  In this paper we examine in detail the procedure of averaging of the local
field-theoretic Poisson brackets proposed by B.A. Dubrovin and S.P. Novikov for
the method of Whitham. The main attention is paid to the questions of
justification and the conditions of applicability of the Dubrovin-Novikov
procedure. Separate consideration is given to special features of single-phase
and multiphase cases. In particular, one of the main results is the
insensitivity of the procedure of bracket averaging to the appearance of
"resonances" which can arise in the multi-phase situation.
