wave equation WaveEquation 2002-11-21 09:12:58 2008-09-15 20:42:01 Definition d'Alembert's solution to the wave equation partial differential equation % 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 The \emph{wave equation} is a partial differential equation which describes certain kinds of waves. It arises in various physical situations, such as vibrating \PMlinkescapetext{strings}, \PMlinkescapetext{sound} waves, and electromagnetic waves. The wave equation in one \PMlinkescapetext{dimension} is $$ \frac{\partial^2 u}{\partial t^2}=c^2\frac{\partial^2 u}{\partial x^2}. $$ The general solution of the one-dimensional wave equation can be obtained by a change of coordinates: $(x,t)\longrightarrow(\xi,\eta)$, where $\xi=x-ct$ and $\eta=x+ct$. This gives $\frac{\partial^2 u}{\partial\xi\partial\eta}=0$, which we can integrate to get \emph{d'Alembert's solution}: $$ u(x,t)=F(x-ct)+G(x+ct) $$ where $F$ and $G$ are twice differentiable functions. $F$ and $G$ represent waves traveling in the positive and negative $x$ directions, respectively, with velocity $c$. These functions can be obtained if appropriate initial conditions and boundary conditions are given. For example, if $u(x,0)=f(x)$ and $\frac{\partial u}{\partial t}(x,0)=g(x)$ are given, the solution is $$ u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\mathrm d s. $$ In general, the wave equation in $n$ \PMlinkescapetext{dimensions} is $$ \frac{\partial^2 u}{\partial t^2}=c^2\nabla^2 u. $$ where $u$ is a function of the location variables $x_1,x_2,\ldots,x_n$, and time $t$. Here, $\nabla^2$ is the Laplacian with respect to the location variables, which in Cartesian coordinates is given by $ \nabla^2=\frac{\partial^2}{\partial x_1^2}+\frac{\partial^2}{\partial x_2^2}+\cdots+\frac{\partial^2}{\partial x_n^2}$.