The wave equation is a partial differential equation that describes the propagation of waves through a medium, such as mechanical waves (such as sound) through air or water, or electromagnetic waves (such as visible light or radio waves) through vacuum or glass.
The two-dimensional wave equation can be used to visualize waves propagating in a two-dimensional medium, such as ripples on the surface of water. Each point in the $xy$-plane has an associated real-numbered value, and this collection of values evolves through time. I've used a color map where red indicates a positive value, blue indicates a negative value, and white indicates zero. The applet below simulates a dipole, where the wave equation has two point sources with opposite phase.
The behavior of the wave surface is determined by a set of boundary conditions, and the wave equation which governs its evolution over time. It's similar to how a pendulum's motion is fully determined by its initial position and velocity (the boundary conditions), and the pendulum's length, mass and strength of gravity (which determine the differential equation).
We can make these two pieces more precise for the wave equation. The values of the heatmap can be represented by a function in three variables: the two spatial variables $x$ and $y$, and the time variable $t \ge 0$. We can denote this by $f(x, y, t)$. The wave equation is the partial differential equation which indicates how the function changes over time and space every inside the domain: $$\left(\frac{\partial}{\partial t}\right)^2f = \left(\frac{\partial}{\partial x}\right)^2f + \left(\frac{\partial}{\partial y}\right)^2f$$ Boundary conditions are locations in space and time where the function does not evolve according to the wave equation, but is forced to evolve some other way. One such example is to specify point sources $(x, y)$ where the function value varies sinusoidally, i.e. goes as $A\cos(\omega \cdot (t - \phi))$ for some amplitude $A$, frequency $\omega$, and phase $\phi$. Such points can be viewed as sources for propagating waves. For example, above we have two point sources which are fully out-of-phase with each other (i.e. phase difference $\pi$).
As mentioned in the opening paragraph, electromagnetic radiation (such as light) and sound propagate as waves. There is another familiar model we use for envisioning light, namely as rays. A wonderful fact about ellipses is as follows: if you shoot a ray out from one focus and it reflects off the ellipse, it will converge on the other focus. Moreover, the total distance traveled from one focus to the other is independent of the initial angle of the ray.
The analogous statement for a parabola, whose second focus lies at infinity, is: multiply rays emanating at the same speed from the focus and reflecting off the parabola will then end up traveling parallel to the axis of symmetry, all lying on a line perpendicular to said axis. This is, for example, how flashlights work, by reflecting light from a point source off a parabolic reflector to convert it into a unidirectional beam. Taking the reverse problem, parabolic satellite dishes and directional microphones work by focusing a distant source of radiation onto a single point by reflecting it off a parabolic reflector.
The question I want to investigate is: does this property of conic sections still hold when the ray-model is replaced by the wave-model?
More generally, how much does qualitative intuition carry over between the two models? There are definitely behaviors which differ between the two models: for example, light and sound cannot bend around a corner in the ray model, but they can in the wave model. (TODO example) The double-slit experiment, which demonstrates the existence of wave interference, is the prototypical example of qualitative behavior which is captured by the wave model and not by the ray model. I've read that when the wavelength is much smaller than any physical features, the ray model is a good approximation of the wave model -- this holds true for visible light at macroscopic scales, but does not hold for sound waves.
First, let's decipher the differential equation above into plain English, by thinking from first principles.
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?