The subject of thermo-electroelasticity involves many complications due to the multiple ways in which the mechanical, thermal and electric fields can interact, some of these involving non-linearities. In extended thermodynamics, an additional difficulty arises due to the requirement of finiteness of the speed of propagation of the thermal disturbances. This implies, as may be observed in the extensive literature on the subject, a re-visiting of the basic postulates of thermodynamics, ultimately leading to the desired generalization. There are only a few nonlinear models dealing with this subject. In order to consider general nonlinear models, it is necessary to study linear ones first, as they represent most of the basic features of the studied phenomena. This is particularly true when the problem is tackled numerically through iteration methods, in which case the starting field equations are linear. Here we study a one-dimensional system of equations of thermo-electroelasticity in extended thermodynamics and in the quasi-electrostatic regime. The nonlinear equations are given for reference only. The mixed character, parabolic-hyperbolic, of the associated linear system is established through the study of the characteristic curves. Two speeds of wave propagation are given in evidence, one for the usual coupled thermoelastic wave, and the other for a second sound. Parabolicity is due to the quasi-static distribution of the electric field. An example concerning the half-space is treated numerically by the Cranck-Nicolson method. The curves presented clearly illustrate the propagation of two types of waves, the usual coupled thermoelastic wave, and a faster wave generated by the second sound. It is hoped that the present study will clarify the basic features of the solution, as a prelude to tackling more sample, nonlinear equations.