Consider the initial-boundary value problem for the nonlinear wave equation $$\begin{cases} u_{tt}-\mu(t)u_{xx}+K|u|^{p-2}u+\lambda|u_t|^{q-2}u_t=F(x,t), \,\, 0 < x < 1, \,\, 0 < t < T,\\ \mu(t)u_x(0,t)=K_0u(0,t)+\int_0^tk(t-s)u(0,s)ds+g(t),\\ -\mu(t)u_x(1,t)=K_1u(1,t)+\lambda_1|u_t(1,t)|^{\alpha-2}u_t(1,t),\\ u(x,0)=\tilde{u}_0(x),\,\, u_t(x,0)=\tilde{u}_1(x),\end{cases}$$ where $p,q,\alpha \geq 2$; $K_0, K_1, K \geq 0$; $\lambda, \lambda_1 > 0$ are given constants and $\mu, F, g, k, \tilde{u}_0, \tilde{u}_1$ are given functions. First, the existence and uniqueness of a weak solution are proved by using the Galerkin method. Next, with $\alpha =2$, we obtain an asymptotic expansion of the solution up to order $N$ in two small parameters $\lambda, \lambda_1$ with error $(\sqrt{\lambda^2+\lambda_1^2})^{N+\frac 12}$.