Summary: We consider hyperbolic conservation laws with relaxation terms. By studying the dispersion relation of the solution of general linearized \(2\times 2\) hyperbolic relaxation systems, we investigate in detail the transition between the wave dynamics of the homogeneous relaxation system and that of the local equilibrium approximation. We establish that the wave velocities of the Fourier components of the solution to the relaxation system will be monotonic functions of a stiffness parameter \(\phi = \varepsilon\xi\), where \(\varepsilon\) is the relaxation time and \(\xi\) is the wave number. This allows us to extend in a natural way the classical concept of the sub-characteristic condition into a more general transitional sub-characteristic condition. We further identify two parameters \(\beta\) and \(\gamma\) that characterize the behavior of such general \(2\times 2\) linear relaxation systems. In particular, these parameters define a natural transition point, representing a value of \(\phi\) where the dynamics will change abruptly from being equilibrium-like to behaving more like the homogeneous relaxation system. Herein, the parameter \(\gamma\) determines the location of the transition point, whereas \(\beta\) measures the degree of smoothness of this transition.