Using hydrostatic balance \(0 = -\frac{1}{\rho_0} \frac{\partial \overline{p}}{\partial z} - \frac{\overline{\rho} g}{\rho_0}\) and Brunt–Väisälä frequency \(N^2 = -\frac{g}{\rho_0} \frac{\partial \overline{\rho}}{\partial z} \) in \(\text{III, IV, V}\) and linearized density equation for small perturbations \(\frac{\partial \rho'}{\partial t} + w \frac{\partial \overline{\rho}}{\partial z} = 0\) \[ \frac{\partial u}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial x}\\ \frac{\partial v}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial y}\\ \frac{\partial w}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial z} - \frac{\rho' g}{\rho_0}\\ \frac{\partial \rho'}{\partial t} - \frac{N^2 \rho_0}{g} w = 0 \] Thus, perturbation variables \( p' \) and \( \rho' \) replace \( p \) and \( \rho \) in the linearized system. Taking the time derivative of the incompressibility condition \(\text{II}\) and using \(\frac{\partial u}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial x}\) and \(\frac{\partial v}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial y}\) \[ \frac{1}{\rho_0} \nabla_H^2 p' = \frac{\partial^2 w}{\partial z \partial t} \] where \( \nabla_H^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \) is the horizontal Laplacian operator. From \(\frac{\partial w}{\partial t} = -\frac{1}{\rho_0} \frac{\partial p'}{\partial z} - \frac{\rho' g}{\rho_0}\) and \(\frac{\partial \rho'}{\partial t} - \frac{N^2 \rho_0}{g} w = 0\), we have \[ \frac{1}{\rho_0} \frac{\partial p'}{\partial t} = \frac{\partial^2 w}{\partial z^2} - N^2 w \] Eliminating \( p' \) by applying \( \nabla_H^2 \) to \(\frac{1}{\rho_0} \frac{\partial^2 \rho'}{\partial t \partial z} = -\frac{\partial^2 w}{\partial z^2} - N^2 w \) and substituting into \(\frac{1}{\rho_0} \nabla_H^2 p' = \frac{\partial^2 w}{\partial z \partial t}\) yields \[ \frac{\partial^2}{\partial t^2} \left( \frac{\partial^2 w}{\partial z^2} \right) = -\nabla_H^2 \left( \frac{\partial^2 w}{\partial t^2} + N^2 w \right) \text{ or } \frac{\partial^2}{\partial t^2} \left( \nabla^2 w \right) + N^2 \nabla_H^2 w = 0 \] where \( \nabla^2 = \nabla_H^2 + \frac{\partial^2}{\partial z^2} \) is the full three-dimensional Laplacian. This final equation governs the vertical velocity \( w \) and is fundamental for deriving the dispersion relation for internal gravity waves