Fixed-end spring-chain determinant and its repeated-root limit ($D_N=k^N{U}_N((2k-m{\omega }^2)/(2k))$)

For $N$ equal masses joined to fixed walls by equal springs, the stiffness matrix has diagonal $2k$ and adjacent entries $-k$. Its characteristic determinant obeys $D_N=d{D}_{N-1}-k^2{D}_{N-2}$, $d=2k-m{\omega }^2$, $D_0=1$, $D_1=d$. If ${\beta }_1+{\beta }_2=d$ and ${\beta }_1{\beta }_2=k^2$, then $D_N=({\beta }_1^{N+1}-{\beta }_2^{N+1})/({\beta }_1-{\beta }_2)$ for distinct roots. At a repeated root $\beta =\pm k$, the correct limit is $(N+1){\beta }^N$, which is nonzero. Thus the apparent ratio solution ${\beta }_1/{\beta }_2=1$ must be excluded. Writing $d=2k\cos q$ gives $D_N=k^N\sin [(N+1)q]/\sin q$, the Chebyshev polynomial of the second kind. There are exactly $N$ distinct positive normal frequencies $2\sqrt{k/m}\sin [j\pi /(2(N+1))]$, $1\le j\le N$.

For $N$ identical masses between fixed walls with equal springs, the dimensionless stiffness matrix has diagonal entries $2$ and adjacent entries $-1$. Its normalized eigenvectors and eigenvalues areExtend each vector by $a_0=a_{N+1}=0$. The sine addition identity gives $2a_i-a_{i-1}-a_{i+1}={\lambda }_j{a}_i$, including the fixed-end rows. The finite sine orthogonality identity supplies the normalization and makes distinct vectors orthogonal. The normal mode angular frequencies are ${\omega }_j=\sqrt{k{\lambda }_j/m}$ for mass $m$ and spring constant $k$. This complements the fixed-end spring-chain determinant and its repeated-root limit with the spatial mode shapes.