Mathematics of Waves: Visualized with Neural Networks

In the physical world we are surrounded by waves of all kinds, each with its own underlying physics. Acoustic waves for example involve the vibrations of the medium which we can hear, while electromagnetic waves involve vibrations of electric fields that require no medium and are detected with specially designed antennas. These vibrations are physically very different and are usually taught in very different courses although they are governed by very similar partial differential equations. Furthermore, the equations are sufficiently opaque that they hide the underlying unity. Our aim is to make this unity apparent through a unifying viewpoint, using neural networks (NN) as a pictorial tool to visualize the underlying mathematics.

All waves can be viewed as a collection of individual oscillators, and so we will start in Week 1 by laying out the mathematical and conceptual framework for oscillatory systems as diverse as mechanical oscillators (mass-spring), electrical oscillators (inductor-capacitor), ferromagnets and quantum spins. The physical variables are very different and even the equations may look different, but we will stress their unifying characteristics, introducing the concept of normal modes and mapping them all to an NN model. In Week 2 , we will show how the coupling of different elementary oscillators leads to different wave systems, with quantitative illustrative examples.

This is a pilot 2-week course where we will illustrate the basic idea for acoustic, Schrödinger, and electromagnetic waves, all in Week 2. In future versions, our aim will be to expand the content with more illustrative examples as well as with more advanced topics such as quantized oscillators and waves.

Week 1: Oscillators

1.1. Math and the Physical World

1.2. Exponential Response

1.3. Oscillatory Response

1.4. More Oscillator Examples

1.5. Impulse Response

1.6 Sinusoidal Excitation

Week 2: Waves

2.1. Oscillators to Waves

2.2. Normal Modes

2.3. Tx Lines and Guitars

2.4. Dipole Antenna