wave equationWaveEquation2007-08-10 13:53:152008-10-14 14:47:01Definitionphysical perturbation' or `signal'types of wave equations% this is the default PlanetMath preamble. as your knowledge
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% define commands hereAny \emph{wave equation} describes the propagation in space-time of a wave (or periodic motion, oscillation, `physical pereturbation' or 'signal') in terms of certain types of differential equations (such as partial differential ones); the solutions of such wave equations--usually with additonal boundary conditions-- are either propagating or stationary waves; there are numerous types of waves, and thus, there are many different types of wave equations.
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 heat transfer;
\item Laplace wave equation;
\item Maxwell's equations for electromagnetic wave propagation;
\item Schr\"odinger 'wave' equation for electrons (see also \PMlinkname{Hamiltonian operator}{QuantumHamiltonianOperator});
\item Heisenberg's quantum dynamic equations (see also \PMlinkname{Hamiltonian operator}{QuantumHamiltonianOperator} and \PMlinkname{quantum harmonic oscillator and Lie algebra}{QuantumHarmonicOscillatorAndLieAlgebra});
\item Dirac relativistic wave equation;
\item Soliton wave equations;
\item Spin wave equations;
\item Einstein's gravitational wave equations;
\end{enumerate}
\subsection{Examples:}
In its simplest form, the wave equation refers to a scalar function $w$ that satisfies:
$ \partial^2 (w) \over {\partial t^2}$ = $c^2 \nabla^2 u,$
where $ \nabla^2$ is the Laplace operator, and where $c$ is a fixed constant equal to the propagation speed of the wave.