= Solution
The <characteristic> roots satisfy
$$
\lambda^2-\frac13\mu^{1/4}\lambda+4\mu^{7/6}=0.
$$
They are a nonreal <complex conjugate>[conjugate pair] precisely when $\mu^{1/2}/9-16\mu^{7/6}<0$, or $\mu^{2/3}>1/144$. Therefore
$$
\boxed{\mu_c=144^{-3/2}=\frac1{1728}.}
$$
In <lake-coupled Röthlisberger drainage instability>, above this value the <real part> is $\mu^{1/4}/6>0$, so nonzero linear perturbations are sinusoidal <oscillations> with an exponentially growing envelope. Their <angular frequency> is $[4\mu^{7/6}-\mu^{1/2}/36]^{1/2}$.
Physically the feedback represents repeated lake filling and <pressure> increase, channel enlargement by melting, rapid drainage, <pressure> reduction and <material creep> closure. Large drainage events are glacial lake outburst floods, or jökulhlaups. \b[The unbounded <amplitude> is a conclusion of the linearized <equations>], not a proof of infinite physical discharge or an indefinitely growing nonlinear solution. Once perturbations are comparable with the <equilibrium> scales, nonlinear dynamics and constraints such as lake depletion must be retained. At the threshold the two real roots coincide; below it the instability is nonoscillatory.
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