  The linear stability of rapid granular flow on a slope under gravity against
the longitudinal perturbation is analyzed using hydrodynamic equations. It is
demonstrated that the steady flow uniform along the flow direction becomes
unstable against the long-wavelength perturbations longitudinal to the flow
direction for certain parameter ranges to form the density wave, in contrast
with the finite wavelength instability against the transverse perturbation
(Forterre & Pouliquen 2002). It is shown that the instability can be understood
as the the long-wave instability of the kinematic waves in a quasi-one
dimensional system. The results are compared with our previous molecular
dynamics simulations (Mitarai & Nakanishi 2001), where the spontaneous density
wave formation has been found.
