I would like to derive an analytic solution for the wave equation with periodic boundary condition. All derivations that I encountered are for infinite space domain or for Dirichlet boundary condition. I'm wondering if a modification for d'alembert formula works or not.
2026-03-25 21:47:27.1774475247
Analytical solution to the 1D wave equation with perodic boundary condition
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Have you thought of using spatial Fourier series? Using the spatial periodicity hypothesis, we may expand $$u(x,t) =\sum_n a_n(t)\cos (nkx) + b_n(t)\sin (nkx)$$ as a spatial Fourier series with period $2\pi/k$. The PDE $u_{tt}=u_{xx}$ leads to a set of second-order ODEs for $a_n$, $b_n$, while the initial conditions of the PDE give the initial conditions $a_n(0)$, $b_n(0)$, $a'_n(0)$, $b'_n(0)$ of the ODEs.
Alternatively, using the method of characteristics, one may be able to develop a modified d'Alembert formula (see chap. 12-* of the book (1)).
(1) R. Habermann, Applied Partial Differential Equations; with Fourier Series and Boundary Value Problems, 5th ed., Pearson Education Inc., 2013.