  By using the multiple scale method with the simultaneous introduction of
multiple times, we study the propagation of long surface-waves in a shallow
inviscid fluid. As a consequence of the requirements of scale invariance and
absence of secular terms in each order of the perturbative expansion, we show
that the Korteweg-de Vries hierarchy equations do appear in the description of
such waves. Finally, we show that this procedure of eliminating secularities is
closely related to the renormalization technique introduced by Kodama and
Taniuti.
