  Sr$_2$RuO$_4$ is a prototypical multi-band superconductor with three bands
crossing the Fermi level. These bands exhibit distinct dimensional
characteristics, with one quasi-2D $\gamma$-band and two quasi-1D $\alpha$- and
$\beta$-bands. This leads to the expectation that the superconductivity on the
$\gamma$-band may be only weakly Josephson-coupled to that on the other two
bands. Based on an explicit microscopic weak coupling calculation appropriate
for Sr$_2$RuO$_4$, we study the collective Leggett modes associated with the
relative phase oscillations between the bands and show that a relatively soft
Leggett mode exists due to the comparatively weaker inter-band Josephson
coupling. These calculations also provide insight into why the superconducting
gap magnitudes may be comparable on all three bands, despite the noticeable
differences between the $\gamma$ and $\alpha, \beta$ bands. The analyses can be
readily applied to other multi-band superconductors.
