Density of States: Varying Boundary Conditions

[PineApple2]

Consider waves in a box. It is customary to calculate the density of states either by enforcing vanishing boundary conditions, then the wave numbers are
k=\frac{n\pi}{L} and we take only positive k,
or using periodic boundary conditions, in which case k=\frac{2n\pi}{L}
and taking all wave numbers.

My question is - why in the case of vanishing boundary conditions do we take only positive wave numbers? and why in the case of periodic boundary conditions do we take both positive and negative?

 

[Bill_K]

In the first case, vanishing boundary conditions, the solutions are standing waves, and the solutions for +k and -k are exactly the same. So we use just positive k to avoid counting them twice.