only solution to wave equation under certain restriction?

Question

Suppose $u_1, u_2: \mathbb{R}^2 \rightarrow \mathbb{R}$ are two solutions to the wave equation \begin{equation} \partial_t^2u_i = \partial_x^2u_i \end{equation} obeying the restriction $\partial_x \left(u_1^2 + u_2^2\right) = 0$. Is it so that $u_1$ and $u_2$ must be of the form \begin{equation} u_1(x,t)=C_1\sin(kz)\sin(kt+\phi_1) + C_2\cos(kz)\sin(kt+\phi_2) \end{equation} \begin{equation} u_2(x,t)=\pm \left( C_1\cos(kz)\sin(kt+\phi_1) - C_2\sin(kz)\sin(kt+\phi_2) \right) \end{equation} EDIT: as is evident from Robert Israel's answer, $u_1(z,t)=0, u_2(z,t)=t$ is a counterexample. But of course it is not of the kind I'm looking for. So impose the extra constraint that $u_1$ and $u_2$ must be globally bounded, what about then?

Answer 1

Counterexample: let $u_1(x,t) = 0$, $u_2(x,t) = t$.