= Solution
For <quasistatic bed stability with suspended sediment>, in the leading approximation, $(hu)_x=0$ and $\eta_x=0$. For spatially varying perturbations at fixed unit <water> <flux> and fixed surface level, write $u=1+U$, $h=1+H$, $c=1+C$, $s=s_*+S_1$. The constraints give $H=-U$ and $S_1=-H=U$. Linearizing the suspended and bed <equations> yields
$$
C_x=4U-C,\qquad U_t+3\beta U_x=-4U+C.
$$
For a <Fourier mode> $e^{\sigma t+i\kappa x}$, the first relation gives $C=4U/(1+i\kappa)$; substitution into the second gives
$$
\boxed{\sigma=-\frac{4\kappa^2}{1+\kappa^2}-i\kappa\left(3\beta+\frac4{1+\kappa^2}\right).}
$$
Thus every nonzero spatial <frequency> has negative growth rate, independently of $\beta$. The parameter changes propagation, not damping. The spatially uniform mode has $\sigma=0$ and is neutral, representing a change among uniform reference states compatible with the leading constraints. Hence \b[there is no growing linear mode in this approximation]; the claim is stability, not strict decay of every homogeneous perturbation. Restoring the small discarded terms or changing <boundary conditions> requires a separate calculation.
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