Talk:PlanetPhysics/D'Alembert and D. Bernoulli Solutions of Wave Equation

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%%% This file is part of PlanetPhysics snapshot of 2011-09-01 %%% Primary Title: d'Alembert and D. Bernoulli solutions of wave equation %%% Primary Category Code: 02.30.Jr %%% Filename: DAlembertAndDBernoulliSolutionsOfWaveEquation.tex %%% Version: 1 %%% Owner: pahio %%% Author(s): pahio %%% PlanetPhysics is released under the GNU Free Documentation License. %%% You should have received a file called fdl.txt along with this file. %%% If not, please write to gnu@gnu.org. \documentclass[12pt]{article} \pagestyle{empty} \setlength{\paperwidth}{8.5in} \setlength{\paperheight}{11in}

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Let's consider the \htmladdnormallink{d'Alembert's solution}{http://planetphysics.us/encyclopedia/WaveEquation.html} \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 \htmladdnormallink{wave equation}{http://planetphysics.us/encyclopedia/WaveEquation.html} 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 expand the given \htmladdnormallink{function}{http://planetphysics.us/encyclopedia/Bijective.html} $f$ 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 solution of D. Bernoulli in the case\, $g(x) \equiv 0$.\\

\textbf{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 \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 \htmladdnormallink{systems}{http://planetphysics.us/encyclopedia/SimilarityAndAnalogousSystemsDynamicAdjointnessAndTopologicalEquivalence.html} one has Schr\"odinger's wave equation whose solutions are different from the above.

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