plane monochromatic waves $$\begin{aligned}
\label{eq:theta:sol1}
\theta_{\mu\nu} & = & A_{\mu\nu}\,e^{-ik_{\mu}x^{\mu}},\quad k^\mu = (\omega,{\bf k}), \;\;\; \omega=|{\bf k}|, \quad k_{\mu}A^{\mu\nu}=0  \\
\label{eq:BD:lin:sol1}
\Phi & = & A\,e^{-il_{\mu}x^{\mu}}, \quad \;\;\;\;\; l^\mu = (\omega,{\bf l}), \;\;\;\; \omega=\sqrt{{\bf l}^2+m^2}\end{aligned}$$ which can be written as $$\begin{aligned}
\label{eq:theta:sol2}
\theta_{\mu\nu} & = & A_{\mu\nu}\,e^{i\omega (t - {{\boldsymbol \Omega}}\cdot{\bf x})}, \qquad {{\boldsymbol \Omega}}={\bf k}/|{\bf k}| \\
\label{eq:BD:lin:sol2}
\Phi & = & A\,e^{i\omega (t - \frac{{{\boldsymbol \Omega}}\cdot{\bf x}}{{\rm v}(\omega)})}, \qquad \;\;\; {{\boldsymbol \Omega}}={\bf l}/|{\bf l}|,\end{aligned}$$ where ${{\boldsymbol \Omega}}$’s are unit vectors along the wave propagation and ${\rm v}$ is the $\omega$–dependent phase velocity, ${\rm v}(\omega)=\frac{|\omega|}{\sqrt{\omega^2-m^2}}$, of the scalar field. (The phase velocity diverges when $\omega$ tends to $m$. This is because the wavelength $\lambda=2\pi/\sqrt{\omega^2-m^2}$ grows then to infinity; in the limit $\omega=m$ the solutions of Eqs. (\[eq:theta:sol2\]), (\[eq:BD:lin:sol2\]) are therefore proportional to space–independent oscillations $e^{imt}$.) At this step we have $h_{\mu\nu}=\theta_{\mu\nu}-\eta_{\mu\nu}(\frac12\,\theta-\Phi)$, where $\theta=\eta^{\mu\nu}\theta_{\mu\nu}$ but the Lorentz condition (\[eq:glor\]) is preserved under the supplementary gauge transformation (\[eq:gfreedom\]) with $\zeta_{\mu}$ satisfying $$\begin{aligned}
\Box_{\eta}\zeta_{\mu} & = & 0, \\
\zeta^{\mu}{}_{,\mu} & = & -\frac12 \theta\end{aligned}$$ rendering the trace of $\theta_{\mu\nu}$ equal to zero and giving $h_{\mu\nu}=\theta_{\mu\nu}+\Phi\,\eta_{\mu\nu}$. The residual gauge freedom $$\begin{aligned}
\Box_{\eta}\zeta_{\mu} & = & 0, \\
\label{eq:tr0}
\zeta^{\mu}{}_{,\mu} & = & 