We investigate the existence of multiple nontrivial solutions $u(x,t)$ for a perturbation $b[({xi}-{eta}+2)^+-2]$ of the hyperbolic system with Dirichlet boundary condition $$(0.1);L{xi}={mu}[({xi}-{eta}+2)^+-2];in;({-{frac{{pi}}{2}}},{frac{{pi}}{2}}){ imes}mathbb{R},\L{eta}={ u}[({xi}-{eta}+2)^+-2];in;({-{frac{{pi}}{2}}},{frac{{pi}}{2}}){ imes}mathbb{R},$$, where $u^+$=max{u,o}, ${mu}$, ${ u}$ are nonzero constants. Here L is the wave operator in $mathbb{R}^2$ and the nonlinearity $({mu}-{ u})[({xi}-{eta}+2)^+-2]$ crosses the eigenvalues of the wave operator.