<p>Hypersonic boundary-layer receptivity to freestream entropy and vorticity waves is investigated using direct numerical simulations for a Mach 6 flow over a 5.08 mm nose radius cone. Two frequencies of 33 kHz and 150 kHz are considered to be representative of the first and second instability modes, respectively. For the first mode, wall pressure fluctuations for both entropy and vorticity wave cases exhibit a strong modulation yet without a growing trend, indicating that the first mode is not generated despite its instability predicted by linear stability theory. The potential reason for this is the absence of postshock slow acoustic waves capable of synchronizing with the first mode. By contrast, for the second mode, a typical three-stage boundary-layer response is observed, consistent with that to slow acoustic waves studied previously. Furthermore, the postshock disturbances outside the boundary layer can be decomposed into the entropy (density/temperature fluctuations) and vorticity components (velocity fluctuations), and the latter is shown to play a leading role in generating the second mode, even for the case with entropy waves where the density/temperature fluctuations dominate the postshock regions.</p>