= Solution
Let the perturbation phase be $\vartheta=\mathbf k(t)\cdot\mathbf r$ and write $\psi=\psi_0+\operatorname{Re}[\Pi(t)e^{i\vartheta}]$. <Incompressible flow> imposes $\mathbf k\cdot\mathbf v=0$. Applying the background advective derivative to the phase gives
$$
(\partial_t+\mathbf u_0\cdot\nabla)\vartheta=\dot{\mathbf k}\cdot\mathbf r-2Axk_y.
$$
Cancel this position-dependent phase by choosing
$$
\boxed{\dot k_x=2Ak_y,\qquad \dot k_y=\dot k_z=0,\qquad k_x(t)=k_x(t_0)+2Ak_y(t-t_0).}
$$
The perturbation acting on the basic flow gives $(\mathbf u'\cdot\nabla)\mathbf u_0=-2Au_x'\mathbf e_y$. Its self-advection vanishes exactly: the physical velocity $\mathbf u'$ is perpendicular to $\mathbf k$, while every spatial derivative of $\mathbf u'$ is proportional to a component of $\mathbf k$. Thus $\mathbf u'\cdot\nabla\mathbf u'=0$, including the products between the complex mode and its conjugate. Substituting into the full <Navier-Stokes equation> therefore gives an <exact incompressible shearing wave>, with no small-amplitude approximation:
$$
\boxed{\dot{\mathbf v}-2Av_x\mathbf e_y+2\Omega\mathbf e_z\times\mathbf v=-i\mathbf k\Pi-\nu k^2\mathbf v,\qquad \mathbf k\cdot\mathbf v=0.}
$$
In components these equations are
$$
\begin{aligned}
\dot v_x-2\Omega v_y&=-ik_x\Pi-\nu k^2v_x,\\
\dot v_y+2(\Omega-A)v_x&=-ik_y\Pi-\nu k^2v_y,\\
\dot v_z&=-ik_z\Pi-\nu k^2v_z.
\end{aligned}
$$
Differentiating the transversality constraint closes the <pressure>:
$$
\boxed{ik^2\Pi=2\Omega k_xv_y+(4A-2\Omega)k_yv_x.}
$$
The coefficient $4A$ includes both the shear term in the momentum equation and $\dot{\mathbf k}\cdot\mathbf v$; treating the <wavevector> as fixed would miss one contribution. Initial data with $\mathbf k(t_0)\cdot\mathbf v(t_0)=0$ remain transverse under this closed evolution. These <shearing waves> are exact in an unbounded local shear flow or with compatible shearing-periodic boundaries; arbitrary rigid <boundary conditions> need not admit a single such wave.
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