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Synopsis

Chains of identical coupled oscillators are often used in undergraduate waves and oscillations courses to connect the standing wave to the spring-mass system. This connection often involves a non-rigorous continuum approximation — that a large number of coupled oscillators essentially act like waves. The classic illustration of this fact is to simulate a large number of transverse coupled oscillators and to notice that their amplitudes line up with the solution to a standing wave. When I was doing my undergraduate degree, I found an analytic solution to the n-coupled oscillator problem. This solution allows for the connection to the standing wave to be rigorously derived using only basic linear algebra and calculus. My talk will discuss the solution, which involves a class of polynomials called the Chebyshev polynomials of the second kind, and how their properties very elegantly give rise to the behaviour of standing waves.