Wave reflection and refraction at the interface between a normal media and a one-dimensional left-handed material (1 DLHM) with a hyperbolic dispersion relationship is studied. It is found that in a special case that the boundary is perpendicular to one asymptotic line of the hyperbola, phase matching cannot be achieved unless the 1 DLHM is regarded to be intrinsically lossy. After introducing a small loss factor to the 1 DLHM, a reasonable solution for the phase matching is obtained. According to the analytical result, a wave confined to a thin layer near the boundary is found, which can be excited at the interface as a reflected wave or a refracted wave attenuating drastically away from the boundary inside the 1 DLHM in both cases.