For the case of a flat plate, approximate analytical results are available for both weak and strong interaction (excluding the neighborhood of the leading edge).
Following Hayes and Probstein  and Cox and Crabtree , the displacement thickness for weak interaction on a flat plate is given by (as discussed before)
Thus, the equivalent body slope due to the displacement effect is
where Rex = poUx/^o.
On the other hand, the pressure ratio for oblique shocks is
where K = M0 d6*/dx.
For (M0 d6*/dx)2 ^ 1, the pressure ratio over a wedge is obtained from
p d6* Ye ( Tw
— = 1 + YM0 = 1 + 0.664 + 1.73—W x (12.232)
P0 dx 2 Te
where x = M03VC/V Rex.
For an insulated surface (Tw/T0 ^ 1) and a very cold wall (Tw/T0 ^ 0), the above results are reduced to
In the case of strong interaction, the pressure formula is based on strong oblique
shock, i. e.
It can be shown that for a consistent solution, the displacement thickness and induced pressure distribution must comply to the following behaviors, S* ~ x-3/4, p ~ x-1/2.
Let p/p0 ~ kx-1/2, then
The role of parameter x is clear for both interaction regions.
Comparison with experimental results are reasonable (see Hayes and Probstein ).
Later, a more comprehensive theory has been developed based on a triple deck structure, see reviews by Stewartson , Rothmayer and Smith , Sychev  and Brown et al. . In this theory, upstream influence (suggested earlier by Lighthill ) as well as flow reversal and separation are studied in details.