  We investigate the criterion for the acoustic mechanism to work successfully
in core-collapse supernovae. The acoustic mechanism is an alternative to the
neutrino-heating mechanism. It was proposed by Burrows et al., who claimed that
acoustic waves emitted by $g$-mode oscillations in proto-neutron stars (PNS)
energize a stalled shock wave and eventually induce an explosion. Previous
works mainly studied to which extent the $g$-modes are excited in the PNS. In
this paper, on the other hand, we investigate how strong the acoustic wave
needs to be if it were to revive a stalled shock wave. By adding the acoustic
power as a new axis, we draw a critical surface, an extension of the critical
curve commonly employed in the context of neutrino heating. We perform both 1D
and 2D parametrized simulations, in which we inject acoustic waves from the
inner boundary. In order to quantify the power of acoustic waves, we use the
extended Myers theory to take neutrino reactions into proper account. We find
for the 1D simulations that rather large acoustic powers are required to
relaunch the shock wave, since the additional heating provided by the secondary
shocks developed from acoustic waves is partially canceled by the neutrino
cooling that is also enhanced. In 2D, the required acoustic powers are
consistent with those of Burrows et al. Our results seem to imply, however,
that it is the sum of neutrino heating and acoustic powers that matters for
shock revival.
