In this paper a mathematical model for the study of the interaction of longitudinal and transversal vibrations in a stretched string is presented. The study implies an existence theory for time periodic transversal vibrations generated by a horizontal excitation of one of the end-points of the string. The conditions for the existence of this parametrically excited time periodic vibrations are evaluated in a practical application.

The innovative character of the results obtained concern the application of an operator method to a system of nonlinearly coupled wave equations modeling the dynamical behaviour of a strectched string where unite elasticity is taken into account.

It may be known that in the literature little attention has been paid to a rigorous analysis of time periodic solutions for systems of partial differential equations.

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