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We report calculations of the intensities of rovibrational transitions of H₂ emitted from C-type shock waves propagating in molecular gas. Attention was paid to the thermal balance of the gas and to the rates of collisional dissociation and ionization of H₂. We found that the maximum shock speeds which can be attained, prior to the collisional dissociation of H₂ (which results in a sonic point in the flow and hence a J-type shock wave), can be much higher than had previously been believed. Thus, adopting the ‘standard’ scaling of the transverse magnetic induction with the gas density, B(μG)=[n_H(cm^-3)]^1/2, we established that the maximum shock speed increased from 20–30 km s^-1 at high pre-shock densities (n_H≥10⁶ cm^-3) to 70–80 km s^-1 at low densities (n_H≤10⁴ cm^-3). The critical shock speed, v_crit, also increases significantly with the transverse magnetic induction, B, at a given pre-shock gas density, n_H …