As an important application and extension of the foregoing ideas, and to obtain a first glimpse of wave phenomena, we consider the following system. Suppose we have N identical particles of mass m in a line, with each particle bound to its neighbors by a Hooke’s law force, with “spring constant” k. Let us assume the particles can only be displaced in one-dimension; label the displacement from equilibrium for the jth particle by qj , j = 1, ...,N. Let us also assume that particle 1 is attached to particle 2 on the right and a rigid wall on the left, and that particle N is attached to particle N 1 on the left and another rigid wall on the right.
linear chain, coupled oscillator, chapter 4, four
Torre, Charles G., "04 Linear Chain of Coupled Oscillators" (2012). Foundations of Wave Phenomena. Book 19.