Computations for sharp cones, using the e N method with N=10, show that the first oblique Tollmien-Schlichting mode is responsible for transition at adiabatic wall conditions for freestream Mach numbers up to about 7. For cold walls, the two-dimensional second mode dominates the transition process at lower hypersonic Mach numbers due to the well-known destabilizing effect of cooling on the second mode
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Mujeeb R. Malik (1989) studied this question.