Compressible Subsonic Flow
The governing equation for this flow is:
(1 — МОТ)фХХ + Fz = °.
Solving as before, we get the result:
ф(х, z) = — = exp — kz^J 1 — M2 cos(kx)
1 — M2a
Hence, we have:
9.25.3 Supersonic Flow
For supersonic flow the governing potential equation is:
(M2 — 1) фхх — Фzz = 0.
For this equation, by Equation (9.196), we have the solution as:
ф(х, z) = f (x — Pz) + g(x + Pz),
where в = cot Ц, = M2 — 1.
From the geometry of the problem under consideration, since the disturbances can move only in the direction of flow, there can be only left-running Mach lines, as shown in Figure 9.41(c). Therefore:
ф(х^) = f (x — Pz), g = 0.
Hence, the perturbation velocity w on the wall is
The ф here is only the disturbance potential, and if the total potential is required, add (Vlx) to ф.