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The simplest periodic functions of the form
 
The simplest periodic functions of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h0465701.png" /></td> </tr></table>
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$$A\sin(\omega x+\phi).$$
  
These functions are encountered in the study of many oscillatory processes. The number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h0465702.png" /> is known as the amplitude, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h0465703.png" /> is known as the frequency, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h0465704.png" /> is known as the initial phase, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h0465705.png" /> is the oscillation period. The functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h0465706.png" /> are, respectively, the second, third, etc., higher harmonics with respect to the fundamental harmonic. In addition to the harmonics themselves, their sums
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These functions are encountered in the study of many oscillatory processes. The number $A$ is known as the amplitude, $\omega$ is known as the frequency, $\phi$ is known as the initial phase, and $T=2\pi/\omega$ is the oscillation period. The functions $\sin(2\omega x+\phi),\sin(3\omega x+\phi),\dots,$ are, respectively, the second, third, etc., higher harmonics with respect to the fundamental harmonic. In addition to the harmonics themselves, their sums
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h0465707.png" /></td> <td valign="top" style="width:5%;text-align:right;">(*)</td></tr></table>
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$$a_0+a_1\sin(\omega x+\phi)+a_2\sin(2\omega x+\phi)+\dots\tag{*}$$
  
are also considered, since a very broad class of functions can be expanded in series of the form (*) in the study of various processes.
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are also considered, since a very broad class of functions can be expanded in series of the form \ref{*} in the study of various processes.
  
  
  
 
====Comments====
 
====Comments====
More generally, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h0465708.png" /> is a compact group, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h0465709.png" /> is a closed subgroup of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h04657010.png" /> and if the regular representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h04657011.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h04657012.png" /> decomposes uniquely into irreducible subrepresentations, then the functions on the homogeneous space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h04657013.png" /> belonging to irreducible subspaces of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h04657014.png" /> are called harmonics, cf. [[#References|[a1]]]. For <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h04657015.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/h/h046/h046570/h04657016.png" />, one finds the classical harmonics.
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More generally, if $G$ is a compact group, $K$ is a closed subgroup of $G$ and if the regular representation of $G$ on $L_2(G/K)$ decomposes uniquely into irreducible subrepresentations, then the functions on the homogeneous space $G/K$ belonging to irreducible subspaces of $L_2(G/K)$ are called harmonics, cf. [[#References|[a1]]]. For $G=O(2)$, $K=O(1)$, one finds the classical harmonics.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Weyl,  "Harmonics on homogeneous manifolds"  ''Ann. of Math.'' , '''35'''  (1934)  pp. 486–499</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Hobson,  "The theory of spherical and ellipsoidal harmonics" , Chelsea, reprint  (1955)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  H. Weyl,  "Harmonics on homogeneous manifolds"  ''Ann. of Math.'' , '''35'''  (1934)  pp. 486–499</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  E.W. Hobson,  "The theory of spherical and ellipsoidal harmonics" , Chelsea, reprint  (1955)</TD></TR></table>

Revision as of 15:46, 29 September 2014

The simplest periodic functions of the form

$$A\sin(\omega x+\phi).$$

These functions are encountered in the study of many oscillatory processes. The number $A$ is known as the amplitude, $\omega$ is known as the frequency, $\phi$ is known as the initial phase, and $T=2\pi/\omega$ is the oscillation period. The functions $\sin(2\omega x+\phi),\sin(3\omega x+\phi),\dots,$ are, respectively, the second, third, etc., higher harmonics with respect to the fundamental harmonic. In addition to the harmonics themselves, their sums

$$a_0+a_1\sin(\omega x+\phi)+a_2\sin(2\omega x+\phi)+\dots\tag{*}$$

are also considered, since a very broad class of functions can be expanded in series of the form \ref{*} in the study of various processes.


Comments

More generally, if $G$ is a compact group, $K$ is a closed subgroup of $G$ and if the regular representation of $G$ on $L_2(G/K)$ decomposes uniquely into irreducible subrepresentations, then the functions on the homogeneous space $G/K$ belonging to irreducible subspaces of $L_2(G/K)$ are called harmonics, cf. [a1]. For $G=O(2)$, $K=O(1)$, one finds the classical harmonics.

References

[a1] H. Weyl, "Harmonics on homogeneous manifolds" Ann. of Math. , 35 (1934) pp. 486–499
[a2] E.W. Hobson, "The theory of spherical and ellipsoidal harmonics" , Chelsea, reprint (1955)
How to Cite This Entry:
Harmonics. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Harmonics&oldid=11587
This article was adapted from an original article by A.I. Barabanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article