翻訳と辞書
Words near each other
・ "O" Is for Outlaw
・ "O"-Jung.Ban.Hap.
・ "Ode-to-Napoleon" hexachord
・ "Oh Yeah!" Live
・ "Our Contemporary" regional art exhibition (Leningrad, 1975)
・ "P" Is for Peril
・ "Pimpernel" Smith
・ "Polish death camp" controversy
・ "Pro knigi" ("About books")
・ "Prosopa" Greek Television Awards
・ "Pussy Cats" Starring the Walkmen
・ "Q" Is for Quarry
・ "R" Is for Ricochet
・ "R" The King (2016 film)
・ "Rags" Ragland
・ ! (album)
・ ! (disambiguation)
・ !!
・ !!!
・ !!! (album)
・ !!Destroy-Oh-Boy!!
・ !Action Pact!
・ !Arriba! La Pachanga
・ !Hero
・ !Hero (album)
・ !Kung language
・ !Oka Tokat
・ !PAUS3
・ !T.O.O.H.!
・ !Women Art Revolution


Dictionary Lists
翻訳と辞書 辞書検索 [ 開発暫定版 ]
スポンサード リンク

cylindrical harmonics : ウィキペディア英語版
cylindrical harmonics
In mathematics, the cylindrical harmonics are a set of linearly independent solutions to Laplace's differential equation, \nabla^2 V = 0, expressed in cylindrical coordinates, ''ρ'' (radial coordinate), ''φ'' (polar angle), and ''z'' (height). Each function ''V''''n''(''k'') is the product of three terms, each depending on one coordinate alone. The ''ρ''-dependent term is given by Bessel functions (which occasionally are also called cylindrical harmonics).
==Definition==
Each function V_n(k) of this basis consists of the product of three functions:
:V_n(k;\rho,\varphi,z)=P_n(k,\rho)\Phi_n(\varphi)Z(k,z)\,
where (\rho,\varphi,z) are the cylindrical coordinates, and ''n'' and ''k'' are constants which distinguish the members of the set from each other. As a result of the superposition principle applied to Laplace's equation, very general solutions to Laplace's equation can be obtained by linear combinations of these functions.
Since all of the surfaces of constant ρ, φ and ''z''  are conicoid, Laplace's equation is separable in cylindrical coordinates. Using the technique of the separation of variables, a separated solution to Laplace's equation may be written:
:V=P(\rho)\,\Phi(\varphi)\,Z(z)
and Laplace's equation, divided by ''V'', is written:
:
\frac+\frac\,\frac+\frac\,\frac+\frac=0

The ''Z''  part of the equation is a function of ''z'' alone, and must therefore be equal to a constant:
:\frac=k^2
where ''k''  is, in general, a complex number. For a particular ''k'', the ''Z(z)'' function has two linearly independent solutions. If ''k'' is real they are:
:Z(k,z)=\cosh(kz)\,\,\,\,\,\,\mathrm\,\,\,\,\,\,\sinh(kz)\,
or by their behavior at infinity:
:Z(k,z)=e^\,\,\,\,\,\,\mathrm\,\,\,\,\,\,e^\,
If ''k'' is imaginary:
:Z(k,z)=\cos(|k|z)\,\,\,\,\,\,\mathrm\,\,\,\,\,\,\sin(|k|z)\,
or:
:Z(k,z)=e^\,\,\,\,\,\,\mathrm\,\,\,\,\,\,e^\,
It can be seen that the ''Z(k,z)'' functions are the kernels of the Fourier transform or Laplace transform of the ''Z(z)'' function and so ''k'' may be a discrete variable for periodic boundary conditions, or it may be a continuous variable for non-periodic boundary conditions.
Substituting k^2 for \ddot/Z , Laplace's equation may now be written:
:
\frac+\frac\,\frac+\frac\frac+k^2=0

Multiplying by \rho^2, we may now separate the ''P''  and Φ functions and introduce another constant (''n'') to obtain:
:\frac =-n^2
:\rho^2\frac+\rho\frac+k^2\rho^2=n^2
Since \varphi is periodic, we may take ''n'' to be a non-negative integer and accordingly, the \Phi(\varphi) the constants are subscripted. Real solutions for \Phi(\varphi) are
:\Phi_n=\cos(n\varphi)\,\,\,\,\,\,\mathrm\,\,\,\,\,\,\sin(n\varphi)
or, equivalently:
:\Phi_n=e^\,\,\,\,\,\,\mathrm\,\,\,\,\,\,e^
The differential equation for \rho is a form of Bessel's equation.
If ''k'' is zero, but ''n'' is not, the solutions are:
:P_n(0,\rho)=\rho^n\,\,\,\,\,\,\mathrm\,\,\,\,\,\,\rho^\,
If both k and n are zero, the solutions are:
:P_0(0,\rho)=\ln\rho\,\,\,\,\,\,\mathrm\,\,\,\,\,\,1\,
If ''k'' is a real number we may write a real solution as:
:P_n(k,\rho)=J_n(k\rho)\,\,\,\,\,\,\mathrm\,\,\,\,\,\,Y_n(k\rho)\,
where J_n(z) and Y_n(z) are ordinary Bessel functions.
If ''k''  is an imaginary number, we may write a real solution as:
:P_n(k,\rho)=I_n(|k|\rho)\,\,\,\,\,\,\mathrm\,\,\,\,\,\,K_n(|k|\rho)\,
where I_n(z) and K_n(z) are modified Bessel functions.
The cylindrical harmonics for (k,n) are now the product of these solutions and the general solution to Laplace's equation is given by a linear combination of these solutions:
:V(\rho,\varphi,z)=\sum_n \int dk\,\, A_n(k) P_n(k,\rho) \Phi_n(\varphi) Z(k,z)\,
where the A_n(k) are constants with respect to the cylindrical coordinates and the limits of the summation and integration are determined by the boundary conditions of the problem. Note that the integral may be replaced by a sum for appropriate boundary conditions. The orthogonality of the J_n(x) is often very useful when finding a solution to a particular problem. The \Phi_n(\varphi) and Z(k,z) functions are essentially Fourier or Laplace expansions, and form a set of orthogonal functions. When P_n(k\rho) is simply J_n(k\rho) , the orthogonality of J_n, along with the orthogonality relationships of \Phi_n(\varphi) and Z(k,z) allow the constants to be determined.
If (x)_k is the sequence of the positive zeros of J_n then:
:\int_0^1 J_n(x_k\rho)J_n(x_k'\rho)\rho\,d\rho = \fracJ_(x_k)^2\delta_
In solving problems, the space may be divided into any number of pieces, as long as the values of the potential and its derivative match across a boundary which contains no sources.

抄文引用元・出典: フリー百科事典『 ウィキペディア(Wikipedia)
ウィキペディアで「cylindrical harmonics」の詳細全文を読む



スポンサード リンク
翻訳と辞書 : 翻訳のためのインターネットリソース

Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.