Connectivity in waves and vibrations: One-to-six, one-to-all, all-to-all and random connections

This paper develops a theory of propagation based on connectivity templates. Connectivity describes how the elastic connections distribute. A visual counterpart is the structure of the stiffness matrix. D'Alembert equation refers to classical elasticity based on closest neighbors connectivity and is characterized by propagation of waves, which can be classified as one-to-six, since each particle of the scheme is connected only with two other particles for each direction. However, very different connectivity schemes can be introduced, e.g. a one-to-all connectivity scheme, in which one particle can be connected with a cluster of particles, or all-to-all where each particle is connected with any other. Moreover, connections are not instantaneous: the information flows is delayed due to the connection length. Waves exhibit unbelievable behaviour changing the system connectivity. Nondissipative structures shows damping. Energy can propagate backwards in respect to wave direction. Waves can stop or localize at some points. Negative mass effect can emerge. These effects will be discussed in the present paper.