An unsteady, two-dimensional, compressible, inviscid flow can be described by the equation

\[\begin{aligned}\frac{\partial^{2} \psi}{\partial t^{2}}+\frac{\partial}{\partial t} & \left(u^{2}+v^{2}\right)+\left(u^{2}-c^{2}\right) \frac{\partial^{2} \psi}{\partial x^{2}} \\& +\left(v^{2}-c^{2}\right) \frac{\partial^{2} \psi}{\partial y^{2}}+2 u v \frac{\partial^{2} \psi}{\partial x \partial y}=0\end{aligned}\]

where \(\psi\) is the stream function, \(u\) and \(v\) are the \(x\) and \(y\) components of velocity, respectively, \(c\) is the local speed of sound, and \(t\) is the time. Using \(L\) as a characteristic length and \(c_{0}\) (the speed of sound at the stagnation point) to nondimensionalize this equation, obtain the dimensionless groups that characterize the equation.