  We use the Discrete Element Method (DEM) to understand the underlying
attenuation mechanism in granular media, with special applicability to the
measurements of the so-called effective mass developed earlier. We consider
that the particles interact via Hertz-Mindlin elastic contact forces and that
the damping is describable as a force proportional to the velocity difference
of contacting grains. We determine the behavior of the complex-valued normal
mode frequencies using 1) DEM, 2) direct diagonalization of the relevant
matrix, and 3) a numerical search for the zeros of the relevant determinant.
All three methods are in strong agreement with each other. The real and the
imaginary parts of each normal mode frequency characterize the elastic and the
dissipative properties, respectively, of the granular medium. We demonstrate
that, as the interparticle damping, $\xi$, increases, the normal modes exhibit
nearly circular trajectories in the complex frequency plane and that for a
given value of $\xi$ they all lie on or near a circle of radius $R$ centered on
the point $-iR$ in the complex plane, where $R\propto 1/\xi$. We show that each
normal mode becomes critically damped at a value of the damping parameter $\xi
\approx 1/\omega_n^0$, where $\omega_n^0$ is the (real-valued) frequency when
there is no damping. The strong indication is that these conclusions carry over
to the properties of real granular media whose dissipation is dominated by the
relative motion of contacting grains. For example, compressional or shear waves
in unconsolidated dry sediments can be expected to become overdamped beyond a
critical frequency, depending upon the strength of the intergranular damping
constant.
