Chapter 7, Problem 7P

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

$\begin{array}{r}\hfill \frac{{\partial }^2\psi }{\partial t^2}+\frac{\partial }{\partial t}\left(u^2+{\upsilon }^2\right)+\left(u^2-c^2\right)\frac{{\partial }^2\psi }{\partial x^2}\\ \hfill +\left({\upsilon }^2-c^2\right)\frac{{\partial }^2\psi }{\partial y^2}+2u\upsilon \frac{{\partial }^2\psi }{\partial x\partial y}=0\end{array}$

where ψ is the stream function, u and υ 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 c₀ (the speed of sound at the stagnation point) to nondimensionalize this equation, obtain the dimensionless groups that characterize the equation.