= Solution
Material conservation of <shallow-water potential vorticity> carries $P=0$ from the reservoir into the channel. At leading order in a long, slowly varying channel, $v$ and $v_x$ are negligible, so $\zeta\simeq-u_y$. Zero <absolute vorticity> therefore implies $u_y=f$. Integration from $y=0$, followed by cross-channel <geostrophic balance>, gives the <zero-potential-vorticity rotating channel flow>:
$$
\boxed{u(x,y)=u_1(x)+fy,\qquad h(x,y)=h_1(x)-\frac f g\left[u_1(x)y+\frac12fy^2\right].}
$$
In particular, $u_2=u_1+fb$ and $h_2=h_1-(fb/g)(u_1+fb/2)$. The velocity increases towards the shallower wall. Differentiating the depth-based specific energy confirms
$$
\partial_y\left(h+\frac{u^2}{2g}\right)=h_y+\frac{u u_y}{g}=-\frac{fu}{g}+\frac{fu}{g}=0.
$$
This cross-channel constancy is also the $P=0$ consequence of $\nabla B=-P\nabla\psi$.
For a reservoir of finite depth, exact $P=0$ requires relative <vorticity> $\zeta=-f$, an anticyclonic rotation cancelling the planetary <vorticity>. A reservoir at rest in the rotating frame has $\zeta=0$ and $P=f/h$, so it does not have exactly zero <shallow-water potential vorticity>. A very deep, nearly quiescent reservoir can nevertheless supply the approximation $P\simeq0$ because $f/h$ is small. The idealization must describe vanishing <absolute vorticity> or this deep-reservoir limit, rather than ordinary finite-depth rest in the rotating frame.
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