Page 141 -

124 3. Further applications

This second condition is rewritten as an evaluation on y = 0, by assuming that
possesses a Taylor expansion (cf. §3.1), so we now obtain

or

and so on. It is not possible to solve, in any simple compact way, for both the upper
and lower surfaces together (as will become clear), so we will consider only the upper
surface, (The problem for the lower surface follows a similar, but different,
construction.)
The general solution of equation (3.26), for the case of supersonic flow
is given by d’Alembert’s solution:

where and F and G are arbitrary functions. The contribution to the
solution from G, in the upper half-plane, extends into x < 0 and so, in order to satisfy
(3.23), we must have (The contribution from F extends into x> 0 for y >0.)
Condition (3.29a) (upper surface) now requires that

and so to within an arbitrary constant, which may be
ignored in the determination of a velocity potential (because such a constant cannot
contribute to the velocity field). Thus we have

in y > 0. (It will be noted that the corresponding problem in the lower half-plane
requires the retention of with
The solution of equation (3.27), for is approached in the same way that we
adopted for the similar exercise in E3.1: we introduce characteristic variables, which
here are With these variables, equation (3.27) becomes