Define the quiescent density \( \overline{\rho}(z) \) and pressure \( \overline{p}(z) \) profiles in the medium as those that satisfy a hydrostatic balance \[ 0 = -\frac{1}{\rho_0} \frac{\partial \overline{p}}{\partial z} - \frac{\overline{\rho} g}{\rho_0} \] When the motion develops, the pressure and density will change relative to their quiescent values \[ p = \overline{p}(z) + p', \quad \text{and} \quad \rho = \overline{\rho}(z) + \rho' \] The density equation \(\text{I}\) then becomes \[ \frac{\partial}{\partial t} \left( \overline{\rho} + \rho' \right) + u \frac{\partial}{\partial x} \left( \overline{\rho} + \rho' \right) + v \frac{\partial}{\partial y} \left( \overline{\rho} + \rho' \right) + w \frac{\partial}{\partial z} \left( \overline{\rho} + \rho' \right) = 0 \] Here, \( \partial \overline{\rho} / \partial t = \partial \overline{\rho} / \partial x = \partial \overline{\rho} / \partial y = 0 \), and the nonlinear terms (namely, \( u \partial \rho' / \partial x \), \( v \partial \rho' / \partial y \), and \( w \partial \rho' / \partial z \)) are also negligible for small-amplitude motions. The linear part of the fourth term \( \boxed{w \partial \overline{\rho} / \partial z} \) must be retained, so the linearized version of \(\text{I}\) is \[ \frac{\partial \rho'}{\partial t} + w \frac{\partial \overline{\rho}}{\partial z} = 0 \] which states that the density perturbation at a point is generated only by the vertical advection of the background density distribution. Now introduce the Brunt–Väisälä frequency, or buoyancy frequency: \[ N^2 = -\frac{g}{\rho_0} \frac{\partial \overline{\rho}}{\partial z} \] \( N(z) \) has units of rad/s and is the oscillation frequency of a vertically displaced fluid particle released from rest in the absence of fluid friction