= Solution
The <dispersion relation> shows that an infinite rod has unstable long-wave modes. A finite rod excludes sufficiently long wavelengths, so stability depends on its length and the precise end conditions. The fourth spatial <derivative> requires two conditions at each end; fixing transverse positions alone does not specify whether slopes are constrained.
First take displacement-fixed ends with free rotation. Setting $\phi(0)=\phi(l)=0$ leaves slope variations free; the bending boundary term in the <first variation> then gives the corresponding <natural boundary conditions on the slope>, $\phi_{xx}(0)=\phi_{xx}(l)=0$. The spatial <normal modes> are $\sin(n\pi x/l)$, and
$$
\boxed{\omega_n^2=\frac\beta\rho\left(\frac{n\pi}{l}\right)^4-\frac\alpha\rho\left(\frac{n\pi}{l}\right)^2.}
$$
The first mode destabilizes first. Consequently
$$
\boxed{l<\pi\sqrt{\frac\beta\alpha}:\ \text{stable oscillatory modes};\qquad l>\pi\sqrt{\frac\beta\alpha}:\ \text{unstable straight rod}.}
$$
At equality, the first mode has zero restoring force and is marginal, with a static component and a possible linear-in-time component in the linear theory. Every integer $n<l\sqrt{\alpha/\beta}/\pi$ gives an exponentially growing mode.
If "fixed" instead means clamped, the <clamped boundary condition> is $\phi=\phi_x=0$ at both ends. The critical compression is then
$$
\boxed{\alpha_c=\frac{4\pi^2\beta}{l^2},\qquad l_c=2\pi\sqrt{\frac\beta\alpha}.}
$$
For example, the static critical shape $1-\cos(2\pi x/l)$ obeys the four clamp conditions and the zero-frequency equation at this load. To establish that it is the first threshold, write $v=\phi_x$: it has zero endpoints and zero mean, and periodic extension gives $\int_0^lv_x^2dx\geq(2\pi/l)^2\int_0^lv^2dx$ by the zero-mean periodic <Poincaré inequality>, with equality at $v\propto\sin(2\pi x/l)$. This proves the critical <Rayleigh quotient>. Both cases are instances of the <compressed Euler-Bernoulli beam and its endpoint-dependent buckling threshold>. The source does not specify the endpoint rotation, so both interpretations are stated explicitly.
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