d'Alembert and D. Bernoulli solutions of wave equationDAlembertAndDBernoulliSolutionsOfWaveEquation2008-09-15 14:53:342008-10-20 06:16:23Derivationwave equation% this is the default PlanetMath preamble. as your knowledge
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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}
\textbf{Remark.}\,
In the case of quantum systems one has \PMlinkname{Schr\"odinger's wave equation}{SchrodingersWaveEquation}
whose solutions are different from the above.