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[parent] d'Alembert and D. Bernoulli solutions of wave equation (Topic)

Let's consider the d'Alembert's solution

$\displaystyle u(x,\,t) \,:=\, \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds$ (1)
of the wave equation in one dimension in the special case when the other initial condition is
$\displaystyle u'_t(x,\,0) \,:=\, g(x) \,\equiv\, 0.$ (2)
We shall see that the solution is equivalent with the solution of D. Bernoulli.

We expand the given function $f$ to the Fourier sine series on the interval  $[0,\,p]$:

$\displaystyle f(y) \,=\, \sum_{n=1}^\infty A_n\sin\frac{n\pi y}{p}$   with$\displaystyle \;\; 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... ...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
$\displaystyle u(x,\,t) = \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)] = \sum_{n=1}^\infty A_n\cos\frac{n\pi ct}{p}\sin\frac{n\pi x}{p},$ (3)
which indeed is the solution of D. Bernoulli in the case  $g(x) \equiv 0$.

Note.  The solution (3) of the wave equation is especially simple in the special case where one has besides (2) the sine-formed initial condition

$\displaystyle u(x,\,0) \,:=\, f(x) \,\equiv\, \sin\frac{\pi x}{p}.$ (4)
Then  $A_n = 0$  for every $n$ except 1, and one obtains
$\displaystyle u(x,\,t) \,= \cos\frac{\pi ct}{p}\sin\frac{\pi x}{p}\,.$ (5)

Remark.  In the case of quantum systems one has Schrödinger's wave equation whose solutions are different from the above.



"d'Alembert and D. Bernoulli solutions of wave equation" is owned by pahio.

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Keywords:  Fourier series

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Cross-references: systems, function, wave equation

This is version 1 of d'Alembert and D. Bernoulli solutions of wave equation, born on 2009-04-18.
Object id is 661, canonical name is DAlembertAndDBernoulliSolutionsOfWaveEquation.
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Physics Classification: 02.30.Jr (Partial differential equations)
 03.75.-b (Matter waves )
 41.20.Jb (Electromagnetic wave propagation; radiowave propagation )
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