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Difference between revisions of "Harmonics"

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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
 
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
  
\begin{equation}a_0+a_1\sin(\omega x+\phi)+a_2\sin(2\omega x+\phi)+\dots\label{*}\end{equation}
+
\begin{equation}a_0+a_1\sin(\omega x+\phi)+a_2\sin(2\omega x+\phi)+\dotsb\label{*}\end{equation}
  
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.
+
are also considered, since a very broad class of functions can be expanded in series of the form \eqref{*} in the study of various processes.
  
  

Latest revision as of 14:31, 14 February 2020

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

\begin{equation}a_0+a_1\sin(\omega x+\phi)+a_2\sin(2\omega x+\phi)+\dotsb\label{*}\end{equation}

are also considered, since a very broad class of functions can be expanded in series of the form \eqref{*} 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=43447
This article was adapted from an original article by A.I. Barabanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article