 Solution

Using the affine parameter and the normalization requested for the geodesic Lagrangian,The coordinate $v$ is cyclic. Its Euler-Lagrange equation gives $\ddot{u}=0$. The transverse coordinate equations give $\ddot{x}+H_x{\dot{u}}^2=0$ and $\ddot{y}+H_y{\dot{u}}^2=0$. For the remaining coordinate,Its Euler-Lagrange equation, after using $\ddot{u}=0$, isCompare with ${\ddot{x}}^a+{\Gamma }_{bc}^a{\dot{x}}^b{\dot{x}}^c=0$. With indices $(1,2,3,4)=(u,v,x,y)$, the complete list of nonzero Christoffel symbols, including symmetry of the lower indices, isAll contracted symbols ${\Gamma }_{ad}^d$ vanish. For $R_{11}$, the first derivative term in the supplied Ricci tensor formula is ${\partial }_v{H}_u+{\partial }_x{H}_x+{\partial }_y{H}_y=H_{xx}+H_{yy}$, since $H$ is independent of $v$. The second derivative term vanishes because the trace symbols vanish. The third term vanishes for the same reason, and every possible product in the last term requires a zero symbol with upper index $1$ or a lower $2$, so it also vanishes. ThusThe Vacuum Einstein equations with zero cosmological constant require $R_{ab}=0$, and therefore $\boxed{H_{xx}+H_{yy}=0}$. The wave profile of this plane-fronted gravitational wave is harmonic in the transverse plane for each fixed $u$.