We show the existence of the unique solution of the following system of the nonlinear wave equations with Dirichlet boundary conditions and periodic conditions under some conditions $U_{tt}-U_{xx}+av^+=s{phi}_{00}+f$ in $(-{frac{pi}{2},{frac{pi}{2}}){ imes}R$, ${upsilon}_{tt}-{upsilon}_{xx}+bu^+=t{phi}_{00}+g$ in $(-{frac{pi}{2},{frac{pi}{2}}){ imes}R$, where $u^+$ = max{u, 0}, s, t ${in}$ R, ${phi}_{00}$ is the eigenfunction corresponding to the positive eigenvalue ${lambda}_{00}$ of the wave operator. We first show that the system has a positive solution or a negative solution depending on the sand t, and then prove the uniqueness theorem by the contraction mapping principle on the Banach space.