Propagation of a Longitudinal Disturbance on a One-Dimensional Lattice

Abstract

Propagation of a particular longitudinal disturbance on a semi-infinite one-dimensional lattice of identical mass particles coupled by identical massless ideal springs is studied. With all but one of the particles at rest and all the connecting springs at equilibrium length at t = 0, the end particle is constrained to move with fixed longitudinal velocity for all t ≥ 0. Use is made of a method in which an infinite system of differential equations of motion is replaced by a single ordinary differential equation; the time-dependent Fourier coefficients of the equation's solution are the respective displacements from equilibrium of the individual lattice particles. These displacements and the corresponding velocities are expressed in terms of definite integrals. The results of velocity computations are presented along with asymptotic velocity distributions on the lattice and are discussed in relation to a simplified propagation description posited in a recent introductory textbook. The case of a semi-infinite elastic continuum subject to essentially the same initial and boundary conditions is worked out as a limit of the solution of the problem for the lattice.