02-1D Periodic Bravais Lattice

The easiest way to picture a Bravais Lattice is to picture lined up atoms attached to each other with springs as shown in Figure:

The atoms have equal mass M, and are separated by distance a. The springs are mass less with a spring constant K. It is easy to imagine vibrations being sent through such an arrangement, as a small displacement in one atom will displace its neighbors, etc.

Periodic boundary conditions allow us discard anomalies that can pop up at end boundaries. To picture such an arrangement, we can simply attach the ends of the chain together into a circle, like so:

This is called Born-von Karman's Boundary Conditions, where the first position is attached to the last.