Reducibility for wave equations of finitely smooth potential with periodic boundary conditions

In the present paper, the reducibility is derived for the wave equations with finitely smooth and time-quasi-periodic potential subject to periodic boundary conditions. More exactly, the linear wave equation $u_{tt}?u_{xx}+Mu+\epsilon (V_0(\omega t)u_{xx}+V(\omega t,x)u)=0,x\in \mathbb{R}/2\pi \mathbb{Z}$ can be reduced to a linear Hamiltonian system with a constant coefficient operator which is of pure imaginary point spectrum set, where V is finitely smooth in $(t,x)$, quasi-periodic in time t with Diophantine frequency $\omega \in {\mathbb{R}}^n$, and $V_0$ is finitely smooth and quasi-periodic in time t with Diophantine frequency $\omega \in {\mathbb{R}}^n$. Moreover, it is proved that the corresponding wave operator possesses the property of pure point spectra and zero Lyapunov exponent.