The stress-temperature variational principle of thermoelasticity with two relaxation times is used to obtain an approximate solution of an one-dimensional initial-boundary value problem for a half-space. It is assumed that initial conditions are homogeneous and a thermoelastic disturbance propagates throughout the half-space due to a smooth heating of its boundary. The variational principle is replaced by an approximate formulation in which a space finite element method is combined with a time approximation due to Gurtin. Numerical examples are given to show the influence of relaxation times on the propagating thermoelastic wave.