Density waves of granular material (sand) flowing through a vertical pipe have been investigated. Clear density waves emerge when the cock attached to the bottom end of the pipe is closed. The FFT power spectra were found to show a stable power-law form P(f)~f^-α. The value of the exponent was evaluated as α4/3. We also introduce a simple one-dimensional model which reproduces α0ex0ex=0ex0ex4/3 from both simulation and theoretical analysis.
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Moriyama et al. (1998) studied this question.
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