Assume that $\partial \overline{\rho}/\partial y$ is constant, and $U=0$ at the surface $z=0$, the background flow is $$ U(z)=U_0\,z/H $$ where $U_0$ is the velocity at the top of the layer of interest, $z=H$

Form the vorticity equation by cross-differentiating and adding the frictionless horizontal momentum equations to eliminate the pressure, then use the continuity equation \(\nabla \cdot \mathbf{u} = 0\) to replace $\partial u/\partial x+\partial v/\partial y$ with $-\partial w/\partial z$ $$ \frac{\partial \zeta}{\partial t}+u\frac{\partial \zeta}{\partial x}+v\frac{\partial \zeta}{\partial y} -(\zeta+f)\frac{\partial w}{\partial z}=0 $$

Assume that the perturbations are large-scale and slow, so that the velocity is nearly geostrophic $$ u' \approx - \frac{1}{\rho_0 f} \frac{\partial p'}{\partial y}, \qquad v' \approx \frac{1}{\rho_0 f} \frac{\partial p'}{\partial x} $$ The perturbation vorticity is found as $$ \zeta' = \frac{1}{\rho_0 f} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) p' = \frac{1}{\rho_0 f} \nabla_H^2 p' $$

Develop an expression for $w'$ in terms of $p'$ using the density Equation $ \frac{\partial}{\partial t} (\overline{\rho} + \rho') + (U + u') \frac{\partial}{\partial x} (\overline{\rho} + \rho') + v' \frac{\partial}{\partial y} (\overline{\rho} + \rho') + w' \frac{\partial}{\partial z} (\overline{\rho} + \rho') = 0$ $ \xrightarrow[\text{and linearize}]{\text{Evaluate derivatives}}\; \frac{\partial \rho'}{\partial t}+U\frac{\partial \rho'}{\partial x}+v'\frac{\partial \overline{\rho}}{\partial y} -\rho_0 N^2 \frac{w'}{g}=0 $ where \(N\) is the buoyancy frequency, \(\boxed{N^{2} \equiv -\frac{g}{\rho_{0}} \frac{d\rho}{dz}}\)