We study the spatiotemporal evolution of small disturbances developing in an unstable dispersive system. At the initial linear stage, the slow modulation of an unstable harmonic is analyzed subject to prescribed initial and boundary conditions. A theoretical model based on a modified linearized Ginzburg–Landau equation predicts the emergence of effectively localized envelope structures that can strongly affect the subsequent nonlinear stage of evolution and provides a simple criterion for the onset of absolute instability. As an illustrative example, we consider wind waves generated by an impulsive forcing in a semi-infinite domain over a viscous shear flow.