have defined $v(\lambda) = {\rm t}_{0}(\lambda) - {{\boldsymbol \Omega}}\cdot{\rm \bf x}_{0}(\lambda)/{\rm v}$.

Comparison of Eqs.(\[eq:tof\_T2a\]) and (\[eq:DT3a\]) suggests that the scalar wave defined in the conformal gauge corresponds to a superposition of scalar transversal and scalar longitudinal waves defined in the synchronous gauge (and in the frame in which the wave propagates in the $-z$ direction) as: $$\begin{aligned}
\label{eq:synch}
ds^2 = -dt^2 + h_{st}\left(t + \frac{z}{{\rm v}}\right)(dx^2 + dy^2) + h_{sl}\left(t + \frac{z}{{\rm v}}\right)dz^2,\end{aligned}$$ with $$\begin{aligned}
\label{eq:hsl:hst:rel}
h_{st} = h_{s},\qquad h_{sl}=\left(1-\frac{1}{{\rm v}^2}\right)h_{s}\end{aligned}$$ This is indeed the case and in the Appendix \[A3\] we give the gauge transformation that connects both gauges. There it is shown that more general result holds: given a scalar mode $h_{s}(t,z)$ that in the conformal gauge satisfies the field equation $\partial_{t}^2h_{s} = F[\partial_{z}]h_{s}$ the corresponding synchronous gauge wave is given by $$\begin{aligned}
\label{eq:gen}
{\rm \bf h}(t,z) & = &
\text{diag}\left(0, h_s(t,z), h_s(t,z), \frac{F\left[\partial_{z}\right] - \partial_{z}^2}{F\left[\partial_{z}\right]}h_{s}(t,z)\right).\end{aligned}$$ Beside the massive Brans–Dicke and $f(R)$ theories this result includes also theories with modified dispersion relation [@MiYuW11] that can arise e.g. as the effective level of some approaches to quantum gravity or in Lorentz–symmetry violating theories [@GAC], [@HL09], [@Jac07]. We note here that usually the effects of modification of the dispersion 