查看“︁拍频”︁的源代码

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'''差頻'''（英文：{{lang|en|beat note}}或{{lang|en|beat frequency}}）一詞源於[[聲學]]上两个[[頻率 (物理學)|频率]]相近但不同的[[声波]]的[[干涉]]，所得到的干涉信号的频率是原先两个声波的频率之差的絕對值，因此叫做差频。這個概念也用到了[[光学]]和[[电子学]]中，指兩個頻率不同的信号進行合波后得到频率为两者之差的新信號。<ref>{{Cite book|title=This is Your Brain on Music: The Science of a Human Obsession|url=https://archive.org/details/thisisyourbraino00levi|last=Levitin|first=Daniel J.|publisher=Dutton|year=2006|isbn= 978-0525949695 |page=[https://archive.org/details/thisisyourbraino00levi/page/22 22]}}</ref>

以兩擁有相同[[振幅]]、無[[相位差]]，但頻率略有差異之[[正弦波]]為例
:<math>y_\mathrm{1} = R \sin(k_\mathrm{1} x - \omega_\mathrm{1} t)</math>
:<math>y_\mathrm{2} = R \sin(k_\mathrm{2} x - \omega_\mathrm{2} t)</math>
且因為頻率只是略有差異，在此假設
:<math>k_\mathrm{1}\doteqdot k_\mathrm{2}\doteqdot k</math>
:<math>\omega_\mathrm{1} \doteqdot \omega_\mathrm{2}\doteqdot \omega</math>
令
:<math>y = y_\mathrm{1} + y_\mathrm{2}</math> 
:<math>y = 2R \sin(\frac{k_\mathrm{1} + k_\mathrm{2}}{2} x - \frac{\omega_\mathrm{1} + \omega_\mathrm{2}}{2} t) \cos(\frac{k_\mathrm{1} - k_\mathrm{2}}{2} x - \frac{\omega_\mathrm{1} - \omega_\mathrm{2}}{2} t)</math>
在此又令:
:<math>k' = \frac{k_\mathrm{1} - k_\mathrm{2}}{2} = \frac{\Delta k}{2}</math>
:<math>\omega' = \frac{\omega_\mathrm{1} - \omega_\mathrm{2}}{2} = \frac{\Delta \omega}{2}</math>
故y可以改寫成
:<math>y = 2R \sin(k x - \omega t) \cos(k' x - \omega' t)</math>

==註釋==
{{Reflist}}

==延伸閱讀==
{{refbegin}}
* {{cite book|last1=Thaut|first1=Michael H.|title=Rhythm, music, and the brain : scientific foundations and clinical applications|url=https://archive.org/details/rhythmmusicbrain0000thau|date=2005|publisher=Routledge|location=New York|isbn=978-0415973700|edition=1st in paperback}}
* {{cite book|editor1-last=Berger|editor1-first=Jonathan|editor2-last=Turow|editor2-first=Gabe|title=Music, science, and the rhythmic brain : cultural and clinical implications|date=2011|publisher=Routledge|isbn=978-0415890595}}
{{refend}}

==外部連結==
* [http://mathlets.org/mathlets/beats/ Java applet] {{Wayback|url=http://mathlets.org/mathlets/beats/ |date=20211106155006 }}, MIT
* [http://www.acs.psu.edu/drussell/Demos/superposition/superposition.html Acoustics and Vibration Animations] {{Wayback|url=http://www.acs.psu.edu/drussell/Demos/superposition/superposition.html |date=20210416111135 }}, D.A. Russell, Pennsylvania State University
* [http://phy.hk/wiki/englishhtm/Beats.htm A Java applet showing the formation of beats due to the interference of two waves of slightly different frequencies] {{Wayback|url=http://phy.hk/wiki/englishhtm/Beats.htm |date=20210308105820 }}
* [http://gerdbreitenbach.de/lissajous/lissajous.html Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference, vibrating strings] {{Wayback|url=http://gerdbreitenbach.de/lissajous/lissajous.html |date=20100827233923 }}
* [https://feynmanlectures.caltech.edu/I_48.html The Feynman Lectures on Physics Vol. I Ch. 48: Beats]

{{Acoustics|state=collapsed}}

[[Category:声学]]