= Solution
For this density the <Euler-Lagrange field equation with second derivatives> gives
$$
\boxed{\rho\phi_{tt}+\alpha\phi_{xx}+\beta\phi_{xxxx}=0.}
$$
Insert a <plane wave> $\phi=Ae^{i(kx-\omega t)}$. The second and fourth spatial <derivatives> give $-k^2$ and $k^4$, respectively, so
$$
-\rho\omega^2-\alpha k^2+\beta k^4=0,\qquad\boxed{\omega^2=\frac{\beta k^4-\alpha k^2}{\rho}.}
$$
For propagating modes, $k^2>\alpha/\beta$ and the <phase velocity> has magnitude $|\omega/k|=\sqrt{(\beta k^2-\alpha)/\rho}$, which depends on the <wave number>. Thus the waves are dispersive. For $0<k^2<\alpha/\beta$, the squared frequency is negative: the corresponding solutions grow or decay exponentially rather than propagate with a real frequency. The uniform $k=0$ displacement is a zero-frequency shift mode.
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