Solution
the density of states for a linear atomic chain is (as previously stated)
D( \omega )=\frac{ L }{ \pi } \frac{ dk }{d \omega}
that in the Debye's assumption of \omega = v k , becomes
D( \omega )=\frac{ L }{ \pi } \frac{ dk }{d \omega} ≈ \frac{ L }{v \pi }
the total...