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| Title of object: |
wave equation |
| Canonical Name: |
WaveEquation |
| Type: |
Definition |
| Created on: |
2002-11-21 09:12:58 |
| Modified on: |
2007-06-26 12:43:15 |
| Classification: |
msc:35L05 |
| Keywords: |
partial differential equation |
| Defines: |
d'Alembert's solution to the wave equation |
Preamble:
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Content:
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 variables $(x,t)\longrightarrow(\xi,\eta)$,
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 travelling in the positive and negative $x$
directions, respectively, with velocity $c$. These functions can be
obtained if appropriate starting or 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}$. |
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