= Solution
{c}
For the <plane gravitational wave in linearized gravity>, the <chain rule> gives $\partial_\kappa\tilde h_{\mu\nu}=n_\kappa\tilde h'_{\mu\nu}$. The <Lorenz gauge in linearized gravity>, called wave gauge in the paper, imposes zero <divergence> of the <trace-reversed metric perturbation>. Hence
$$
0=\partial^\mu\overline{\tilde h}_{\mu\nu}
=n^\mu\tilde h'_{\mu\nu}-\frac12n_\nu\tilde h',
\qquad
\boxed{n^\mu\tilde h'_{\mu\nu}=\frac12\tilde h'n_\nu.}
$$
The <wave equation> itself holds for any smooth profile of $u$, because $\Box\tilde h_{\mu\nu}=(n^\kappa n_\kappa)\tilde h''_{\mu\nu}=0$ for the constant <null vector> $n$.
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