How the homepage wave works

The ripple field on the front page is not an animation loop. It is a small solver for the 2D scalar wave equation, stepped about 120 times per second in your browser.

Model

\[\frac{\partial^2 u}{\partial t^2} = c^2 \left( \frac{\partial^2 u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2} \right) + s(x, y, t)\]

discretized with the explicit leapfrog scheme and a five-point Laplacian on square cells:

\[u^{n+1}_{ij} = 2u^n_{ij} - u^{n-1}_{ij} + r^2 \left( u^n_{i+1,j} + u^n_{i-1,j} + u^n_{i,j+1} + u^n_{i,j-1} - 4u^n_{ij} \right)\]

with \(r = c\,\Delta t / \Delta x = 0.5\), inside the 2D stability limit \(r \le 1/\sqrt{2}\). Units are grid cells and time steps — there is no mapping to SI and none is implied.

No continuous source on the home page: the field rests until something taps it. Until you do, a ghost hand left of the slits alternates a tap and a swipe, feeding the field exactly what a real tap or mouse move would. The slit itself is fixed here; moving it is the lab’s job. (The lab keeps two coherent oscillators with \(T = 24\) steps by default, so the wavelength is \(cT = 12\) cells; a slider changes it.)
Moving the mouse leaves a faint wake of Gaussian bumps whose amplitude scales with pointer speed (scrolling with the mouse resting on the film counts as moving); click adds one strong bump; dragging adds a rate-limited trail.
The barrier is two cells of Dirichlet wall, \(u = 0\), with a six-cell graded damping layer on its upstream face, so the jaws mostly soak up what hits them, like tungsten X-ray slit blades. This is an approximation, not a perfect absorber: some reflection remains. (The lab can switch the jaws to ideal reflectors.) Each slit is 7 cells wide, narrower than the 12-cell wavelength, so diffraction is strong.
Edges use a quadratic sponge layer, \(u^{n+1} = \left[ 2u^n - (1-\sigma)u^{n-1} + r^2 \nabla^2 u^n \right] / (1+\sigma)\), to suppress border reflections. It is an approximation, not a perfectly matched layer.

The detector line

The curve on the right is an exponential moving average of \(u^2\) along one column, with a time constant of about six periods, normalized to its own maximum. It is a relative intensity from the solver arrays only — the glow and bloom are drawn on top and never feed into it. In the lab, drag a slit and the fringes on the detector respond after a short lag.

One honest caveat: this is an in-plane 2D field. A real double slit lit in 3D produces a 2D pattern on a screen; here the slit is a gap in a line, and the detector sees a 1D profile. The fringe physics — spacing shrinks as the slits move apart — is the same.

Colors

The field is drawn like exposed film on a grey panel: crests develop toward the ink colour, troughs toward the opposite tone. The shading is a visual encoding of the sign of \(u\), not the colour of any real wavelength.

The solver is about 100 lines of plain JavaScript (wave.js), with a regression test that checks stability, absorption at the edges, that a closed barrier blocks transmission, and that the absorbing jaws send back less than a quarter of the energy reflecting ones do.