We consider the fourth-order wave equation
\begin{equation}
\varphi_{tt} + \varphi_{xxxx}+f(\varphi)=0,~~~
(x,t)\in {R} \times {R}, \nonumber
\end{equation}
with a nonlinearity $f$ vanishing at $0$.
Traveling waves $\varphi(x,t)=u(x-ct)$ satisfy the ODE
\begin{equation}
u^{\prime \prime \prime \prime}+c^2 u^{\prime \prime}
+f(u)=0~~~{\rm on}~{R} , \nonumber
\end{equation}
and for the case $f(u)=e^u-1$,
the existence of at least 36 solitary travelling waves was proved
in [1] by computer assisted means.
We investigate the orbital stability of these solutions via computation of their Morse indices and using results from [2] and [3].
In order to achieve our results we make use of both analytical and computer-assisted techniques.