Analytical studies of particle dynamics in bending waves in planetary   rings
  Particles inside a planetary ring are subject to forcing due to the central planet, moons in inclined orbits, self-gravity of the ring and other forces due to radiation drag, collisional effects and Lorentz force due to magnetic field of the planet. We write down the equations of motion of a single particle inside the ring and solve them analytically. We find that the importance of the shear caused by variation of the radial velocity component with local vertical direction cannot be ignored and it may be responsible for damping of the bending waves in planetary rings as observed by the Voyager data. We present the wave profile resulting from the dissipation. We estimate that the surface mass density of the C ring to be of the order of $\sigma \sim 1.2-1.6$gm cm$^{-2}$, and the height $h \sim 2.2-2.4$m. These theoretical results are in agreement with observations. 
A. Bhattacharyya and Sandip K. Chakrabarti