ABSTRACT
Consider a mechanical system consisting of a linear array of N identical masses m that are free to slide in one dimension over a frictionless horizontal surface. Suppose that the masses are coupled to their immediate neighbors via identical light springs of unstreched length a and force constant K. (Here, we employ the symbol K to denote the spring force constant, rather than k, because k is already being used to denote wavenumber.) Let x measure distance along the array (from the left to the right). If the array is in its equilibrium state then the x-coordinate of the ith mass is xi = i a, for i = 1,N. Consider longitudinal oscillations of the masses, namely, oscillations for which the x-coordinate of the ith mass is
xi = i a + ψi(t), (5.1)
where ψi(t) represents longitudinal displacement from equilibrium. It is assumed that all of the displacements are relatively small, that is, |ψi| a, for i = 1,N.
