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In supernovae neutrinos are emitted from a region with a width $r_{\rm eff}$of a few kilometers (rather than from a surface of infinitesimal width). Westudy the effect of integration (averaging) over such an extended emissionregion on collective oscillations. The averaging leads to additionalsuppression of the correlation (off-diagonal element of the density matrix) bya factor $ \sim 1/r_{\rm eff} V_e \sim 10^{-10}$, where $V_e$ is the matterpotential. This factor enters the initial condition for further collectiveoscillations and, consequently, leads to a delay of the strong flavourtransitions. We justify and quantify this picture using a simple example ofcollective effects in two intersecting fluxes. We have derived the evolutionequation for the density matrix elements integrated over the emission regionand solved it both numerically and analytically. For the analytic solution wehave used linearized equations. We show that the delay of the development ofthe instability and the collective oscillations depends on the suppressionfactor due to the averaging (integration) logarithmically. If the instabilitydevelops inside the production region, the integration leads not only to adelay but also to a modification of the exponential grow.

Effect of extended neutrino production region on collective oscillations in supernovae