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and so on. We will take the solution of (4.65) to be (cf. (4.59))

and this requires

The equation for (4.66), will include terms generated by which involve
or —and all these must be removed if is to be periodic. The terms in
give

and those in

leaving

The two equations, (4.69) and (4.70), which ensure the removal of secular terms,
look rather daunting, but quite a lot can be done with them. However, we first need
to introduce two familiar properties of a propagating wave. One is the speed at which
the underlying wave—the carrier wave—travels, usually called the phase speed. This is
defined as the speed at which lines move i.e. lines such that

so

The other—and for us, the far more significant—property is the speed at which the
energy propagates (and therefore, for example, the speed at which the amplitude
modulation moves); this is defined as the group speed. From our result
in (4.68), we have