  It is shown that spatially periodic one-dimensional surface waves in shallow
water behave almost linearly, provided large part of the energy is contained in
sufficiently high frequencies. The amplitude is not required to be small (apart
from the shallow water approximation assumption) and the near-linear behavior
occurs on a much longer time scale than might be anticipated based on the
amplitude size. Heuristically speaking, this effect is due to the nonlinearity
getting averaged by the dispersive action.
  This result is obtained by an averaging procedure, which is briefly outlined,
and is also confirmed by numerical simulations.
