Curvature of a diagonal plane wave in Rosen coordinates ($\mathrm{Ric}=-(A^{\prime \prime }/A+B^{\prime \prime }/B)du\otimes du$)

For $g=-d{t}^2+A(u{)}^{2}d{x}^2+B(u{)}^{2}d{y}^2+d{z}^2$, $u=t-z$, with nowhere-zero smooth $A,B$, take $e^0=dt$, $e^1=Adx$, $e^2=Bdy$, $e^3=dz$, $\ell =du$. In signature $(-,+,+,+)$, Cartan's first structure equation and metric compatibility give the lower-index connection 1-forms ${\omega }_{01}={\omega }_{13}=-(A^{\prime }/A)e^1$ and ${\omega }_{02}={\omega }_{23}=-(B^{\prime }/B)e^2$. Cartan's second structure equation gives ${\Theta }_{01}={\Theta }_{13}=-(A^{\prime \prime }/A)\ell \wedge e^1$ and ${\Theta }_{02}={\Theta }_{23}=-(B^{\prime \prime }/B)\ell \wedge e^2$. All other independent curvature 2-forms vanish: in particular the $12$ component cancels between the time and longitudinal terms. Contracting $R^a{}_{bad}$ yields $\mathrm{Ric}=-(A^{\prime \prime }/A+B^{\prime \prime }/B)du\otimes du$. Thus the Vacuum Einstein equations are $A^{\prime \prime }/A+B^{\prime \prime }/B=0$, while nonzero $A^{\prime \prime }/A=-B^{\prime \prime }/B$ gives opposite transverse tidal curvatures and plus polarization relative to these axes.