  We investigate the properties of the gravitational waves (GWs) generated
during a strongly first order electroweak phase transition (EWPT) in models
with the classical scale invariance (CSI). Here, we distinguish two parameter
space regions that correspond to the cases of (1) light dilaton and (2) purely
radiative Higgs mass (PRHM). In the CSI models, the dilaton mass, or the Higgs
mass in the PRHM case, in addition to some triple scalar couplings are fully
triggered by the radiative corrections (RCs). In order to probe the RC effects
on the EWPT strength and on the GW spectrum, we extend the standard model by a
real singlet to assist the electroweak symmetry breaking and an additional
scalar field $Q$ with multiplicity $N_Q$ and mass $m_Q$. After imposing all
theoretical and experimental constraints, we show that a strongly first order
EWPT with detectable GW spectra can be realized for the two cases of light
dilaton and PRHM. We also show the corresponding values of the relative
enhancement of the cross section for the di-Higgs production process, which is
related to the triple Higgs boson coupling. We obtain the region in which the
GW spectrum can be observed by different future experiments such as LISA and
DECIGO. We also show that the scenarios (1) and (2) can be discriminated by
future GW observations and measurements of the di-Higgs productions at future
colliders.
