In the article, the traveling wave solutions to a hydrodynamic model for relaxing media are considered. When fluctuations in the model are absent these solutions satisfy the planar dynamical system, which has been studied by means of the qualitative analysis methods. Now we are interested in the structure of the wave solutions, namely the stationary and periodic regimes, when the model parameters are disturbed by Gaussian noise. In this case, the characteristics of fixed points and limit cycles of the corresponding stochastic dynamical system are studied. In particular, we analyze the stability of the fixed point with the help of the top Lyapunov exponent and found out the displacement of the Andonov-Hopf bifurcation with respect to the noise intensity and the wave velocity. Stochastic limit cycles are studied by means of the sensitivity function determining the dispersion of trajectories in the vicinity of the deterministic limit cycle. This function is derived from the deterministic differential equation by the shooting method. It is shown that trajectories nearby the limit cycle undergo the most dispersion in the vicinity of the saddle fixed point.