How far can a light be seen?
Two limits apply and the smaller one wins. The geometric range is the horizon: for a light at height hL and an eye at he above the sea, d = √(2ReffhL) + √(2Reffhe) with Reff = R/(1−k) and standard refraction k ≈ 0.13 (a typical value). The sailor's rule is d[nmi] ≈ 2.08(√hL + √he). With the lamp 38 m above the sea (a 14 m headland and a 24 m tower in this model) and an eye at 5 m, that is about 17.4 nmi.
The luminous range comes from Allard's law: the illuminance at distance d is E = I · Td / d², with I the intensity in candela and T the transmissivity per nautical mile. The light is seen when E > Et. The threshold is 2×10−7 lx at night (the usual nominal-range value) and 1×10−6 lx in the blue hour (a model value, there is more sky light).
Fog
Fog is described by the meteorological optical range V, where the contrast of a dark object falls to 5 %: V = 3.0/σ and T = 0.051/V per unit of V. (The older Koschmieder 2 % threshold gives 3.9/σ.) The slider maps to V = 20 · 10−f/0.6 nmi: 20 nmi at f = 0 and 2 nmi at the default 0.6. This is why fog matters more than darkness: darkness only sets the threshold, while fog multiplies the light by an exponential at every mile. Doubling the height of a tower extends the horizon, but it does not push the light through fog.
The lens and the flash
A Fresnel lens collects the lamp's light into a beam. The divergence is about δ ≈ 2 atan(s/2f) for a source of size s at focal length f: an 80 mm mantle at f = 920 mm gives 4.98°. With four panels turning once in 20 s, a ship sees a flash every 5 s, each lasting t = (δ/360°) Trot = 0.28 s: the characteristic Fl W 5s (a white flash every 5 seconds). A ship identifies the light after two full periods. The drawn beam is wider than the real 5° so that it can be seen.
Waves
The sea is a sum of deep-water waves with ω² = gk, so the wavelength is L = gT²/2π and the speed c = gT/2π (about 12.5 m/s at T = 8 s). Their energy follows a Pierson-Moskowitz shape scaled so that the standard deviation of the surface is Hs/4. Wind sets the sea state: fully developed Hs ≈ 0.21 U²/g, or the fetch-limited JONSWAP law Hs = 1.6×10−3 √(gF/U²) U²/g; Beaufort wind is U ≈ 0.836 B3/2 m/s. A small boat has its eye only 2 m above the water, so a crest between it and the lamp can hide the light even inside the horizon: the page checks the line of sight over the same wave field that moves the boats.
Honest simplifications
Waves are linear (no breaking, no shoaling or refraction near the headland; the surf is illustrative). The fog is uniform. The sighting threshold is fixed and the flash is treated at its peak intensity (a real flash is judged by its effective intensity, which is lower for a short flash). Ships are kinematic, steer by a simple rule and are time-compressed (about 20 times faster than real so that a game lasts minutes); the horn's audible range is illustrative; the lamp intensities, the blue-hour threshold and the focal lengths of the lens orders are model values. The light's characteristic, the headland and the lighthouse are fictional ("Penhallow Point"). The beam, ranges and sea state are for teaching and are not for navigation. Independent educational model, not affiliated with Trinity House or any lighthouse authority.