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-variantnonsmooth lower order terms. We do not need any unique continuation property of the linearequation a priori, since this is actually one of the by-products of our analysis. Furthermore, the constantin the observability inequality is estimated by an explicit function of the norm of the involvedcoefficients in the equation. Also, we apply our observability estimates to exact controllability forwave equations.