Waves: interference, diffraction and refraction

The tank holds one number per cell: the height of the wave there. Each step, a cell accelerates towards the average of its four neighbours. That is the two-dimensional wave equation:

It describes small ripples on shallow water and sound in air exactly, and light well enough for interference and diffraction, though it leaves out polarisation. The grid steps it with the leapfrog scheme: the next height of a cell comes from its current height, its previous one, and the curvature around it. Walls hold the height at zero, so waves bounce off them. Glass is a region where the wave speed is c/n, with n the refractive index.

Two details keep the tank honest. Its edges are a sponge: a band where the waves are damped gradually, so they leave instead of reflecting back across the picture. And a wavelength needs about a dozen cells or more; on a coarser grid, waves travel at slightly the wrong speed, differently along the diagonals.

Two waves at the same place simply add. Where crest meets crest the water rises twice as high; where crest meets trough it stays flat. Choose Two sources: the calm lines fan out from between the sources along curves where the distances to the two differ by half a wavelength, one and a half, two and a half, and so on.

The Brightness view averages the height squared over the last two periods, which is what a light meter or your ear measures. It turns the moving pattern into a still one.

Thomas Young showed in 1803 that light passing through two narrow openings makes bands of light and dark, and took it as proof that light is a wave. Choose Double slit. Each slit becomes a source of its own, and the two interfere just as before. On the screen, the bright fringes sit where the paths from the two slits differ by a whole number of wavelengths:

with d the distance between the slits, L the distance to the screen and Δy the spacing of neighbouring fringes. The dotted lines on the chart mark these positions. A grating of seven slits puts its bright fringes in the same places but makes them much narrower, because seven waves agree only very near those angles. That is how a spectrometer separates the colours of light. A single slit wider than a wavelength makes a broad central band with dark gaps where $a\sin\theta = m\lambda$.

In 1690 Christiaan Huygens proposed that every point on a wavefront acts as a small source of new wavelets, and the wavefront a moment later is where those wavelets add up. Augustin Fresnel added in 1818 that the wavelets interfere, and that turned Huygens’ idea into a way to calculate diffraction.

The dashed line on the screen chart is that calculation done literally. Each open cell in the slits is treated as a source of circular wavelets, falling off with distance as $1/\sqrt{r}$ in two dimensions; the wavelets are added at each point of the screen and the sum is squared. It uses the wavelength the grid actually carries, and no approximation for a distant screen. The tank, which knows nothing about wavelets, should agree with it closely.

There is a twist. In three dimensions a sharp pulse passes and leaves silence behind it. In two dimensions it does not: the wavelets from a pulse keep arriving after the front, so a splash leaves a fading wake. Choose Single pulse and look behind the ring. Strictly, Huygens’ principle in its clean form holds only in an odd number of dimensions, which is lucky for anyone who wants to hear speech clearly.

A wave entering glass at an angle slows on one side first, so its front swings round. The angles obey Snell’s law,

first written down by Ibn Sahl in Baghdad around 984 and rediscovered by Willebrord Snell in 1621. The Refraction experiment draws the predicted direction inside the glass. A lens is thicker in the middle, so it delays the middle of a wave more than the edges and bends the front into a curve that closes on a focus.

Going the other way, from glass into empty space, a wave meeting the face at a shallow enough angle cannot get out at all: $\sin\theta$ would have to exceed 1. Past the critical angle, $\sin\theta_c = 1/n$, it is totally reflected. An optical fibre uses this to carry light round bends.

A parabola sends every ray parallel to its axis to one point, its focus, which is why satellite dishes and big telescope mirrors are parabolic. An ellipse does something different: every path from one focus, bounced once off the wall, reaches the other focus having travelled the same distance. All the echoes arrive together, which is how a whispering gallery lets you hear someone across the room.

A source moving through the water crowds its crests together ahead of it and spreads them out behind. Christian Doppler described the effect for sound and light in 1842: the pitch rises as a siren approaches and falls as it leaves. A source moving faster than its own waves outruns them altogether. The waves it has made pile up along a cone, the shock wave, whose half-angle obeys $\sin\mu = c/v$. That is the sonic boom of a supersonic aircraft, and the blue Cherenkov glow around fast particles in a nuclear reactor’s cooling water. (A duck’s V-shaped wake looks similar but is a different effect: water waves of different lengths travel at different speeds, and the wake keeps the same angle at any speed.)

A row of sources all in step sends a beam straight ahead. Delay each source a little more than the one before, and the beam swings towards the delayed end, because waves from the later sources need a shorter path to arrive in step. For sources a distance d apart with a phase step $\Delta\varphi$, the beam points at $\sin\theta = \Delta\varphi\,\lambda/(2\pi d)$. Radar, 5G masts and medical ultrasound probes steer their beams this way, with no moving parts.

• Colour. Each experiment runs at one wavelength, so a glass prism here refracts but cannot split white light into a spectrum. That needs a refractive index that changes with wavelength, and several wavelengths at once.
• Polarisation, and everything else that makes light a vector wave rather than a number.
• The third dimension. Waves spread over circles here, not spheres, so they fade more slowly with distance: as $1/\sqrt{r}$ instead of $1/r$.
• Units. The heights are arbitrary, and the colours rescale themselves to stay readable.

• In the double slit, halve the wavelength and watch the fringes close up: $\lambda L/d$ in action.
• Paint a wall over one slit with the brush. The fringes vanish into a single-slit pattern.
• Raise the glass index in the Refraction experiment until part of the wave reflects off the face as well as passing through.
• In the resonant box, slide the wavelength slowly and watch for the moments the pattern locks.
• Add three or four sources with the Source tool and look for the calm lines between them.

• R. P. Feynman, R. B. Leighton and M. Sands, The Feynman Lectures on Physics, Volume I, chapters 29 and 30, interference and diffraction.
• T. Young (1804), Experiments and calculations relative to physical optics, Phil. Trans. R. Soc..
• C. Huygens (1690), Traité de la lumière, the wavelet principle.
• A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, Artech House; the standard text on solving wave equations on grids.