The excitation of Alfven and fast magneto-acoustic waves in coronal loops driven by footpoint motions is studied in linear, ideal MHD. The analysis is restricted to azimuthally polarized footpoint motions so that only Alfven waves are directly excited to couple to fast magneto-acoustic waves at later times. In the companion paper De Groof et al. (2002) (hereafter referred to as Paper I), the behaviour of the MHD waves is studied in case of a monochromatic driver. In the present study, the effects of a more realistic random driver are investigated. First, we consider loops of equal length and width in order to limit the number of quasi-modes in the frequency range of the driver so that the influence of quasi-modes in the system can easily be detected. In contrast to the single resonant surface which was found in case of a periodic driver (see Paper I), a random pulse train excites a variety of resonant Alfven waves and consequently the small length scales built up are spread over the whole width of the loop. The specific effects of the quasi-modes are not so easily recognized as for radial footpoint motions (De Groof & Goossens 2000) since the resonances corresponding to directly and indirectly excited Alfven waves are mixed together. In the second part of the paper, longer loops are considered. Since more quasi-modes are involved, the resonant surfaces are more numerous and widely spread throughout the whole loop volume. On the other hand, it takes more time for the MHD waves to cross the loop and to form standing waves. Nevertheless this negative effect does not have too much impact since the simulations show that after a small time interval, resonant surfaces are created all over the loop, with length scales which are short enough for effective dissipation.