Transient motions of bi-lattices

We develop formal solutions for the propagation of transient pulses on a variety of bi-lattice models. The lattices are composed of a finite homogeneous chain connected in series with a different semi-infinite homogeneous chain at a common location occupied by a single mass which is different from the masses of both chains. Exact analytic solutions of this general case are not possible. Some analytic solutions are, however, possible for a variety of special cases. The general solutions are illustrated by numerically inverting the Laplace transform functions. The exact solutions are found to correlate very well with the numerical inversion scheme. Such correlations give confidence in the numerical scheme's predictions of the solutions of the more complicated chains.