 Solution

For the plane gravitational wave in linearized gravity, the chain rule gives ${\partial }_{\kappa }{\tilde{h}}_{\mu \nu }=n_{\kappa }{\tilde{h}}_{\mu \nu }^{\prime }$. The Lorenz gauge in linearized gravity, called wave gauge in the paper, imposes zero divergence of the trace-reversed metric perturbation. HenceThe wave equation itself holds for any smooth profile of $u$, because $\square {\tilde{h}}_{\mu \nu }=(n^{\kappa }{n}_{\kappa }){\tilde{h}}_{\mu \nu }^{\prime \prime }=0$ for the constant null vector $n$.