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We have investigated the effects of a smooth transition layer at the contact discontinuity on the growth of the Richtmyer-Meshkov instability (RMI) by hydrodynamic numerical simulations, and we derived an empiricalcondition for the suppression of the instability. The transition layer has little influence on the RMI when the thickness L is narrower than the wavelength of an interface modulation lambda. However, if the transition layer becomes broader than lambda, the perturbed velocity associated with the RMI is reduced considerably. The suppression condition is interpreted as the cases in which the shock transit time through the transition layer is longer than the sound crossing time of the modulation wavelength. The fluctuation kinetic energy decreases as L−p with p = 2.5, which indicates that the growth velocity of the RMI decreases in proportion to L−p/2 by the presence of the transition layer. This feature is found to be quite universal and appeared in a wide range of shock-interface interactions.