In this paper, by means of Carleman estimates and the usual energy estimate, we obtain directly two observability inequalities for the linear wave equation with time-variant nonsmooth lower order terms. We do not need any unique continuation property of the linear equation a priori, since this is actually one of the by-products of our analysis. Furthermore, the constant in the observability inequality is estimated by an explicit function of the norm of the involved coefficients in the equation. Also, we apply our observability estimates to exact controllability for wave equations.