This is a simulation of the wave equation in two dimensions. We use a color map where positive values appear in red, and negative values in blue.
The value at each point $(x, y)$ at time $t$ is represented by a real number $u(x, y, t)$. The function $u(x, y, t)$ obeys the differential equation $$\left(\frac{\partial}{\partial t}\right)^2u = \left(\frac{\partial}{\partial x}\right)^2u + \left(\frac{\partial}{\partial y}\right)^2u$$ at almost all points $(x, y)$. The points where this relationship is violated come in two flavors: (1) stationary points where the function value is held at 0 for all $t$, which act as reflective surfaces for waves, and (2) wave point sources where the function value goes as $A\cos(\omega t)$ for some amplitude $A$ and frequency $\omega$.
In the simulation below, the reflective surface is chosen to be a conic section, and one focus is chosen as a point source.
TODO Why am I investigating this? Conic section property (applet). Colloquially described in terms of light rays. However, light behaves like waves (applet?). Are the wave description and ray description the same? I don't think so: think of light bending around a corner (applet).
Example 0: A single vibrating spring. Force is proportional to displacement. TODO Add explanation and force arrows.
Now let's consider a sequence of point masses in a horizontal line, connected by springs. TODO Example 1: A sequence of point masses connected by springs. Let $x_i$ be the displacement of the $i$-th mass. Displacement can be either transversal, or longitudinal (show both examples). Think about force on each mass, and draw with arrows (which can be turned on or off).
TODO Example 2: Go to the limit of an infinite number of point masses which are infinitely close, and see how we get a differential equation, which is the 1D wave equation (draw as a curve). Global behavior: 2a: If fixed at both ends, we get standing waves, and stable states are given by linear combinations of sine waves with discrete frequencies. 2b: If open (e.g. at one end) then stable states are again given by sine waves. Show examples of driven behavior.
TODO Example 3: Extend example 1 to 2 dimensions. Lattice of point masses connected by springs. Again, depict the force diagram.
TODO Go back to one of the original examples and make the spring stiffness interactively change-able. See that this determines propagation speed.
TODO Show refraction in 2D with a change of materials and a plane wave.
TODO Show that a boundary condition of 0 is effectively a reflective surface. First show in 1D, then in 2D.
TODO Figure out what to depict after this?