  The teleparallel equivalent of higher order Lagrangians like $L_{\Box
R}=-R+a_{0}R^{2}+a_{1}R\Box R$ can be obtained by means of the boundary term
$B=2\nabla_{\mu}T^{\mu}$. In this perspective, we derive the field equations in
presence of matter for higher-order teleparallel gravity considering, in
particular, sixth-order theories where the $\Box$ operator is linearly
included. In the weak field approximation, gravitational wave solutions for
these theories are derived. Three states of polarization are found: the two
standard $+$ and $\times$ polarizations, namely 2-helicity massless transverse
tensor polarizations, and a 0-helicity massive, with partly transverse and
partly longitudinal scalar polarization. Moreover, these gravitational waves
exhibit four oscillation modes related to four degrees of freedom: the two
classical $+$ and $\times$ tensor modes of frequency $\omega_{1}$, related to
the standard Einstein waves with $k^{2}_{1}=0$; two mixed
longitudinal-transverse scalar modes for each frequencies $\omega_{2}$ and
$\omega_{3}$, related to two different 4-wave vectors, $k^{2}_{2}=M_{2}^{2}$
and $k^{2}_{3}=M^{2}_{3}$. The four degrees of freedom are the amplitudes of
each individual mode, i.e. $\hat{\epsilon}^{(+)}\left(\omega_{1}\right)$,
$\hat{\epsilon}^{(\times)}\left(\omega_{1}\right)$,
$\hat{B}_{2}\left(\mathbf{k}\right)$, and $\hat{B}_{3}\left(\mathbf{k}\right)$.
