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where is the horizontal velocity component in the flow, and
is the surface wave; cf. Q3.4. Find the first terms in the near-field expansions
of and as and then obtain the equation for the leading term in
valid in the far-field You should consider only right-going
waves. (The equation that you obtain here is a Korteweg-de Vries-Burgers
(KVB) equation; see Johnson, 1997.)
Q3.9 Supersonic, thin-aerofoil theory: characteristic approach. The characteristics for equa-
tion (3.22) can be defined by the equation where is the
streamline direction (so that and is the inclination of
the characteristic relative to the streamline (so that tan where
M is the local Mach Number). Show that

and hence deduce that, on the characteristics,

Finally, since to leading order show that

and confirm that this is recovered from equation (3.39).
Q3.10 Thin aerofoil in a transonic flow. Show that the asymptotic expansion (3.35) is not
uniformly valid as
(a) Set write and and hence deduce that a
scaling consistent with equations (3.22) and (3.24) is
and that then satisfies, to leading order,

(b) Given that use the scaling in (a) to show that there is a
distinguished limit in which what now is the equation for
to leading order?
Q3.11 Thin aerofoil in a hypersonic flow. See (3.35); show that this expansion breaks
down as when Introduce leave x un-
scaled and write show that terms from both the left-
hand and right-hand sides of equation (3.22) are of the same order in the
case for a particular choice of What is the resulting