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122 3. Further applications

impose initial data at in terms of matching (first term to first term), this is
equivalent to solving (3.21) with as (where F is a suitable arbitrary
function) and hence, for the right-going wave, we select and the solution
is uniformly valid for i.e. This uniform validity generated
by the far-field solution is an unlooked-for bonus, but it should be remembered that
the basic expansion (3.11) is not uniformly valid, and so we certainly have a singular
perturbation problem.

Other examples of wave propagation problems, which exhibit a breakdown and con-
sequent rescaling, are set as exercises in Q3.4–3.8. In addition, we present one further
example which embodies this same mathematical structure, but which is a little more
involved. However, this is a classical problem which should appear in any standard text
and, further, it has various different limits that are of practical interest (and some of
these will be discussed later; see also Q3.10 and Q3.11).
E3.2 Supersonic, thin-aerofoil theory
We consider irrotational, steady, supersonic flow of a compressible fluid past a (two-
dimensional) thin aerofoil. The equations of mass and momentum conservation can
be reduced to the single equation

where a is the local sound speed in the gas, and the velocity is
The corresponding energy equation (Bernoulli’s equation) is

where and as and is a constant describing the na-
ture of the gas: pressure (an isentropic gas). The aerofoil is described by
with where the upper/lower surfaces are
denoted by +/–. Using U and to non-dimensionalise the variables, eliminating
and then writing the resulting non-dimensional velocity potential as we
obtain

where is the Mach number of the oncoming flow from infinity. The
aerofoil is now written as (which defines for this problem)
and so ensures that we have a thin aerofoil. (Thus implies that there is
no aerofoil present, and then the non-dimensional velocity potential is simply x i.e.