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Fourier Method for $1$D Wave equation-Visualization

Dirichlet-Neumann

For boundary value problem $$\begin{aligned} &u_{tt}-c^2u_{xx}=0,\\ &u|_{x=0}=u_x|_{x=l}=0 \end{aligned}$$ there are simple solutions in the form $$u(x,t)= \cos(\frac{(n-\frac{1}{2})\pi ct}{l})\cos(\frac{(n-\frac{1}{2})\pi x}{l})$$

and general solutions of in the form $$u(x,t)= \sum_{n=1}^{\infty}\bigl[A_n\cos(\frac{(n-\frac{1}{2})\pi ct}{l})+B_n\sin (\frac{(n-\frac{1}{2})\pi ct}{l})\bigr]\sin(\frac{(n-\frac{1}{2})\pi x}{l})$$