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 , has oscillating solutions (and N gives our angular frequency).
After revision: the displacementWe 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 , has oscillating solutions (and N gives our angular frequency).