= Solution
For a nonzero plane-wave amplitude, substituting $u=Ae^{i(kx-\omega t+\theta_0)}$ gives $u_{tt}=-\omega^2u$ and $u_{xx}=-k^2u$. Therefore
$$
\boxed{\omega^2=c^2k^2+a,\qquad\omega(k)=\sqrt{c^2k^2+a}\text{ on the positive-frequency branch}.}
$$
This <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.
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