instability, regardless of the value of the dimensionless parameters $\tau_s$ and $f_p$, with a subdynamical growth rate scaling like $\Omega\tau_s$ in the tight-coupling limit. For a given $\tau_s$ and $f_p$, growth was maximised in the $k_x$-$k_z$ plane in a long-wavelength ridge (with $k_z\sim k_x^2
t_{\mathrm{stop}}/(\Omega f_g)$) and a short-wavelength branch with $k_z \gg
k_x$ ($k_x$ having a preferred value). Growth was strongly suppressed in the former for dust-to-gas ratio near unity ($f_p\sim f_g\sim 0.5$). YG05 found that the wave speed $\omega_\Re/k_x$ was bounded by $U_{\mathrm{sum}}$ defined as the sum of the background radial velocities of the dust and the gas, respectively, and growth rates (generally smaller than $|k_x|U_{\mathrm{sum}}/2$) were maximised when the said wave speed was about $U_{\rm sum}/2$.

Terminal velocity approximation
-------------------------------

To gain physical insight on the streaming instability, we need some simplifications to make the problem tractable analytically. We shall adopt the “terminal velocity approximation” (YG05), that is take the left-hand-side of equation (\[Euler-DeltaV\]) to be zero, such that the relative drift is given by: $$\Delta\mathbf{V}=\frac{\nabla P}{\rho}t_{\mathrm{stop}}.
\label{TVA}$$ This approximation holds provided (i) the perturbation and dynamical timescales ($\omega^{-1}$ and $\Omega^{-1}$) are longer than $f_gt_{\mathrm{stop}}=\rho_sa/(\rho v_T)$ (where the equality holds in the Epstein drag regime) and (ii) 