  Gravitational waves (GWs) induced by primordial fluctuations can be affected
by the modification of the sound speed $c^2_{\rm s}$ and the equation of state
parameter $w$ once the curvature fluctuations reenter the cosmological horizon.
That softening can also significantly boost the production of Primordial Black
Holes (PBHs) at the mass scale where the softening arises. In this work, we
consider a hypothetical softening of $w$ and $c^2_{\rm s}$ caused by a smooth
crossover beyond Standard Model theories, for what we numerically compute the
secondary induced GW considering the case of a flat scale-invariant power
spectrum. We find that if the amplitude of the power spectrum is sufficiently
large, the characteristic feature of the GW signal caused by the smooth
crossover can be detected by future space-based gravitational wave
interferometers and differentiated from the pure radiation case. At the same
time, depending on the mass scale where the crossover occurs, such a scenario
can have compatibility with PBHs being all the dark matter when $\mathcal{A}
\sim \mathcal{O}(10^{-3})$, with a mass function very sharply peaked around the
horizon mass scale of the minimum of the sound speed. Our results show that the
GW signal can be used to resolve the existing degeneracy of sharply peaked mass
function caused by peaked power spectrums and broad ones in the presence of
softenings of $w$ and $c^2_{\rm s}$.
