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 slider controls the distance between the two sources, which in turn affects the resulting interference pattern. The two buttons allow you to pause and reset the simulation, respectively.
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:
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. 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.
Let's simplify the two-dimensional wave equation by first considering the zero-dimensional wave equation. For a function $f(t)$ with a single variable $t$ representing time, the zero-dimensional wave equation has the form
This equation describes the horizontal position of a point mass attached to a (idealized) spring, where the force (second-derivative) is proportional and opposite to the displacement from equilibrium. For a spring with spring constant $k$ (representing the stiffness), the equation becomes (we have normalized mass to 1) $$f''(t) = -k f(t)$$ In the applet below, the point mass is represented by a black square. Try clicking and dragging the mass to change its position. Its equilibrium position is depicted by a dashed line, and the force acting on the mass is shown as a red arrow. By clicking the play button, you can observe how the mass moves over time. To its right, the position $f(t)$ is graphed through time. This forward-evolution through time is analogous to the forward-evolution of the wave equation. The spring constant $k$ can be adjusted with the first slider below. Loss of energy to friction (a reality in the physical world) can be simulated by adding a damping term to the equation, which is proportional to the velocity of the mass ; this is controlled by the second slider.
Next, let's consider a sequence of point masses, connected by springs. The leftmost and rightmost masses are fixed in place at positions $L$ and $R$, while the middle masses (let's say there are $N$ of them) are free to move up and down along vertical tracks. Their displacements from their equilibrium positions (now depicted vertically, instead of horizontally as before) are denoted by $f_1(t), f_2(t), f_3(t), \ldots, f_N(t)$. Note that the equilibrium state of the system is where the points form a straight line stretching from the left side to the right side, because the shortest path between two points is a line.
Each point mass experiences a force from both sides. If we suppose that the equilibrium length of each spring is zero, then the force exerted from the left on the $i$-th mass has vertical component equal to $-k(f_i(t) - f_{i-1}(t))$, where $k$ is the spring constant. Similarly, the force exerted from the right on the $i$-th mass has vertical component equal to $-k(f_i(t) - f_{i+1}(t))$. Added together, we get the equations
The same simulation as before is set up below, where point masses can be clicked and dragged to change their positions.
Now imagine the limit of the scenario above as the number $N$ of point masses goes to infinity.1 This is a mental model for an elastic object such as a rubber band or a spring. Instead of $N$ displacement functions $f_1(t), f_2(t), \ldots, f_N(t)$, we now have an entire continuum of them, represented by a two-variable function $f(x,t)$. Writing $f''(x, t)$ is now vague: do we mean the second derivative with respect to $t$, or respect to $x$? We therefore write these two possible second derivatives as $\frac{\partial^2}{\partial t^2}f(x,t)$ and $\frac{\partial^2}{\partial x^2}f(x,t)$, respectively. The sequence of $N$ separate differential equations for the displacement functions $f_1, f_2, \ldots, f_N$ become a single partial differential equation: the one-dimensional wave equation.
This is a consequence of the fact that for a function $g(x)$, the second derivative can be expressed as $$g''(x) = \lim\limits_{h \to 0}\frac{g(x+h) - 2g(x) + g(x-h)}{h^2}$$
In the simulation below, one can click-and-drag to change the view, and a slider for zooming in to a microscopic scale has been added. It's also worth noting that in a continuous medium, the "stiffness" is naturally analogous to the square of the wave propagation speed.
The one-dimensional wave equation exhibits much richer behavior than the zero-dimensional wave equation. On the left, we see that an oscillating point source in the middle of an unbound string generates waves which propagate outwards. On the right, we see that on a bound string (where both endpoints are held at fixed positions) waves reflect off the ends.
With the above examples in mind, we can now visualize the two-dimensional wave equation. The sequence of point masses used in the one-dimensional case is replaced by a two-dimensional grid of point masses, with each mass connected to its horizontal and vertical neighbors and allowed to move in the third dimension orthogonal to the plane of the grid.
The $z$-position of the point mass at position $(i, j)$ can be written as $f_{i, j}(t)$, and satisfies the differential equation $$f_{i, j}''(t) = k(f_{i-1, j}(t) - 2f_{i, j}(t) + f_{i+1, j}(t)) + k(f_{i, j-1}(t) - 2f_{i, j}(t) + f_{i, j+1}(t))$$ For each point mass, the force from the springs to its left and right neighbors produce the first term, while the springs to its top and bottom neighbors produce the second term. Letting the number of point masses in each direction tend to infinity, and replacing $f_{i, j}(t)$ with $f(x, y, t)$, we obtain the two-dimensional wave equation $$\left(\frac{\partial}{\partial t}\right)^2f = \left(\frac{\partial}{\partial x}\right)^2f + \left(\frac{\partial}{\partial y}\right)^2f$$
Below are a few visual demonstrations. On the left is a simulation of a single point isolated point source in a medium with constant $k$ value, with no reflective materials. In the middle is the same simulation where a reflective conic section (with the point source at its focus) has been added. On the right is a simulation of two point sources separated by distance $d$, recreating the visual interference patterns revealed in the famous double-slit experiment.