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164 4. The method of multiple scales
where and is a constant independent of cf. Q3.24. We seek an
asymptotic solution in the form
where
and note that we have omitted the term in accordance with our earlier observa-
tion. Equation (4.18) becomes
which gives the sequence of equations
and so on. The initial conditions, (4.19), become
as far as terms The solution to (4.20a), with (4.21) and (4.22), is
The problem for can be written
where with
