  We revisit the effects of an early matter-dominated era on gravitational
waves induced by scalar perturbations. We carefully take into account the
evolution of the gravitational potential, the source of these induced
gravitational waves, during a gradual transition from an early matter-dominated
era to the radiation-dominated era, where the transition timescale is
comparable to the Hubble time at that time. Realizations of such a gradual
transition include the standard perturbative reheating with a constant decay
rate. Contrary to previous works, we find that the presence of an early
matter-dominated era does not necessarily enhance the induced gravitational
waves due to the decay of the gravitational potential around the transition
from an early matter-dominated era to the radiation-dominated era.
