Geodesic
On the spectrum of the discrete inhomogeneous wave equation, and vibrations of a discrete string
Sbornik. Mathematics, Tome 75 (1993) no. 2, pp. 317-331 Cet article a éte moissonné depuis la source Math-Net.Ru

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Explicit analytic expressions are found for the spectrum and solutions of the discrete, inhomogeneous wave equation $$ {d^2 q_n \over d t^2}-a_n(q_{n+1}-2q_n+q_{n-1})+\delta_n q_n=0 $$ with boundary conditions $q_0(t) = q_N(t) = 0$, where $n=0,\,1,\,\dots,\,N$, $a_n>0$, and $\delta_n \geqslant 0$. As a corollary a solution is given of the classical problem of finding an explicit analytic expression describing the vibrations of a string all the mass of which is concentrated at a finite number of equidistant points, which was the object of detailed study by Euler, D'Alembert, D. Bernoulli, Lagrange, Sturm, Routh, and others, who gave a solution of it in the particular case where the masses of all points are the same. The general solution of the problem turns out to be connected with a generalized quaternion algebra and properties of certain of its ideals, and this connection is used in an essential way in the proofs of the theorems. 
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     title = {On the spectrum of the~discrete inhomogeneous wave equation, and vibrations of a~discrete string},
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}
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L. D. Pustyl'nikov. On the spectrum of the discrete inhomogeneous wave equation, and vibrations of a discrete string. Sbornik. Mathematics, Tome 75 (1993) no. 2, pp. 317-331. http://geodesic.mathdoc.fr/item/SM_1993_75_2_a1/

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