  As potential candidates of dark matter, primordial black holes (PBHs) are
within the core scopes of various astronomical observations. In light of the
explosive development of gravitational wave (GW) and radio astronomy, we
thoroughly analyze a stochastic background of cosmological GWs, induced by over
large primordial density perturbations, with several spikes that was inspired
by the sound speed resonance effect and can predict a particular pattern on the
mass spectrum of PBHs. With a specific mechanicsm for PBHs formation, we for
the first time perform the study of such induced GWs that originate from both
the inflationary era and the radiation-dominated phase. We report that, besides
the traditional process of generating GWs during the radiation-dominated phase,
the contribution of the induced GWs in the sub-Hubble regime during inflation
can become significant at critical frequency band because of a narrow resonance
effect. All contributions sum together to yield a specific profile of the
energy spectrum of GWs that can be of observable interest in forthcoming
astronomical experiments. Our study shed light on the possible joint probe of
PBHs via various observational windows of multi-messenger astronomy, including
the search for electromagnetic effects with astronomical telescopes and the
stochastic background of relic GWs with GW instruments.
