拍频

差頻（英文：beat note或beat frequency）一詞源於聲學上两个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。^[1]

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

y_{\mathrm {1} }=R\sin(k_{\mathrm {1} }x-\omega _{\mathrm {1} }t)

y_{\mathrm {2} }=R\sin(k_{\mathrm {2} }x-\omega _{\mathrm {2} }t)

且因為頻率只是略有差異，在此假設

k_{\mathrm {1} }\doteqdot k_{\mathrm {2} }\doteqdot k

\omega _{\mathrm {1} }\doteqdot \omega _{\mathrm {2} }\doteqdot \omega

令

y=y_{\mathrm {1} }+y_{\mathrm {2} }

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)

在此又令:

k'={\frac {k_{\mathrm {1} }-k_{\mathrm {2} }}{2}}={\frac {\Delta k}{2}}

\omega '={\frac {\omega _{\mathrm {1} }-\omega _{\mathrm {2} }}{2}}={\frac {\Delta \omega }{2}}

故y可以改寫成

y=2R\sin(kx-\omega t)\cos(k'x-\omega 't)

註釋

^ Levitin, Daniel J. This is Your Brain on Music: The Science of a Human Obsession. Dutton. 2006: 22. ISBN 978-0525949695.

Thaut, Michael H. Rhythm, music, and the brain : scientific foundations and clinical applications 1st in paperback. New York: Routledge. 2005. ISBN 978-0415973700.
Berger, Jonathan; Turow, Gabe (编). Music, science, and the rhythmic brain : cultural and clinical implications. Routledge. 2011. ISBN 978-0415890595.

Java applet （页面存档备份，存于互联网档案馆）, MIT
Acoustics and Vibration Animations （页面存档备份，存于互联网档案馆）, D.A. Russell, Pennsylvania State University
A Java applet showing the formation of beats due to the interference of two waves of slightly different frequencies （页面存档备份，存于互联网档案馆）
Lissajous Curves: Interactive simulation of graphical representations of musical intervals, beats, interference, vibrating strings （页面存档备份，存于互联网档案馆）
The Feynman Lectures on Physics Vol. I Ch. 48: Beats