Before revision: We make a linear approximation to , and move to the RHS: The above second-order differential equation has straightforward solutions of : where the Brunt–Väisälä frequency is: For negative , the displacement has oscillating solutions (and N gives our angular frequency).
After revision: We make a linear approximation to , and move to the RHS: The above second-order differential equation has the following solution : where the Brunt–Väisälä frequency is: For negative , the displacement has oscillating solutions (and N gives our angular frequency).