= Solution
Eliminate the transport amplitudes in the <linear stability analysis>. For a general <bed shear response> $\mathcal T(k)$, the <dispersion relation> is
$$
\boxed{\lambda=\sigma-ikc=-\frac{ikC}{1+ikL_{\mathrm{sat}}}
\left[\mathcal T(k)-\frac{ik\tau_{\mathrm{th},0}}{\mu_s}\right]}.
$$
This is the most specific result available without adding the missing hydrodynamic closure. With $\mathcal T(k)=\tau_0k(A+iB)$, set
$$
a=\tau_0A,\qquad b=\tau_0B-\tau_{\mathrm{th},0}/\mu_s,\qquad x=kL_{\mathrm{sat}},\qquad k>0.
$$
Multiplication by $1-ix$ gives the <growth rate> and migration <velocity>
$$
\boxed{\sigma(k)=Ck^2\frac{b-akL_{\mathrm{sat}}}{1+(kL_{\mathrm{sat}})^2},\qquad
c(k)=Ck\frac{a+bkL_{\mathrm{sat}}}{1+(kL_{\mathrm{sat}})^2}}.
$$
The packed-bed volume fraction cancels. For a real physical closure the negative <wavenumber> response is the complex conjugate; the displayed $k>0$ formulas cover the independent <normal modes>.
If $A>0$, $B>0$ are treated as constants, the upstream <shear stress> phase lead promotes growth, while the <inclined-bed sediment threshold> reduces it and the finite <saturation length> delays sediment transport. \b[There is an unstable band precisely when $b>0$: $0<k<b/(aL_{\mathrm{sat}})$.] Within this band $c>0$, so the growing <bedforms> migrate downstream. The fastest <normal mode> satisfies
$$
\boxed{(k_{\max}L_{\mathrm{sat}})^3+3k_{\max}L_{\mathrm{sat}}=2b/a}.
$$
This last maximization assumes constant $A,B$; if the hydrodynamic coefficients vary with <wavenumber>, their derivatives also enter the selection condition. When $b\leq0$ and $a>0$, all nonzero <normal modes> decay.
Just above the <sediment entrainment threshold>,
$$
b\simeq\tau_{\mathrm{th},0}(B-1/\mu_s).
$$
Thus the common regime $B<1/\mu_s$ is stable near onset: the slope penalty overwhelms the hydrodynamic phase lead. This conclusion is conditional on that inequality, not universal. For constant $B>0$, instability begins only when $\tau_0>\tau_{\mathrm{th},0}/(\mu_sB)$ as well as $\tau_0>\tau_{\mathrm{th},0}$. The transport sensitivity $C=\chi\gamma\Delta^{\gamma-1}$ vanishes at onset for $\gamma>1$, is finite for $\gamma=1$, and is singular for $0<\gamma<1$. Exactly at threshold, the positive-part transport law needs separate treatment; the above <linearization> with perturbations small compared with $\Delta$ is unavailable.
Far above the <sediment entrainment threshold>, $b\simeq\tau_0B$. If $A,B>0$, the unstable cutoff tends to $B/(AL_{\mathrm{sat}})$, and the dominant wavelength is set by the <saturation length>. Both <growth rate> and migration <velocity> scale with $C\tau_0\sim\chi\gamma\tau_0^\gamma$, apart from their length factors. Their numerical values and any detailed dependence on <wavenumber> still require a flow closure.
For example, prescribing uniform <shear stress> independently of the bed gives $\mathcal T=0$ and
$$
\sigma=-\frac{C\tau_{\mathrm{th},0}}{\mu_s}\frac{k^2}{1+(kL_{\mathrm{sat}})^2}<0.
$$
This equally admissible closure illustrates why instability cannot be asserted from the printed sediment equations alone.
\Image[/past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2018/iii/paper-345-bedform.png]
{title=Conditional bedform growth and migration}
The plots use constant $a>0$ and three illustrative ratios $b/a$. They show how the sign of the shear phase lead minus the slope correction determines stability; they are not numerical predictions for unspecified flow conditions.
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