How to use and what to try
The web starts dry. Press Dry web to restart, or pick a preset: in Clear still night the thread is colder than the dew point, a water film builds up, breaks into beads (Rayleigh–Plateau), the beads merge and grow and finally some drip. In Dry air nothing forms. Drag to pan, use the wheel or pinch to zoom: at high zoom every drop shows an upside-down image of the meadow behind it, as a real dew drop does. Things to try: (1) raise the wind and watch the chill and the condensation shrink, while the web shakes; (2) lower the spiral radius – the drip limit falls, so drops are smaller; (3) lower the radial tension and press Gust: the modes slow down, especially once dew has loaded the web (graph 3); (4) pick a single mode in Web vibration shown and read the nodal diameters in the inset; (5) set the time-lapse to 1 to see real-time growth. Space pauses, R restarts.
The science
1. When does dew form? The saturation vapour pressure follows the Magnus formula es(T) = 6.112 exp(17.62 T/(243.12 + T)) hPa. Air at temperature T and relative humidity RH has the dew point Td = 243.12 γ/(17.62 − γ), γ = ln(RH/100) + 17.62 T/(243.12 + T). A thin silk thread radiates heat to the clear sky and is a few kelvin colder than the air; wind carries heat to it, so the page lets the chill fall off as exp(−wind/1.4 m/s) (an illustrative law). Water condenses when the thread is colder than Td, driven by the vapour-density excess Δρ = RH · ρsat(T) − ρsat(Tthread).
dV'/dt = 2π η D Δρ / ρw (V' = water volume per metre of thread, D = 2.5×10-5 m²/s)
η = 0.1 is the fraction of the diffusion-limited flux that really arrives (boundary layer, radiative balance); it is a tuning constant, not a measured value, so absolute times are only indicative – the time-lapse slider covers that.
2. Rayleigh–Plateau beading. A water film of radius R on a thread is a liquid cylinder; surface tension σ makes it unstable to wavelengths longer than its circumference. For an inviscid cylinder the growth rate ω of a mode of wavenumber k obeys
ω² = (σ/ρR³) · x I1(x)/I0(x) · (1 − x²), x = kR
with the maximum at x = 0.697, i.e. a beading wavelength λ = 2πR/0.697 = 9.01 R. Volume conservation then fixes the bead radius: (4/3)π r3 n = V' L for n beads on a segment of length L, which gives r ≈ 1.9 R for a thick film. The rate (graph 2 and the readout) is of order 104–105 per second for micrometre films, so beading is practically instantaneous compared with condensation. As V' grows, λ grows and neighbouring beads merge (coarsening); the page shows this as fewer, larger drops.
3. Why drops drip. A bead hangs from a thread of radius rt and is held by capillary force of order 2π rt σ (a Tate-law estimate). It falls when its weight (4/3)π r3 ρ g exceeds that, so rdrip = [3 rt σ / (2 ρ g)]1/3 ≈ 0.24 mm for a 1.2 µm thread. The capillary length √(σ/ρg) = 2.7 mm sets the largest drops of any kind. When a segment drips the page removes 70 % of its water and starts again.
4. Catenary-style loading. A thread of length L under tension T carrying a distributed weight w per length sags in a parabola h = wL²/(8T) (small-sag limit of the catenary), for a point load m g at the middle h = mgL/(4T). Both are checked in the page. Gravity acts in the plane of the web, so only the component of g perpendicular to each thread bends it. The sag drawn on screen is multiplied by the exaggeration slider; the real values are fractions of a millimetre for spiral threads and about a millimetre for the long radials when heavily loaded.
5. Vibration modes. The web is approximated by a tensioned circular membrane of radius aeq (the same area as the real frame polygon) with tension per unit length Tm = N Tradial/(2π a) and surface density ρs = (silk + dew mass)/area. Out-of-plane modes are w = Jn(jnm r/a) cos(nθ) with frequencies
fnm = (jnm/2πa) √(Tm/ρs)
(j01 = 2.405, j11 = 3.832, ...). Dew adds mass but no tension, so all frequencies fall by √(ρdry/ρwet) – graph 3. Wind drives each mode as a randomly kicked damped oscillator (damping ratio 0.03); the page plays the motion in slow motion and with an exaggerated amplitude.
Honest simplifications. The web is a regular hexagonal-frame orb web with 24 radials and 21 spiral turns; real webs are irregular and the viscid spiral is slack and stretchy. Real capture silk is coated with glue that already beads by the same instability. The membrane model treats a net of threads as a continuum (good for the lowest modes, poor for high ones), and absolute amplitudes under wind are illustrative. Drops are spheres of the computed volume; the contact angle on silk and the flattening of drops are ignored. Condensation uses a single tuned efficiency. Drop sizes and beading spacing follow the physics; absolute durations do not.
Checks run in the page. (1) a taut string with a drop at the middle solved with a tridiagonal system equals mgL/4T; (2) the numerical maximum of the Rayleigh–Plateau growth rate is x = 0.697; (3) bead volumes add up to the film volume; (4) the first frequency of a string carrying a heavy bead, from the root of (M/μ) k tan(kL/2) = 2, equals the lowest eigenfrequency of a 400-node mass-spring chain; (5) the finite-difference eigenvalue of the circular membrane equals the Bessel zero j01; (6) the Bessel integral vanishes at that zero; (7) the Magnus dew point for 20 C and 50 % RH matches tables; (8) the capillary length is 2.7 mm.