Wings in Supersonic Flows
The nature of supersonic flow is different from the subsonic one. The governing equations are hyperbolic, rather than elliptic, representing wave propagation. In two
dimensional flows, the linearized small disturbance equation admits the d’Alembert solution (which is a special case of the method of characteristics). The corresponding solution of the three dimensional supersonic flow is based on Kirchoff’s formula, used for example in acoustics, see Garrick . It is given by
Interchanging ф and ф and subtracting yields
(фЬ(ф) — фЬ(ф)’) d9
if Ь(ф) = 0 and Ь(ф) = 0, we obtain
„( й — ф£) dS = 0
In carrying the integration in the above formulas, the domain of influence and the domain of dependency must be taken into consideration.
Panel methods for supersonic flows are available, see Evvard  and Puckett and Stewart .
For nonlinear supersonic flows, Ь(ф) = 0, again volume integrals are required and the problem is solved iteratively provided the local Mach number is always greater than one.