The transmission of waves (at least 1-dimensional waves) through a crystal lattice is analogous to electrical waves propagating down a transmission line. The lumped-element model of a parallel wire transmission line where the wires are each a long series string of inductors, and where capacitors bridge across the wires, displays dispersion, frequency cutoff, and other characteristics of waves in a solid. If two different values of inductance or capacitance are used at alternating positions in the electrical model, then it displays the acoustic and optic branches and band gap that you also see in a diatomic solid. Leon Brillouin has a detailed and beautiful description in his book "Wave Propagation in Periodic Structures," where he also gives mechanical analogs of same. I think it's out of print, but because it was reprinted by Dover there are a ton of copies floating around. You should find it at your university library.
Can't help you now, I'm traveling for a couple of weeks.