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

Let's consider the [[../WaveEquation/|d'Alembert's solution]]

$\begin{array}{c}u(x,t):=\frac{1}{2}[f(x-ct)+f(x+ct)]+\frac{1}{2c}\underset{x-ct}{\overset{x+ct}{\int }}g(s)ds\end{array}$

of the [[../WaveEquation/|wave equation]] in one dimension in the special case when the other initial condition is

$\begin{array}{c}u{\text{'}}_t(x,0):=g(x)\equiv 0.\end{array}$

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

We expand the given [[../Bijective/|function]] $f$ to the Fourier sine series on the interval \,$[0,p]$: $f(y)=\sum _{n=1}^{\mathrm{\infty }}{A}_n\sin \frac{n\pi y}{p}\text{with}{A}_n=\frac{2}{p}\underset{0}{\overset{p}{\int }}f(x)\sin \frac{n\pi x}{p}dx(n=1,2,\dots )$ Thus we may write

$\begin{array}{c}\{\begin{array}{c}f(x-ct)=\sum _{n=1}^{\mathrm{\infty }}{A}_n\sin (\frac{n\pi x}{p}-\frac{n\pi ct}{p})=\sum _{n=1}^{\mathrm{\infty }}{A}_n(\sin \frac{n\pi x}{p}\cos \frac{n\pi ct}{p}-\cos \frac{n\pi x}{p}\sin \frac{n\pi ct}{p}),\\ f(x+ct)=\sum _{n=1}^{\mathrm{\infty }}{A}_n\sin (\frac{n\pi x}{p}+\frac{n\pi ct}{p})=\sum _{n=1}^{\mathrm{\infty }}{A}_n(\sin \frac{n\pi x}{p}\cos \frac{n\pi ct}{p}+\cos \frac{n\pi x}{p}\sin \frac{n\pi ct}{p}).\end{array}\end{array}$

Adding these equations and dividing by 2 yield

$\begin{array}{c}u(x,t)=\frac{1}{2}[f(x-ct)+f(x+ct)]=\sum _{n=1}^{\mathrm{\infty }}{A}_n\cos \frac{n\pi ct}{p}\sin \frac{n\pi x}{p},\end{array}$

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

$\begin{array}{c}u(x,0):=f(x)\equiv \sin \frac{\pi x}{p}.\end{array}$

Then \,$A_n=0$\, for every $n$ except 1, and one obtains

$\begin{array}{c}u(x,t)=\cos \frac{\pi ct}{p}\sin \frac{\pi x}{p}.\end{array}$

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