On the spectrum of the~discrete inhomogeneous wave equation, and vibrations of a~discrete string

L. D. Pustyl'nikov

Sbornik. Mathematics, Tome 75 (1993) no. 2, pp. 317-331

Résumé

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.