Energetic properties of the elastic half-space loaded by a periodic distribution of vibrating punches: The study of the phase shift influence

We develop an analytical approach to study the wave properties of an elastic half-space subjected to harmonic vibrations applied on its free surface by a periodic array of rigid punches. Contrary to previous investigations, it is assumed that any neighbouring pair of these oscillates with a (common) phase shift, thus implying rather specific behaviours of the structure. Starting from integral equations for the contact stress and representation formulas for the wave field in both the anti-plane and in-plane problems, suitable mild approximations on the kernels allow analytical solutions of some related (auxiliary) integral equations in a given range of (not too high) frequency. The explicit formulas thus obtained for the wave field are reflected through some figures and enable us to investigate the energetic properties of the structure with respect to different phase shifts. A direct numerical solution of the original integral equations confirms the precision of the analytical solution.