How sandpiles spill: Sandpile problem in a thick flow regime

We obtain an analytical solution of a one-dimensional sandpile problem in a thick flow regime, when it can be formulated in terms of linear equations. It is shown that a space periodicity takes place during the sandpile evolution even for a sandpile of only one type of particles. Similar periodicity was observed previo usly for many component sandpiles. Space periods are proportional to an input flow of particles $r_0$. We find that the surface angle $\theta$ of the pile reaches it's final critical value ($\theta_f$) from lower values only at long times. The deviation ($\theta_f - \theta$) behaves as $(t/r_{0})^{-1/2}$.