Calculating Density of States and Occupied States in 1D Chain of Atoms

[Eats Dirt]

Homework Statement

The system is a chain of atoms in 1D length L and number of atoms N. and
$$\epsilon_k=\hbar c_s k$$
a) What is the density of states?
b)The number of states that can be occupied (use boundary conditions)
c) Determine w_d(I think this is the debye frequency) in terms of N,L,k

The Attempt at a Solution

A. For the density of states I determined dk/(2pi) then changed variables to energy.
$$\frac{1}{c_s\hbar 2\pi}$$

B. From the state density using boundary conditions I really don't have a good idea of how to find the number of occupied states. I think I should start with my solution from part A but I can't seem to make the connection between the physical boundary conditions and occupation.

C. Again I am quite stuck

 

[_coyote_]

here. The only thing I can think of is to use the energy density and somehow find the debye frequency in terms of N,L,k. Any help is greatly appreciated!