  We provide a quantitative characterization of dissipative effects in
one-dimensional granular crystals. We use the propagation of highly nonlinear
solitary waves as a diagnostic tool and develop optimization schemes that allow
one to compute the relevant exponents and prefactors of the dissipative terms
in the equations of motion. We thereby propose a quantitatively-accurate
extension of the Hertzian model that encompasses dissipative effects via a
discrete Laplacian of the velocities. Experiments and computations with steel,
brass, and polytetrafluoroethylene reveal a {\em common} dissipation exponent
with a material-dependent prefactor.
