><p>For example, see Marco Fanciulli (2009). "§1.1 Envelope function approximation". Electron Spin Resonance and Related Phenomena in Low-Dimensional Structures. Springer. pp. 224 ff. ISBN 978-3540793649. <a href="978-3540793649" target="_blank">978-3540793649</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li> <li id="fn:11"><p>Christian Schüller (2006). "§2.4.1 Envelope function approximation (EFA)". Inelastic Light Scattering of Semiconductor Nanostructures: Fundamentals And Recent Advances. Springer. p. 22. ISBN 3540365257. <a href="3540365257" target="_blank">3540365257</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li> <li id="fn:12"><p>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 (eds.). Handbook of physics. Springer. pp. 306 ff. ISBN 0387952691. <a href="0387952691" target="_blank">0387952691</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li> <li id="fn:13"><p>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 (eds.). Handbook of physics. Springer. pp. 306 ff. ISBN 0387952691. <a href="0387952691" target="_blank">0387952691</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li> <li id="fn:14"><p>"Envelope Extraction - MATLAB & Simulink". MathWorks. 2021-09-02. Retrieved 2021-11-16. <a href="https://www.mathworks.com/help/signal/ug/envelope-extraction-using-the-analytic-signal.html" target="_blank">https://www.mathworks.com/help/signal/ug/envelope-extraction-using-the-analytic-signal.html</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li> </ol>