Analog Study of Supersonic Conical Flows
The governing equation of the perturbation potential for linearized three-dimensional supersonic flow is given by
2 д2ф д2ф д2у
дх2 ду2 dz2
where в2 = M2 – 1.
When the flow is conical, p depends only on two variables as follows. Let
у = r cos в, z = r sin в, x = firX
then, any of the three velocity components, u, v, w depend only on в and X. For X = cos £, a Laplace equation is obtained for the flow interior to the mach cone
For X > 1 (exterior of the mach cone), with X = cosh £, the wave equation is obtained
For more details, see Stewart .
To study the flow inside the Mach cone, the complex variable Z = gel6, where
Either u, v,w may be identified with the electrical potential in a plane circular tank.
Germain  analyzed in detail the analog representation for a flat plate inside the Mach cone. he also studied wings with different cross sections and with dihedral angles. The study of wings in supersonic flow regime, steady and unsteady, was carried out by Enselme .