Assume the equation that governs quasi-geostrophic perturbations on an eastward flow $U(z)$ has traveling wave solutions $$ p'=\hat{p}(z)\exp\{i(kx+ly-\omega t)\} $$ confined between horizontal planes at $z=0$ and $z=H$ that are unbounded in $x$ and $y$. Boundedness in $y$ sets up normal modes $\sin(n\pi y/L)$

Substitute into the perturbation vorticity equation, reduces it to an ordinary differential equation for $\hat{p}$ $$ \frac{d^{2}\hat{p}}{dz^{2}}+\alpha^{2}\hat{p}=0, \quad \alpha^{2}\equiv \frac{N^{2}}{f^{2}}(k^{2}+l^{2}) \qquad \Rightarrow \qquad \hat{p}=A\cosh[\alpha(z-H/2)]+B\sinh[\alpha(z-H/2) $$

The boundary conditions on $p'$ corresponding to $w'=0$ at $z=0$, and $z=H$ are found from \(w'=-\frac{1}{\rho_0 N^2}\!\left[\left(\frac{\partial}{\partial t}+U\frac{\partial}{\partial x}\right)\frac{\partial p'}{\partial z} -\frac{dU}{dz}\frac{\partial p'}{\partial x}\right]\) and $U(z)=U_{0}z/H$ $$ \left(\frac{\partial}{\partial t}+\frac{U_{0}z}{H}\frac{\partial}{\partial x}\right)\frac{\partial p'}{\partial z} -\frac{U_{0}}{H}\frac{\partial p'}{\partial x}=0 \quad\text{at } z=0 \text{ and } z=H $$

In particular, these two boundary conditions are $$ \frac{\partial^{2}p'}{\partial t\,\partial z}-\frac{U_{0}}{H}\frac{\partial p'}{\partial x}=0 \quad\text{at } z=0 $$ $$ \frac{\partial^{2}p'}{\partial t\,\partial z}+U_{0}\frac{\partial^{2}p'}{\partial x\,\partial z} -\frac{U_{0}}{H}\frac{\partial p'}{\partial x}=0 \quad\text{at } z=H $$