= Solution
The <Exner equation> is solid-volume <mass conservation>: a downstream increase in <sediment transport> flux removes material and lowers the bed. The factor $\phi_b$ converts bed-height change into solid-volume change. The <saturation length> equation models downstream adjustment of actual <sediment transport> to the <equilibrium sediment flux>. Grain acceleration and entrainment/deposition need a finite distance, so $q$ lags $q_{\mathrm{sat}}$. The last relation is an empirical transport law above the <sediment entrainment threshold>, with $\chi>0$ and normally $\gamma>0$; below threshold it must be interpreted with a <positive part> rather than raising negative excess stress to an arbitrary power. It is a constitutive closure, not a consequence of <mass conservation>.
Take the physical parameters $L_{\mathrm{sat}}>0$, $\phi_b>0$, $\chi>0$ and $\gamma>0$. Linearize about a horizontal, uniformly transporting bed with $\Delta=\tau_0-\tau_{\mathrm{th},0}>0$ and $q_0=\phi_b\chi\Delta^\gamma$. Set $\lambda=\sigma-ikc$ and use the real parts of
$$
\eta=\hat\eta e^{\lambda t+ikx},\qquad
q=q_0+\hat q e^{\lambda t+ikx},\qquad
q_{\mathrm{sat}}=q_0+\hat q_{\mathrm{sat}}e^{\lambda t+ikx},\qquad
\tau=\tau_0+\hat\tau e^{\lambda t+ikx}.
$$
The locally planar approximation uses the small instantaneous bed slope to compute the <inclined-bed sediment threshold>. Since $\alpha=\eta_x+O(\eta_x^3)$,
$$
\boxed{\hat\tau_{\mathrm{th}}=\frac{ik\tau_{\mathrm{th},0}}{\mu_s}\hat\eta}.
$$
A sinusoidal disturbance has both uphill and downhill slopes: the printed $0<\alpha\ll1$ describes the uphill derivation, and its first-order continuation applies to both signs. The zero-slope reference is the one implicit in the printed decomposition with constant term $\tau_{\mathrm{th},0}$; a finite mean inclined bed would require a shifted base threshold.
Define $C=\chi\gamma\Delta^{\gamma-1}$. The linearized transport equations are
$$
\boxed{\phi_b\lambda\hat\eta=-ik\hat q,\qquad
(1+ikL_{\mathrm{sat}})\hat q=\hat q_{\mathrm{sat}},\qquad
\hat q_{\mathrm{sat}}=\phi_bC\left(\hat\tau-\frac{ik\tau_{\mathrm{th},0}}{\mu_s}\hat\eta\right)}.
$$
The <linearization> requires $|\eta_x|\ll1$ and $|\hat\tau-\hat\tau_{\mathrm{th}}|\ll\Delta$.
\b[A fluid-dynamical shear-response closure is missing from the printed question.] The three sediment equations and local slope correction do not determine $\hat\tau$ from $\hat\eta$. Write the general <bed shear response> as $\hat\tau=\mathcal T(k)\hat\eta$. A frequently used scale-invariant closure for $k>0$ is
$$
\boxed{\mathcal T(k)=\tau_0k(A+iB)}.
$$
Here $A$ is the component in phase with bed height and $B$ represents an upstream phase lead in bed <shear stress>. They require an independent flow model and can depend on $k$. Introducing them explicitly makes the <linear stability analysis> complete conditional on a specified flow response; it does not turn them into data supplied by the question. An example of this hydrodynamic closure is given in https://arxiv.org/abs/0805.3417[Fourrière, Claudin and Andreotti's bedform-instability analysis].
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