The study of oscillating norm behavior in linear systems is addressed. The analysis is carried out by considering a model equation that naturally arises in the context of the linearized formulation of some convection-dominated systems over finite length domains. The occurrence of oscillating patterns in the energy evolution of the solutions is linked to the structure of the spectrum and to the non-normal character of the linear operator involved. A distinction between transient and asymptotic oscillations is made and some relations between the frequency signatures shown in both cases and global characteristics of the spectrum are highlighted. The physical importance of such oscillating behaviors is stressed by reconsidering a model for the linear stability of a falling liquid curtain studied by other authors in a previous work.