Ricci tensor of a Brinkmann plane wave ($R_{uu}=H_{xx}+H_{yy}$)

For $d{s}^2=-2dudv+d{x}^2+d{y}^2-2H(u,x,y)d{u}^2$, the nonzero connections are ${\Gamma }_{uu}^v=H_u$, ${\Gamma }_{ux}^v=H_x$, ${\Gamma }_{uy}^v=H_y$, ${\Gamma }_{uu}^x=H_x$, and ${\Gamma }_{uu}^y=H_y$, with symmetric lower indices. Their traces and quadratic Ricci products vanish, leaving only the displayed Ricci component. Hence the vacuum equations require a harmonic transverse profile. The sign is tied to the metric's $-2Hd{u}^2$ convention.