4/3-Law of Granular Particles Flowing through a Vertical Pipe
  Density waves of granular material (sand) flowing through a vertical pipe have been investigated. Clear density waves emerge when the cock attached to bottom end of the pipe is closed. The FFT power spectra were found to show a stable power-law form $ P(f) \sim f^{-\alpha}. $ The value of the exponent was evaluated as $\alpha \cong 4/3$. We also introduce a simple one-dimensional model which reproduces $\alpha = 4/3$ from both simulation and theoretical analysis. (to be published in Phys.Rev.Lett.) 
Osamu Moriyama, Naoya Kuroiwa and Mitsugu Matsushita (Chuo Univ.)   Hisao Hayakawa (Kyoto Univ.)