Normal modes of a uniformly loaded finite string (${\omega }_j^2=\frac{4\tau }{m\ell }{\sin }^2\frac{j\pi }{2(N+1)}$)

Take $N$ equal point masses $m>0$ on a massless string, with fixed ends, equal horizontal spacing $\ell >0$ and constant tension $\tau >0$. Small transverse displacements obey $m{\ddot{z}}_i=(\tau /\ell )(z_{i-1}-2z_i+z_{i+1})$, with $z_0=z_{N+1}=0$. The normal modes arewith the displayed positive frequencies. Substitution uses $\sin ((i-1)q)+\sin ((i+1)q)=2\cos q\sin (iq)$. The endpoint conditions quantize $q=j\pi /(N+1)$. These are the discrete sine transform modes and form an orthogonal basis, so each initial displacement and velocity decomposes into independent harmonic oscillations. For three masses, convenient relative shapes are $(1,\sqrt{2},1)$, $(1,0,-1)$ and $(1,-\sqrt{2},1)$, with $m\ell {\omega }^2/\tau =2-\sqrt{2},2,2+\sqrt{2}$ respectively.