|
|
|
Viewing Version
5
of
'd'Alembert and D. Bernoulli solutions of wave equation'
|
[ view 'd'Alembert and D. Bernoulli solutions of wave equation'
|
back to history
]
| Title of object: |
d'Alembert and D. Bernoulli solutions of wave equation |
| Canonical Name: |
DAlembertAndDBernoulliSolutionsOfWaveEquation |
| Type: |
Derivation |
| Created on: |
2008-09-15 14:53:34 |
| Modified on: |
2008-09-19 12:05:38 |
| Classification: |
msc:35L05, msc:35L15 |
Revision comment (for changes between this and next version):
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
\theoremstyle{definition}
\newtheorem*{thmplain}{Theorem}
|
Content:
Let's consider the \PMlinkname{d'Alembert's solution}{WaveEquation}
\begin{align}
u(x,\,t) \,:=\, \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds
\end{align}
of the wave equation in one dimension in the special case when the other initial condition is
\begin{align}
u'_t(x,\,0) \,:=\, g(x) \,\equiv\, 0.
\end{align}
We shall see that the solution is equivalent with the solution of D. Bernoulli.\\ \\
We \PMlinkescapetext{expand} the given function $f$ to the Fourier sine series on the interval \,$[0,\,p]$:
$$
f(y) \,=\, \sum_{n=1}^\infty A_n\sin\frac{n\pi y}{p} \quad \mbox{with}\;\;
A_n = \frac{2}{p}\int_0^pf(x)\sin\frac{n\pi x}{p}\,dx \quad (n = 1,\,2,\,\ldots)
$$
Thus we may write
\begin{align*}
\begin{cases}
f(x\!-\!ct) = \sum_{n=1}^\infty A_n\sin\!\left(\frac{n\pi x}{p}-\frac{n\pi ct}{p}\right)=
\sum_{n=1}^\infty A_n\left(\sin\frac{n\pi x}{p}\cos\frac{n\pi ct}{p}-\cos\frac{n\pi x}{p}\sin\frac{n\pi ct}{p}\right),
\\ f(x\!+\!ct) = \sum_{n=1}^\infty A_n\sin\!\left(\frac{n\pi x}{p}+\frac{n\pi ct}{p}\right)=
\sum_{n=1}^\infty A_n\left(\sin\frac{n\pi x}{p}\cos\frac{n\pi ct}{p}+\cos\frac{n\pi x}{p}\sin\frac{n\pi ct}{p}\right).
\end{cases}
\end{align*}
Adding these equations and dividing by 2 yield
\begin{align}
u(x,\,t) = \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)]
= \sum_{n=1}^\infty A_n\cos\frac{n\pi ct}{p}\sin\frac{n\pi x}{p},
\end{align}
which indeed is the \PMlinkname{solution of D. Bernoulli}{SolvingTheWaveEquationByDBernoulli} in the case\, $g(x) \equiv 0$.\\
\textbf{Note.}\, The solution (3) of the wave equation is especially \PMlinkescapetext{simple} in the special case where one has besides (2) the sine-formed initial condition
\begin{align}
u(x,\,0) \,:=\, f(x) \,\equiv\, \sin\frac{\pi x}{p}.
\end{align}
Then \,$A_n = 0$\, for every $n$ except 1, and one obtains
\begin{align}
u(x,\,t) \,= \cos\frac{\pi ct}{p}\sin\frac{\pi x}{p}\,.
\end{align}
|
|
|
|
|
|
