  We study the second order hyperbolic equations with initial conditions, a
nonhomogeneous Dirichlet boundary condition and a source term. We prove the
solution possesses $H^1$ regularity on any piecewise $C^1$-smooth non-timelike
hypersurfaces. We generalize the notion of energy to these hypersurfaces, and
establish an estimate of the difference between the energies on the
hypersurface and on the initial plane where the time $t = 0$. The energy is
shown to be conserved when the source term and the boundary datum are both
zero. We also obtain an $L^2$ estimate for the normal derivative of the
solution. In the proofs we first show these results for $C^2$-smooth solutions
by using the multiplier methods, and then we go back to the original results by
approximation.
