Show that for an arbitrary isentropic flow through a nozzle (i.e., one that is not necessarily choked) that the mass flow rate is given by:

\[ \dot{m}=A_{t} \psi \sqrt{2 p_{0} ho_{0}} \quad \text { where } \quad \psi=\left\{\begin{array}{c} \sqrt{\frac{\gamma}{2}\left(\frac{2}{\gamma+1}\right)^{\frac{\gamma+1}{\gamma-1}}} \\ \left.\sqrt{\frac{\gamma}{\gamma-1}\left[\left(\frac{p}{p_{0}}\right)^{\frac{2}{\gamma}}-\left(\frac{p}{p_{0}}\right)^{\frac{\gamma+1}{\gamma}}\right]} \frac{p_{0}}{p}<\left(\frac{\gamma+1}{\gamma}\right)^{\frac{\gamma}{\gamma-1}}\right)^{\frac{\gamma}{\gamma-1}} \end{array}\right. \]

where \(p\) is the pressure at the nozzle throat, \(p_{0}\) and \(ho_{0}\) are the total pressure and density upstream of the nozzle, and \(A_{t}\) is the area of the throat. Your task is to find \(\psi\).