Chapter 15, Problem 70CP

Parcels of air (small volumes of air) in a stable atmosphere (where the temperature increases with height) can oscillate up and down, due to the restoring force provided by the buoyancy of the air parcel. The frequency of the oscillations are a measure of the stability of the atmosphere. Assuming that the acceleration of an air parcel can be modeled as $\frac{{\partial }^{2}z\text{'}}{\partial t^2}=\frac{g\partial \rho \left(z\right)}{{\rho }_0\partial z}z\text{'}$ , prove that $z\text{'}=z_0{}^{\text{'}}{e}^{t\sqrt{-N^2}}$ is a solution, where N is known as the Brunt-Väisälä frequency. Note that in a stable atmosphere, the density decreases with height and parcel oscillates up and down.