  Collective oscillations of supernova neutrinos above the neutrino sphere can
be completely described by the propagation of individual neutrinos in external
potentials and are in this sense a linear phenomenon. An effective theory of
collective oscillations can be developed based on certain assumptions about
time dependence of these potentials. General conditions for strong flavor
transformations are formulated and these transformations can be interpreted as
parametric resonance effects induced by periodic modulations of the potentials.
We study a simplified and solvable example, where a probe neutrino is
propagating in a flux of collinear neutrinos, such that $\nu \nu-$ interactions
in the flux are absent. Still, this example retains the main feature - the
coherent flavor exchange. Properties of the parametric resonance are studied,
and it is shown that integrations over energies and emission points of the flux
neutrinos suppress modulations of the potentials and therefore strong
transformations. The transformations are also suppressed by changes in
densities of background neutrinos and electrons.
