# Difference between revisions of "Harmonics"

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The simplest periodic functions of the form | 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 | + | 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 | + | 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==== | ====Comments==== | ||

− | More generally, if | + | 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> |

## 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=11587