The interface cohesive behaviour and interface delamination in a bi-material structure consisting of two plates and material interface with zero thickness under time harmonic load is studied. Previously, the authors studied the elasto-brittle interface behaviour, from both sides of a crack, initially normal to the interface, when the elastic-brittle interface debonding appeared. Now, it is again assumed that the restriction for the ratio of energy release rates of the second plate and interface allowing the occurrence of an interface cohesive delamination before the initiation of the normal crack in the second plate is satisfied. The shear lag model is adopted and applied to find the dynamic response of the considered structure, assuming the cohesive interface behaviour, accompanied before of the elastic-brittle one. In both cases, the growth of debond length is not considered e.g. at a given loading condition the corresponding single debond length is found. The inertia forces of the already debonded interface elasto-brittle cracks (mode II) are neglected. The appropriate contact conditions are proposed in order to fit together both elastic and cohesive solutions. The Laplace inverse transform is applied to obtain the original of cohesive debond length by the aid of the already obtained elastic value of debond length at the same loading condition. Parametric analysis of the results obtained is illustrated by examples of the modern ceramic-metal composite on metal substrate. The influence of frequencies and amplitude fluctuations on the cohesive debond length and the interface shear stress distribution are discussed.