This paper is a sequel to Takahashi et al., in which the authors investigated the influences of fluctuations in preshock 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 postshock accretion flows, the present paper should be regarded as an investigation of 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 (LESA). In this paper, we conduct the linear stability analysis of the standing accretion shock commonly encountered in CCSNe on the basis of Laplace transform. We find that the acoustic power enhances the standing accretion shock instability (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 influence 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 ν e and is the key ingredient to production of the self-sustained steady perturbed configuration.
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