  The signal generated by a weak impulse propagates in an oscillatory way and
dispersively in a gravitationally compacted granular chain. For the power-law
type contact force, we show analytically that the type of dispersion follows
power-laws in depth. The power-law for grain displacement signal is given by
$h^{-1/4(1-1/p)}$ where $h$ and $p$ denote depth and the exponent of contact
force, and the power-law for the grain velocity is $h^{-1/4({1/3}+1/p)}$. Other
depth-dependent power-laws for oscillation frequency, wavelength, and period
are given by combining above two and the phase velocity power-law
$h^{1/2(1-1/p)}$. We verify above power-laws by comparing with the data
obtained by numerical simulations.
