  We present a numerical study of the global modes of oscillation of thick
accretion discs around black holes. We have previously studied the case of
constant distributions of specific angular momentum. In this second paper, we
investigate (i) how the size of the disc affects the oscillation
eigenfrequencies, and (ii) the effect of power-law distributions of angular
momentum on the oscillations. In particular, we compare the oscillations of the
disc with the epicyclic eigenfrequencies of a test particle with different
angular momentum distributions orbiting around the central object. We find that
there is a frequency shift away from the epicyclic eigenfrequency of the test
particle to lower values as the size of the tori is increased. We have also
studied the response of a thick accretion disc to a localized external
perturbation using non constant specific angular momentum distributions within
the disc. We find that in this case it is also possible (as reported previously
for constant angular momentum distributions) to efficiently excite internal
modes of oscillation. In fact we show here that the local perturbations excite
global oscillations (acoustic p modes) closely related to the epicyclic
oscillations of test particles. Our results are particularly relevant in the
context of low mass X-ray binaries and microquasars, and the high frequency
Quasi-Periodic Oscillations (QPOs) observed in them. Our computations make use
of a Smooth Particle Hydrodynamics (SPH) code in azimuthal symmetry, and use a
gravitational potential that mimics the effects of strong gravity.
