Wave propagation and complexity: a transfer operator approach

We consider wave dynamics on networks of beams/plates coupled along 1D joints. This set-up can be mapped onto the wave dynamics on graphs and is introduced here as an extension to generic wave graph systems such as studied in quantum graph theory. In particular, we consider the elastic case which entails different mode-types (bending, longitudinal and shear waves) which propagate at different wave speeds and can mix at interfaces. The bending modes are described in terms of 4th order equations introducing an always evanescent wave component into the system. The scattering matrices describing reflection/transmission at interfaces thus contain both propagating (open) and evanescent (closed) channels. As a result, the scattering matrices and the transfer operator are no longer unitary; the consequences of this non-unitaritiness on secular equations and the Weyl law will be discussed. The findings are of relevance to describing complex engineering structures. We note that existing methods used to solve wave propagation problems often provide average solutions. As well as the aforementioned extension of quantum graphs to the elastic case, we consider fluctuations about this mean solution. This is done by propagating correlation functions on graphs; it turns out that this provides a suitable wave analogue of ray methods. This approach allows us to investigate response statistics and distributions; these properties are of real significance in, for example, the automotive industry