New results on electron heat wave propagation using ion cyclotron resonance heating power modulation in the Joint European Torus (JET) [P. H. Rebut , Nucl. Fusion 25, 1011 (1985)] plasmas characterized by internal transport barriers (ITBs) are presented. The heat wave generated outside the ITB, and traveling across it, always experiences a strong damping in the ITB layer, demonstrating a low level of transport and loss of stiffness. In some cases, however, the heat wave is strongly inflated in the region just outside the ITB, showing features of convective-like behavior. In other cases, a second maximum in the perturbation amplitude is generated close to the ITB foot. Such peculiar types of behavior can be explained on the basis of the existence of a critical temperature gradient length for the onset of turbulent transport. Convective-like features appear close to the threshold (i.e., just outside the ITB foot) when the value of the threshold is sufficiently high, with a good match with the theoretical predictions for the trapped electron mode threshold. The appearance of a second maximum is due to the oscillation of the temperature profile across the threshold in the case of a weak ITB. Simulations with an empirical critical gradient length model and with the theory based GLF23 [R. E. Waltz , Phys. Plasmas, 4, 2482 (1997)] model are presented. The difference with respect to previous results of cold pulse propagation across JET ITBs is also discussed. (C) 2007 American Institute of Physics.