The response of a thick, sinusoidally distorted interface between two gases of different density to impulsive acceleration by a sequence of shock waves propagating normal to the plane of the undistorted interface has been studied experimentally and theoretically. A simple analytical model is proposed for the effect of the finite thickness on the linear growth of the disturbances while the amplitude is small. Experiments in a vertical shock tube show that the growth of distortions intially of order 0.1 mm on 1.8 cm thick light‐heavy interfaces induced by weak (1.12<Ms<1.66) shock waves is not measurable, but growth after the first and second reshocks is. The measured growth rates are presented and the initial distortions are inferred from the theory. Several effects of the wall boundary layers are described.