Talk:PlanetPhysics/Wave Equation

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: wave equation %%% Primary Category Code: 02.30.Jr %%% Filename: WaveEquation.tex %%% Version: 5 %%% Owner: bloftin %%% Author(s): bci1, bloftin %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Any \emph{wave equation} describes the propagation in \htmladdnormallink{space-time}{http://planetphysics.us/encyclopedia/SR.html} of a \htmladdnormallink{wave}{http://planetphysics.us/encyclopedia/CosmologicalConstant2.html} (or periodic \htmladdnormallink{motion}{http://planetphysics.us/encyclopedia/CosmologicalConstant.html}, oscillation, `physical perturbation' or `signal') in terms of certain \htmladdnormallink{types}{http://planetphysics.us/encyclopedia/Bijective.html} of \htmladdnormallink{differential equations}{http://planetphysics.us/encyclopedia/DifferentialEquations.html} (such as partial differential ones); the solutions of such wave equations--usually with additonal \htmladdnormallink{boundary}{http://planetphysics.us/encyclopedia/PiecewiseLinear.html} conditions-- are either propagating or stationary waves; there are numerous types of waves, and thus, there are many different types of \htmladdnormallink{wave equations}{http://planetphysics.us/encyclopedia/TransversalWave.html}. The following is a short list of such wave equations, that is however not intended to be comprehensive.

\subsection{Types of Wave Equations:}

\begin{enumerate}

\item Elastic wave equation and Hook's Law

\item Equation for sound wave propagation

\item Wave equation for \htmladdnormallink{heat}{http://planetphysics.us/encyclopedia/Heat.html} transfer;

\item Laplace wave equation;

\item \htmladdnormallink{Maxwell's equations}{http://planetphysics.us/encyclopedia/FluorescenceCrossCorrelationSpectroscopy.html} for electromagnetic wave propagation;

\item Schr\"odinger 'wave' equation for electrons (see also \htmladdnormallink{Hamiltonian operator}{http://planetphysics.us/encyclopedia/QuantumHamiltonianOperator.html});

\item Heisenberg's quantum \htmladdnormallink{dynamic}{http://planetphysics.us/encyclopedia/MathematicalFoundationsOfQuantumTheories.html} equations (see also \htmladdnormallink{Hamiltonian operator}{http://planetphysics.us/encyclopedia/QuantumHamiltonianOperator.html} and \htmladdnormallink{quantum harmonic oscillator and Lie algebra}{http://planetphysics.us/encyclopedia/QuantumHarmonicOscillatorAndLieAlgebra.html});

\item Dirac relativistic wave equation;

\item \htmladdnormallink{soliton}{http://planetphysics.us/encyclopedia/Soliton.html} wave equations;

\item \htmladdnormallink{spin}{http://planetphysics.us/encyclopedia/QuarkAntiquarkPair.html} wave equations;

\item \htmladdnormallink{Einstein's}{http://planetphysics.us/encyclopedia/AlbertEinstein.html} gravitational wave equations;

\end{enumerate}

\subsection{Examples:}

In its simplest form, the wave equation refers to a \htmladdnormallink{scalar}{http://planetphysics.us/encyclopedia/Vectors.html} \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $w$ that satisfies:

$ \partial^2 (w) \over {\partial t^2}$ = $c^2 \nabla^2 u,$

where $ \nabla^2$ is the \htmladdnormallink{Laplace operator}{http://planetphysics.us/encyclopedia/LaplaceOperator.html}, and where $c$ is a fixed constant equal to the propagation \htmladdnormallink{speed}{http://planetphysics.us/encyclopedia/Velocity.html} of the wave.

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