Periodic strings: A mechanical analogy to photonic and phononic crystals

We present a theoretical study of a periodic vibrating string composed of a finite sequence of string segments connected periodically, with each segment characterized by a constant linear mass density. The main purpose is to provide a model that can mimic the properties of photonic or phononic crystals. This system displays frequency intervals for which wave propagation is not allowed (frequency bandgaps), in close analogy to photonic and phononic crystals. We discuss the behavior of these bandgaps when varying physical parameters, such as the values of the linear mass densities, the oscillation frequency, and the number of string segments constituting the entire system.