equation , we see that $k_x u$ is subdominant here if[^3] $$\begin{aligned}
k_x\, L \ll 1, \qquad \text{and}\qquad \left(\frac{k_x}{k_z}\right)\left(\frac{\omega}{\Omega}\right)\ll 1,
\label{condition}\end{aligned}$$ which can be satisfied if $k_x H \sim 1$ (see equation ) and if $k_x/k_z\lesssim 1$.

These approximations give us a simplified set of dynamical equations: the equations for the vertical centre of mass velocity, particle density conservation, and the incompressibility of the gas, in which drag effects enter implicitly. These are now

$$-i\omega w +ik_zh =0
\label{quadratic system Euler z}$$

$$ik_zw -i\omega\delta=0
\label{quadratic system COM continuity}$$

$$ik_zw -ik_xg_e(f_p-f_g)t_{\mathrm{stop}}\delta +f_pk^2t_{\mathrm{stop}}h =0
\label{quadratic system incompressibility}$$

yielding the following quadratic dispersion relation : $$-if_pt_{\mathrm{stop}}\left(\frac{k}{k_z}\omega\right)^2 + \omega +
k_xg_et_{\mathrm{stop}}(f_g-f_p)=0,
\label{quadratic-reduced}$$ from which we recover the secular mode and its growth rate as given in equation ! The corresponding eigenvector reads: $$\left[\begin{array}{r}
w \\ \delta \\ h
\end{array}\right]
=\delta
\left[\begin{array}{r}
\omega/k_z \\ 1 \\ \left(\omega/k_z\right)^2
\end{array}\right]
\label{eigenvector}$$ The second root corresponds to a damping $s=-\left(k/k_z\right)^2/(f_pt_{\mathrm{stop}})$ but violates the condition $\omega t_{\rm stop}\ll 1$ of validity of the terminal velocity approximation and will thus not be discussed further here.

It is also possible, if less rigorous, to see this reduction directly by inspection of the quartic dispersion relation (\[quartic\]). If we have $k_xg_et_{\mathrm{stop}}^2(k_x/k_z)^2\ll 1$ 