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

Let's consider the d'Alembert's solution $$\begin{matrix} u(x,\,t) \,:=\, \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)]+\frac{1}{2c}\int_{x-ct}^{x+ct}g(s)\,ds \end{matrix}$$ of the wave equation in one dimension in the special case when the other initial condition is $$\begin{matrix} u'_t(x,\,0) \,:=\, g(x) \,\equiv\, 0. \end{matrix}$$ 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 \,$$[0,\,p]$$: $$ f(y) \,=\, \sum_{n=1}^\infty A_n\sin\frac{n\pi y}{p} \quad \mbox{with}\;\; A_n = \frac{2}{p}\int_0^pf(x)\sin\frac{n\pi x}{p}\,dx \quad (n = 1,\,2,\,\ldots) $$ Thus we may write $$\begin{matrix} \begin{cases} f(x\!-\!ct) = \sum_{n=1}^\infty A_n\sin\!\left(\frac{n\pi x}{p}-\frac{n\pi ct}{p}\right)= \sum_{n=1}^\infty A_n\left(\sin\frac{n\pi x}{p}\cos\frac{n\pi ct}{p}-\cos\frac{n\pi x}{p}\sin\frac{n\pi ct}{p}\right), \\ f(x\!+\!ct) = \sum_{n=1}^\infty A_n\sin\!\left(\frac{n\pi x}{p}+\frac{n\pi ct}{p}\right)= \sum_{n=1}^\infty A_n\left(\sin\frac{n\pi x}{p}\cos\frac{n\pi ct}{p}+\cos\frac{n\pi x}{p}\sin\frac{n\pi ct}{p}\right). \end{cases} \end{matrix}$$ Adding these equations and dividing by 2 yield $$\begin{matrix} u(x,\,t) = \frac{1}{2}[f(x\!-\!ct)+f(x\!+\!ct)] = \sum_{n=1}^\infty A_n\cos\frac{n\pi ct}{p}\sin\frac{n\pi x}{p}, \end{matrix}$$ 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 $$\begin{matrix} u(x,\,0) \,:=\, f(x) \,\equiv\, \sin\frac{\pi x}{p}. \end{matrix}$$ Then \,$$A_n = 0$$\, for every $$n$$ except 1, and one obtains $$\begin{matrix} u(x,\,t) \,= \cos\frac{\pi ct}{p}\sin\frac{\pi x}{p}\,. \end{matrix}$$

Remark. \, In the case of quantum systems one has Schr\"odinger's wave equation whose solutions are different from the above.