= Middle-mass-independent mode of a fixed three-mass spring chain
{title2=$q=(1,0,-1)^T,\quad\omega^2=2k/m$}
Three consecutive <masses> $m,M,m$ joined by four identical <springs> of constant $k$ between fixed walls have an antisymmetric <normal mode> with zero central <displacement>. The two adjacent spring forces cancel on the central mass, while each outer mass experiences restoring force $-2kx$. This proves that the displayed <angular frequency> is independent of $M$ for every positive central mass. The two symmetric modes satisfy $(2k-m\omega^2)(2k-M\omega^2)-2k^2=0$ and generally do depend on $M$. For $M=m$, all three <mode shapes> are $(1,\sqrt2,1)$, $(1,0,-1)$ and $(1,-\sqrt2,1)$, with squared frequencies $(2-\sqrt2)k/m$, $2k/m$, $(2+\sqrt2)k/m$.
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