  This work is concerned with the study of a scalar delay differential equation
\begin{equation*} z^{\prime\prime}(t)=h^2\,V(z(t-1)-z(t))+h\,z^\prime(t)
\end{equation*} motivated by a simple car-following model on an unbounded
straight line. Here, the positive real $h$ denotes some parameter, and $V$ is a
so-called \textit{optimal velocity function} of the traffic model involved. We
analyze the existence and local stability properties of solutions
$z(t)=c\,t+d$, $t\in\mathbb{R}$, with $c,d\in\mathbb{R}$. In the case
$c\not=0$, such a solution of the differential equation forms a wavefront
solution of the car-following model where all cars are uniformly spaced on the
line and move with the same constant velocity. In particular, it is shown that
all but one of these wavefront solutions are located on two branches
parametrized by $h$. Furthermore, we prove that along the one branch all
solutions are unstable due to the principle of linearized instability, whereas
along the other branch some of the solutions may be stable. The last point is
done by carrying out a center manifold reduction as the linearization does
always have a zero eigenvalue. Finally, we provide some numerical examples
demonstrating the obtained analytical results.
