Lattice Waves: Atom Chain Fixed at Ends or Free is Space?

dimensionless

My textbook treats lattice waves as if they are waves propagating through a mass-spring chain. Kind of like this:

M -/\/\/- M -/\/\/- M -/\/\/- M

The book talks about the motion of the nth atom. It says that the motion of the nth atom has been simplified by only considering the motion of its nearest neighbors. Should I treat the chain of atoms as though it is floating free in space(as shown above)? Should I instead consider the chain as if the ends were fixed to walls by springs(as shown below)?

|||-/\/\/- M -/\/\/- M -/\/\/- M -/\/\/- M -/\/\/-|||

Should I not use either of these simplifications? I am investigating the dispersion curves.

Dr Transport

Usually the chain is infinite and the ends are not considered. If they are, you need to worry about the BC's and then you get standing waves.