 Solution

For a nonzero plane-wave amplitude, substituting $u=A{e}^{i(kx-\omega t+{\theta }_0)}$ gives $u_{tt}=-{\omega }^{2}u$ and $u_{xx}=-k^{2}u$. ThereforeThis dispersion relation is not proportionality between frequency and wavenumber: for $a>0$, the phase speed depends on $k$. Thus a superposition generally experiences dispersion. The negative-frequency branch is its opposite; $A=0$ is the trivial solution and imposes no constraint on the unused phase parameters.