= Solution
The linearized equations are homogeneous and have constant coefficients in $x$ and $t$. Consequently, spatial <Fourier modes> evolve independently, and the <superposition principle> reconstructs arbitrary sufficiently regular initial disturbances from them. It is therefore enough to seek the <normal modes>
$$
\eta(x,t)=\operatorname{Re}\{\eta_0e^{ikx+\sigma t}\}.
$$
Here the real $k$ is the <wavenumber>, $\sigma$ is the possibly complex temporal exponent, and taking the <real part> produces a physical displacement.
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