  This paper is a sequel to Takahashi et al. (2016), in which the authors
investigated the influences of fluctuations in pre-shock accreting matter on
the linear stability of the standing accretion shock in core-collapse
supernovae (CCSNe). If one understands that this was concerning the effect of
the outer boundary condition for the post-shock accretion flows, the present
paper should be regarded as an investigation on the possible influences of the
inner boundary conditions. More specifically, we impose a time-dependent,
oscillating condition at the inner boundary, having in mind the injection of
acoustic power by an oscillating proto-neutron star. We also consider possible
correlations between the inner and outer boundary conditions as invoked in the
argument for Lepton-number Emission Self-sustained Asymmetry, or LESA. In this
paper, we conduct the linear stability analysis of the standing accretion shock
commonly encountered in CCSNe based on Laplace transform. We find that the
acoustic power enhances the standing accretion shock instability, or SASI,
especially when the luminosity is low. On the other hand, the correlation
between the fluctuations of neutrino luminosity at the neutrino sphere has
little influences on the instability, changing the amplitudes of eigenmodes
only slightly. We further investigate steady solution of perturbation
equations, being motivated by LESA, and conclude that not the difference but
the sum of the fluxes of electron-type neutrinos and anti-neutrino is the key
ingredient to production of the self-sustained steady perturbed configuration.
