Solution of initial-value problems for some infinite chains of harmonic oscillators

Abstract

Procedures are given for solving the equations of motion of infinite and half-infinite chains of linear spring-mass oscillators with nearest-neighbor coupling. Arbitrary initial displacements and initial velocities may be prescribed for any finite number of the masses in the chains, and external forces may be applied to any finite number of them. The solutions appear as integrals with integrands involving orthogonal polynomials generated by three-term recurrence relations whose coefficients are determined from the equations of motion. Tables of the polynomicals needed for solving the equations of motion of a number of special chains are included.