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178 4. The method of multiple scales

we obtain

Before we proceed with the details, we should make an important observation. The
transform, (4.61), defining the fast scale (usually called the phase in these problems),
implies a consistency condition that must exist if is a twice-differentiable function,
namely

this additional equation is called the conservation of waves (or of wave crests), and it
arises quite naturally from an elementary argument. Consider (one-dimensional) waves
entering and leaving the region the number of waves, per unit time, crossing
into the region is given as and the number leaving, across x, we will write as
The total number of waves (wave crests) between and x is if the number of
waves does not change (which is what is typically observed, even if they change shape)
then

or, upon allowing differentiation with respect to x,

which immediately recovers (4.63) if the dependence on (x, t) is via (X, T).
Returning to equation (4.62), we seek a solution

with

and then we obtain the sequence of equations