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for any value of $n$! That is the entropy of each state is the same, independant of the level of excitation.  When adjusting for waves of any amplitude $\alpha$ we get \begin{equation}  \int_0^\pi \mathrm{ln}\;\alpha\mathrm{sin}^{-2\mathrm{sin}^2(nx)}(nx)\;dx=\frac{\pi}{2}(\mathrm{ln}(4\alpha^2)-1) \alpha^2sin^2(nx)\mathrm{ln} \alpha^2sin^2(nx) \; dx =\frac{\pi\alpha^2}{2}(1-\mathrm{ln}(4\alpha^2))  \end{equation}