When deriving the equation describing the transverse motion of a one-dimensional vibrating elastic string, introductory physics textbooks often assume constant tension and/or small amplitude vibrations. However, these simplifying assumptions are not only unnecessary, but they overlook the elastic nature of the tension and yield an inconsistent derivation of the potential energy density. Because of these assumptions, the derivation of the wave equation and the potential energy density use two different levels of mathematical approximation. In addition, students often get confused as of how a string can carry elastic potential energy if the underlying assumption is that the tension is constant. In this work, we present a mathematically consistent derivation of the wave equation and potential energy density for the vibrating string. Our approach is adequate for physics and engineering introductory courses. We emphasize throughout the derivations the role of elasticity and we propose a simple experiment where students can use wave theory to predict the elastic properties of strings. We also use our framework to illustrate under which conditions longitudinal waves can be neglected for strings that obey Hooke's law of elasticity. We show that a small transverse amplitude vibration does not immediately justify neglecting longitudinal motion.