Neumann cosine modes of a one-dimensional wave equation ($\psi (x,t)=A_0+B_{0}t+{\sum }_{n\ge 1}[A_n\cos (cn\pi t/L)+B_n\sin (cn\pi t/L)]\cos (n\pi x/L)$)

For the wave equation ${\psi }_{tt}=c^2{\psi }_{xx}$ on $0<x<L$, impose homogeneous Neumann boundary conditions ${\psi }_x(0,t)={\psi }_x(L,t)=0$, where $c,L>0$. Separation of variables gives $X^{\prime \prime }+\lambda X=0$ and $T^{\prime \prime }+c^2\lambda T=0$. Integration by parts gives ${\int }_0^L(X^{\prime }{)}^{2}dx=\lambda {\int }_0^L{X}^{2}dx$, so every nontrivial spatial eigenfunction has $\lambda \ge 0$.

For $\lambda =0$, $X$ is constant and $T=A_0+B_{0}t$. For $\lambda >0$, the boundary conditions select ${\lambda }_n=(n\pi /L{)}^2$ and $X_n=\cos (n\pi x/L)$. The superposition principle gives the displayed expansion when the initial data have sufficient regularity. The constant zero mode must be retained: homogeneous Neumann boundary conditions do not force the spatial average of the displacement to vanish.

With zero initial velocity, all $B_n$ vanish, including $B_0$. Initial displacement $a+b\cos (n\pi x/L)$ then evolves as $a+b\cos (cn\pi t/L)\cos (n\pi x/L)$, a standing wave above a constant offset.