Unofficial/Short Course Title: Waves
Spring 2025, Deep Springs College, Prof. Brian Hill
The text we will start off with is freely available on-line, so you do not need a physical copy:
If you want a physical copy, get the 3rd edition (Amazon link).
For the physics theory, I will prepare handouts.
In order to do mathematical modeling, we need to learn a serious programming language, and we will use Mathematica for this purpose. Mathematica is a program that you interact with using the “Wolfram Language.” We will learn the language by studying the first half of An Elementary Introduction to the Wolfram Language, 3rd Edition, by Stephen Wolfram. The printed edition is over 300 pages divided into 48 sections. If we do 2-3 sections a class we can do most of the material up to Section 40 during Term 4, at which point you will be in an extremely good position to apply Mathematica to any problem that interests you. In parallel with learning the language we will be learning oscillatory motion and then waves.
What we will do with Mathematica in this course is fundamental physics that all theoretical physicists know very well: oscillations and waves. After we deal with constant motion and constant acceleration, we will animate simple oscillation of a single particle. The classic physical system that exhibits simple oscillation is a mass hanging from a spring. There is a significant increase in complexity when you next put the mass on the end of a pendulum rod.
The next level of complexity is to step it up to two particles. If the two particles are connected, even weakly connected, this leads to all sorts of complex behavior that was not present for either particle separately. The most common example is known as the coupled pendulum. Since Kel asked about chaos, I will see if we can code up some chaotic motion using the compound pendulum (which is not the same as the coupled pendulum).
The next level of complexity is to step it up to N particles. After that, we take the limit that N goes to infinity! Waves appear! They appear completely naturally from laws governing a finite but ever larger number of ever more closely spaced and ever smaller particles.
Waves first show up in a single dimension, such as waves on a string. But then we can step up the complexity yet again and consider waves in two dimensions, such as waves on a drumhead. Finally the highest level of complexity we can hope to get to in a one-semester course, starting with no significant prerequisites, is a taste of what quantum-mechanical waves look like in three dimensions.