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'Helmholtz differential equation'
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| Title of object: |
Helmholtz differential equation |
| Canonical Name: |
HelmholtzDifferentialEquation |
| Type: |
Definition |
| Created on: |
2002-11-13 06:17:25.283025-05 |
| Modified on: |
2002-11-13 07:18:13.949427-05 |
| Classification: |
msc:26B12, msc:35-00 |
Preamble:
% this is the default PlanetMath preamble. as your knowledge
% of TeX increases, you will probably want to edit this, but
% it should be fine as is for beginners.
% almost certainly you want these
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{amsfonts}
% used for TeXing text within eps files
%\usepackage{psfrag}
% need this for including graphics (\includegraphics)
\usepackage{graphicx}
% for neatly defining theorems and propositions
%\usepackage{amsthm}
% making logically defined graphics
%\usepackage{xypic}
% there are many more packages, add them here as you need them
% define commands here
\newcommand{\vA}{\mathbf{A}}
\newcommand{\vB}{\mathbf{B}}
\newcommand{\vx}{\mathbf{x}}
\newcommand{\vN}{\mathbf{N}}
\newcommand{\bV}{\mathbf{V}}
\newcommand{\vnabla}{\nabla} |
Content:
It is a partial differential equation which, in scalar form is $$\vnabla^2f+k^2f = 0,$$ or in vector form is $$\vnabla^2\vA+k^2\vA = 0,$$ where $\vnabla^2$ is the Laplacian.
The solutions of this equation represent the solution of the wave equation, which is of great interest in physics.
Consider a wave equation $$\frac{\partial^2\psi}{\partial t^2} = c^2\vnabla^2\psi$$
with wave speed $c$. If we look for time harmonic standing waves of frequency $\omega$,
$$\psi(\vx,t) = e^{-j\omega t}\phi(\vx)$$
we find that $\phi(x)$ satisfies the Helmholtz equation:
$$(\vnabla^2+k^2)\phi = 0$$
where $k=\omega/c$ is the wave number.
Usually Helmholtz equation is solved by seperation of variables method, in cartesian, spherical or cylindrical coordinates. |
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