= Normal modes of a uniformly loaded finite string
{title2=$\omega_j^2=\frac{4\tau}{m\ell}\sin^2\!\frac{j\pi}{2(N+1)}$}
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> are
$$
v_i^{(j)}=\sin\frac{ij\pi}{N+1},\qquad j=1,\ldots,N,
$$
with 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 $(1,\sqrt2,1)$, $(1,0,-1)$ and $(1,-\sqrt2,1)$, with $m\ell\omega^2/\tau=2-\sqrt2,2,2+\sqrt2$ respectively.
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