Download Full Text (490 KB)
Our example of a chain of oscillators is nice because it is easy to visualize such a system, namely, a chain of masses connected by springs. But the ideas of our example are far more useful than might appear from this one simple mechanical model. Indeed, many materials (including solids, liquids and gases) have some aspects of their physical response to (usually small) perturbations behaving just as if they were a bunch of coupled oscillators — at least to a first approximation. In a sense we will explore later, even the electromagnetic field behaves this way! This “harmonic oscillator” response to perturbations leads — in a continuum model — to the appearance of wave phenomena in the traditional sense. We caught a glimpse of this when we examined the normal modes for a chain of oscillators with various boundary conditions. Because the harmonic approximation is often a good first approximation to the behavior of systems near equilibrium, you can see why wave phenomena are so ubiquitous. The key di↵erence between a wave in some medium and the examples of §4 is that wave phenomena are typically associated with propagation media (stone, water, air, etc.) which are modeled as continuous rather than discrete. As mentioned earlier, our chain of oscillators in §4 can be viewed as a discrete model of a continuous (one-dimensional) material. We now want to introduce a phenomenological description of the material in which we ignore the atomic discreteness of matter.
continuum limit, wave equation, chain oscillator, chapter 5, five
Torre, Charles G., "05 The Continuum Limit and the Wave Equation" (2014). Foundations of Wave Phenomena. 18.