Motivated by topologically protected states in wave physics, we study localized eigenmodes in one-dimensional periodic media with defects. The Su–Schrieffer–Heeger model (the canonical example of a one-dimensional system with topologically protected localized defect states) is used to demonstrate the method. Our approach can be used to describe two broad classes of perturbations to periodic differential problems: those caused by inserting a finite-sized piece of arbitrary material and those caused by creating an interface between two different periodic media. The results presented here characterize the existence of localized eigenmodes in each case and, when they exist, determine their eigenfrequencies and provide concise analytic results that quantify the decay rate of these modes. These results are obtained using both high-frequency homogenization and transfer matrix analysis, with good agreement between the two methods.