\[ \begin{array}{rl} \left\{ \begin{array}{l} \displaystyle \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} = 0 \quad \text{(I)} \\[8pt] \displaystyle \frac{\partial u}{\partial t} - fv = -\frac{1}{\rho_0} \frac{\partial p'}{\partial x}, \quad \frac{\partial v}{\partial t} + fu = -\frac{1}{\rho_0} \frac{\partial p'}{\partial y} \quad \text{(II)} \\[8pt] \displaystyle \frac{\partial p'}{\partial z} = \rho_0 \frac{N^2}{g} w \quad \text{(III)} \\[8pt] \displaystyle \frac{\partial w}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial z} - \frac{g \rho'}{\rho_0} \quad \text{(IV)} \end{array} \right. & \begin{array}{l} \text{Apply } \partial / \partial t \text{ to (II-1)} \\[4pt] \text{Multiply (II-2) by } f \\[4pt] \text{Add results} \\[8pt] \text{Then:} \\[4pt] \left( \frac{\partial^2}{\partial t^2} + f^2 \right) u = -\frac{1}{\rho_0} \left( \frac{\partial^2 p'}{\partial t \partial x} + f \frac{\partial p'}{\partial y} \right) \quad \text{(V)} \\[10pt] \text{Repeat with } -f \text{ and } \partial/\partial t \\[4pt] \left( \frac{\partial^2}{\partial t^2} + f^2 \right) v = -\frac{1}{\rho_0} \left( \frac{\partial^2 p'}{\partial t \partial y} - f \frac{\partial p'}{\partial x} \right) \quad \text{(VI)} \end{array} \end{array} \]

Apply $\partial / \partial x$ to (V), $\partial / \partial y$ to (VI), and add: $$ \left( \frac{\partial^2}{\partial t^2} + f^2 \right) \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) \overset{\text{(I)}}{=} \left( \frac{\partial^2}{\partial t^2} + f^2 \right) \left( -\frac{\partial w}{\partial z} \right) \tag{VII} $$ $$ \text{RHS} = -\frac{1}{\rho_0} \left( \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \right) \frac{\partial p'}{\partial t} = -\frac{1}{\rho_0} \nabla_H^2 \frac{\partial p'}{\partial t} \Rightarrow \left( \frac{\partial^2}{\partial t^2} + f^2 \right) \frac{\partial w}{\partial z} = \frac{1}{\rho_0} \nabla_H^2 \frac{\partial p'}{\partial t} \tag{VIII} $$ Take $\partial / \partial t$ of (IV) and substitute (III) into the result: $$ \frac{\partial^2 w}{\partial t^2} = -\frac{1}{\rho_0} \frac{\partial^2 p'}{\partial t \partial z} - \frac{g}{\rho_0} \frac{\partial \rho'}{\partial t} \overset{\text{(III)}}{=} -\frac{1}{\rho_0} \frac{\partial^2 p'}{\partial t \partial z} - N^2 w \Rightarrow \left( \frac{\partial^2}{\partial t^2} + N^2 \right) w = \frac{1}{\rho_0} \frac{\partial^2 p'}{\partial t \partial z} \tag{IX} $$ Apply $\partial / \partial z$ to (VIII), and $\nabla_H^2$ to (IX), then add: $$ \left( \frac{\partial^2}{\partial t^2} + f^2 \right) \frac{\partial^2 w}{\partial z^2} + \nabla_H^2 \left( \frac{\partial^2 w}{\partial t^2} + N^2 w \right) = \frac{1}{\rho_0} \nabla_H^2 \frac{\partial^2 p'}{\partial t \partial z} + \frac{1}{\rho_0} \nabla_H^2 \frac{\partial^2 p'}{\partial t \partial z} = 0 $$ $$ \because \nabla^2 = \nabla_H^2 + \frac{\partial^2}{\partial z^2} \quad \therefore \boxed{ \nabla^2 \frac{\partial^2 w}{\partial t^2} + f^2 \frac{\partial^2 w}{\partial z^2} + N^2 \nabla_H^2 w = 0 } $$