  The leading order "temperature" of a dense two dimensional granular material
fluidised by external vibrations is determined. An asymptotic solution is
obtained where the particles are considered to be elastic in the leading
approximation. The velocity distribution is a Maxwell-Boltzmann distribution in
the leading approximation. The density profile is determined by solving the
momentum balance equation in the vertical direction, where the relation between
the pressure and density is provided by the virial equation of state. The
predictions of the present analysis show good agreement with simulation results
at higher densities where theories for a dilute vibrated granular material,
with the pressure-density relation provided by the ideal gas law, are in error.
The theory also predicts the scaling relations of the total dissipation in the
bed reported by McNamara and Luding (PRE v 58, p 813).
