In this paper, we study the periodic solutions to two-dimensional wave equation ytt−Δy+ay+g(y)=f(x,t) on (0,π)×(0,π)×R under the periodic conditions y(x,t+T)=y(x,t), yt(x,t+T)=yt(x,t), and the Sturm–Liouville boundary conditions a11y(0,x2,t)−b11yx1(0,x2,t)=0, a12y(π,x2,t)+b12yx1(π,x2,t)=0, a21y(x1,0,t)−b21yx2(x1,0,t)=0, and a22y(x1,π,t)+b22yx2(x1,π,t)=0 (aij2+bij2≠0 for i,j=1,2). Our main concept is that of the “weak solution,” which will be given in Sec. II. Then, by means of spectral analysis, we investigate some important properties of the wave operator with Sturm–Liouville boundary conditions. Finally, by using the properties we established and the duality principles in the work of Brézis [Bull. Am. Math. Soc. 8, 409 (1983)], the existence of periodic solutions is obtained.