the two gauges by explicitly giving the gauge transformation.

Greek indices $\mu,\nu,\ldots$ run from $0$ to $3$, Latin indices denote spatial coordinates: $i,j,k,\ldots = 1,2,3$; $(x^0,x^1,x^2,x^3)$ are denoted also as $(t,x,y,z)$; colon “$:$” denotes contraction of tensors.

Massive Brans-Dicke gravity {#s:1}
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In this section we give a brief account of the theory in which the gravitational interaction is mediated by two fields, the standard metric tensor field and the massive scalar field. We rederive solutions of the linearized field equations for the general gravitational wave comprising two massless tensor modes and one massive scalar mode.

We recall that a class of massive scalar–tensor theories described by the action $$\begin{aligned}
\label{eq:BDaction}
S[g_{\mu\nu},\phi,\psi_m] & = & S_{g}[g_{\mu\nu},\phi] + S_{m}[g_{\mu\nu},\psi_{\rm m}], \\
S_{g}[g_{\mu\nu},\phi] & = & \frac{1}{16\pi}\int d^4x\,\sqrt{-g}\left[ \phi\,R -
\frac{\omega(\phi)}{\phi}\partial^{\mu}\phi\partial_{\mu}\phi + V(\phi) \right], \\
S_{m}[g_{\mu\nu},\psi_{\rm m}] & = & \int d^4x\,\sqrt{-g}L_{\rm m}[\psi_{\rm m},g_{\mu\nu}],\end{aligned}$$ where $g_{\mu\nu}$ is a metric field, $\phi$ is a scalar field and $\psi_{\rm m}$ denotes collection of matter fields were introduced in [@Berg68], [@Wag70]. Here $\omega$ and $V$ are two coupling functions and $R$ is the scalar curvature of the metric. The effects of the function $\omega$ on the dynamics of compact binaries have been extensively studied in 