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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 s_n=-\int_0^\pi  \alpha^2sin^2(nx)\mathrm{ln} \alpha^2sin^2(nx) \; dx =\pi\alpha^2\left(\mathrm{ln}\left(\frac{2}{\alpha}\right)-\frac{1}{2}\right) \end{equation}