The propagation of wave-trains resulting from a local external trigger inside a network described by a metric-graph is analyzed using quantum graph theory. The external trigger is a finite-time perturbation imposed at one vertex of the graph, leading to a consecutive wave-train into the network, supposedly at rest before the applied external perturbation. A complete analytical solution for the induced wave-train is found having a specific spectrum as well as mode's amplitudes. Furthermore the precise condition by which the external trigger can transfer a maximal energy to any specific natural mode of the quantum graph is derived. Finally, the wave damping associated with boundary-layer dissipation is computed within a multiple times-scale asymptotic analysis. Exponential damping rates are explicitly found related to their corresponding mode's eigenvalue. Each mode energy is then obtained, as well as their exponential damping rate. The relevance of these results to the physics of waves within networks are discussed.