# d’Alembert and D. Bernoulli solutions of wave equation

Let’s consider the d’Alembert’s solution (http://planetmath.org/WaveEquation)

 $\displaystyle u(x,\,t)\,:=\,\frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)]+\frac{1}{2c}% \int_{x-ct}^{x+ct}g(s)\,ds$ (1)

of the wave equation in one dimension in the special case when the other initial condition is

 $\displaystyle u^{\prime}_{t}(x,\,0)\,:=\,g(x)\,\equiv\,0.$ (2)

We shall see that the solution is equivalent with the solution of D. Bernoulli.

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

 $f(y)\,=\,\sum_{n=1}^{\infty}A_{n}\sin\frac{n\pi y}{p}\quad\mbox{with}\;\;A_{n}% =\frac{2}{p}\int_{0}^{p}f(x)\sin\frac{n\pi x}{p}\,dx\quad(n=1,\,2,\,\ldots)$

Thus we may write

 $\displaystyle\begin{cases}f(x\!-\!ct)=\sum_{n=1}^{\infty}A_{n}\sin\!\left(% \frac{n\pi x}{p}-\frac{n\pi ct}{p}\right)=\sum_{n=1}^{\infty}A_{n}\left(\sin% \frac{n\pi x}{p}\cos\frac{n\pi ct}{p}-\cos\frac{n\pi x}{p}\sin\frac{n\pi ct}{p% }\right),\\ f(x\!+\!ct)=\sum_{n=1}^{\infty}A_{n}\sin\!\left(\frac{n\pi x}{p}+\frac{n\pi ct% }{p}\right)=\sum_{n=1}^{\infty}A_{n}\left(\sin\frac{n\pi x}{p}\cos\frac{n\pi ct% }{p}+\cos\frac{n\pi x}{p}\sin\frac{n\pi ct}{p}\right).\end{cases}$

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 (http://planetmath.org/SolvingTheWaveEquationByDBernoulli) in the case  $g(x)\equiv 0$.

Note.  The solution (3) of the wave equation is especially 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 (http://planetmath.org/SchrodingersWaveEquation) whose solutions are different from the above.

Title d’Alembert and D. Bernoulli solutions of wave equation DAlembertAndDBernoulliSolutionsOfWaveEquation 2013-03-22 18:23:15 2013-03-22 18:23:15 pahio (2872) pahio (2872) 11 pahio (2872) Derivation msc 35L15 msc 35L05 AdditionFormulasForSineAndCosine SchrodingersWaveEquation