Oscillations and Waves

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Daily Schedule Term 5

See also Daily Schedule Term 4

Week 8 — Torsion Pendulum — Coupled Torsion Pendula — The Second Derivative — Torsion Waves

As a way of refreshing on material that is three weeks old due to the Term 4-5 break, our eleventh and twelfth notebooks (Torsion Pendulum and Coupled Torsion Pendula) are straightforward recapitulations of our third and eighth notebooks (Mass on a Spring and Coupled Harmonic Oscillators). Similarly, our thirteenth notebook (Torsion Waves) will be a straightforward recapitulation of our ninth notebook (Many Harmonic Oscillators), but we will press on the initial conditions harder and get obvious traveling waves to appear. Also, we are about to start making a connection to an important calculus idea, “The Second Derivative.” Waves typically show up in systems for which Newton’s Laws involve second derivatives with respect to both space and time.

Week 9 — Drumheads — Two-Dimensional Grids of Masses

Rectangular Drumhead Mallet Strike

Week 10 — Three-Dimensional Grid of Masses — Exam 2

Transverse Wave Transiting With Periodic Boundary Conditions

Longitudinal (Compression) Wave

We have a little less than one-third of the course left, and we are going to blast into the stratosphere: instead of doing large numbers of particles — like 72 rods or a grid of 18 by 24=432 masses — we are going to take the limit that n→∞ and start specifying problems to Mathematica using the notation of derivatives. You might reasonably ask that if it is too time-consuming for Mathematica to do, say, a thousand particles for ten-thousand time steps, how is Mathematica going to cope with continuous systems which have an infinite number of particles? The answer is that deep under the hood, it breaks continuous systems up into little chunks and little time steps just as we have been chunking up the world with grids and time steps. In other words, Mathematica also turns continuous problems back into problems with a large but finite number of chunks and time steps. However, from now on in this course, Mathematica is going to hide the need to break continuous systems up into little chunks from us. Mathematica draws graphs and creates animations by choosing the chunks and the time steps to be so small that we don’t generally perceive them unless we blow up the graphics to a large size. Sometimes, the chunkiness leaks through enough to be obvious, but if that happens and we are not satisfied, we can order Mathematica to use finer chunks and finer time steps. Of course, the processor in your laptop may not appreciate this, but if you are patient with your computer’s processor, and also if your computer has enough memory to hold the intermediate results, there is no limit in principle to how accurately computers can do simulations — the only exceptions being simulations with chaos or singularities, and even in such situations, simulations are often informative and indicative of what happens in the real world. All that said, the limits of computer power are real, and are part of the reason why we do not yet have compelling simulations for things like the formation of the solar system, even using the largest supercomputers available.

Week 11 — Continuous Systems from the n→∞ Limit — Oscillators and Guitar Strings

Driven Harmonic Oscillator Just Above Resonance

Special projects being presented in Weeks 12 and 13

Each special project presentation should include a Mathematica notebook that has a description of what you are solving, is interleaved with explanations of how the code you are presenting solves it, and ideally, has activities for the rest of us to do to help us become familiar with the problem and the solution technique. Email your notebook to the class before your presentation. Use notebook chapters, section, and subsections, and other typesetting features to make your presentation organized and professional-looking. It is fine to rely heavily on already-existing resources. The goal is a clear explanation of a problem and its solution, not to achieve something original.

Week 12 — Drumheads — Diffusion of Heat

Lotke-Volterra Graphs for Two Species

Week 13 — Diffusion of Pollutants in Three Dimensions — Schrodinger’s Equation

Week 14 — Exam 3 — Atomic Wave Functions