We present a minimal one-dimensional continuum model for the transition from cracklike to pulselike propagation of frictional rupture. In its nondimensional form, the model depends on only two free parameters: the nondimensional prestress and an elasticity ratio that accounts for the finite height of the system. The model predicts stable slip pulse solutions for slip boundary conditions, and unstable slip pulse solutions for stress boundary conditions. The results demonstrate that a mechanism based solely on elastic relaxation and redistribution of initial prestress can cause pulselike rupture, without any particular rate or slip dependences of dynamic friction. This means that pulselike propagation along frictional interfaces is likely a generic feature that can occur in systems of finite thickness over a wide range of friction constitutive laws.