= Solution
The basic <quasi-geostrophic streamfunction> $\bar\psi=-\Lambda yz$ gives $\bar u=\Lambda z$, $\bar v=0$ and constant <quasi-geostrophic potential vorticity> $\bar Q=f$. Define
$$
q'=\nabla_h^2\psi'+\frac{f^2}{N^2}\psi'_{zz}.
$$
Because the background has no <potential-vorticity gradient>, the linear <potential-vorticity equation> is simply
$$
(\partial_t+\Lambda z\partial_x)q'=0.
$$
Its <characteristics> have constant $y,z,x-\Lambda zt$, so the general solution is $q'=\mathcal F(y,z,x-\Lambda zt)$. In particular, $q'=F(y,z)G(x-\Lambda zt)$ is the requested arbitrary separated family; one product is not the most general initial field.
Linearizing the vertical <velocity> requires retaining advection of $\bar\psi_z=-\Lambda y$ by $v'=\psi'_x$. Therefore
$$
w'=-\frac f{N^2}\left[(\partial_t+\Lambda z\partial_x)\psi'_z-\Lambda\psi'_x\right].
$$
The impermeable boundary at $z=0$ imposes
$$
\boxed{\partial_t\psi'_z-\Lambda\psi'_x=0\quad(z=0).}
$$
This is conservation of boundary <buoyancy>, not merely a condition that $\psi'_z$ vanish.
For zero interior <potential vorticity>, write $\psi'=\operatorname{Re}\{\phi(z,t)e^{i(kx+ly)}\}$. With $K_h=\sqrt{k^2+l^2}>0$, <stratified quasi-geostrophic inversion> gives
$$
\phi_{zz}-\frac{N^2K_h^2}{f^2}\phi=0.
$$
The decaying solution is
$$
\boxed{\phi=A(t)e^{-\mu z},\qquad\mu=\frac{N K_h}{f}.}
$$
No time dependence has been assumed to obtain this vertical structure. Applying the boundary condition gives $-\mu\dot A-i\Lambda kA=0$, hence
$$
\boxed{A=A_0e^{-i\omega t},\qquad
\omega=\frac{\Lambda k}{\mu}=\frac{f\Lambda k}{N\sqrt{k^2+l^2}}.}
$$
Thus the <Eady edge wave> travels eastward with <horizontal phase velocity> $\omega/k=\Lambda/\mu$ for $k\ne0$.
The propagation mechanism can be made explicit using boundary parcels. <Thermal wind> gives the background <buoyancy gradient> $\bar b_y=f\bar\psi_{zy}=-f\Lambda$. A small meridional parcel <displacement> $d$ produces boundary <buoyancy perturbation> $b'=-d\bar b_y=f\Lambda d$. Meanwhile the decaying <potential-vorticity inversion> gives $b'=f\psi'_z=-f\mu\psi'$ at the boundary. Consequently $\psi'=-\Lambda d/\mu$ there, and the parcel equation $d_t=v'=\psi'_x$ becomes
$$
d_t+\frac\Lambda\mu d_x=0.
$$
A displaced buoyancy contour therefore induces a <velocity field> that advances that displacement eastward. No interior <potential-vorticity gradient> is needed because the boundary buoyancy supplies the invertibility data.
<Vortex stretching> is what sets the induced flow's vertical reach. In the interior, the zero-<potential vorticity> condition requires
$$
\underbrace{\nabla_h^2\psi'}_{\text{relative vorticity}}=-K_h^2\psi',\qquad
\underbrace{\frac{f^2}{N^2}\psi'_{zz}}_{\text{stretching}}=+K_h^2\psi'.
$$
Boundary <isopycnal displacement> decreases with height as $e^{-\mu z}$, so fluid columns are stretched or compressed over depth $\mu^{-1}=f/(NK_h)$. Conservation of <potential vorticity> makes this stretching balance the <relative vorticity>. The resulting inversion factor $b'=-f\mu\psi'$ gives the propagation speed: stronger <density stratification> reduces penetration and speed, while larger $f$ increases them. Zero interior <potential vorticity> therefore does not imply zero interior flow.
For imposed boundary vertical <velocity>, retain the same decaying structure and write
$$
w'(0)=\operatorname{Re}\{\epsilon e^{i(kx+ly-\omega_0t)}\}.
$$
The boundary relation becomes the <forced Eady edge wave> equation
$$
\dot A+i\omega A=C e^{-i\omega_0t},\qquad C=\frac{N^2\epsilon}{f\mu}.
$$
The <integrating factor> $e^{i\omega t}$ gives, for $\omega_0\ne\omega$,
$$
\boxed{A(t)=B e^{-i\omega t}+\frac{C}{i(\omega-\omega_0)}e^{-i\omega_0t},\qquad
\psi'=\operatorname{Re}\{A(t)e^{-\mu z+i(kx+ly)}\}.}
$$
The arbitrary homogeneous coefficient $B$ sets the initial disturbance. With $A(0)=0$, this is $A=C(e^{-i\omega_0t}-e^{-i\omega t})/[i(\omega-\omega_0)]$.
At <resonance>, $\omega_0=\omega$, integration instead yields
$$
\boxed{A(t)=(B+Ct)e^{-i\omega t}.}
$$
An initially undisturbed fluid has $\psi'=Ct e^{-\mu z}\cos(kx+ly-\omega t)$. The imposed pumping reinforces the same boundary <normal mode> every cycle, with no dissipative loss, so the linear amplitude grows secularly. The divergence describes <resonance> of the undamped linear model; sufficiently large <wave amplitude> invalidates the assumed small-disturbance approximation.
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