Plan a Light

This is the capstone, so for once we build instead of learn. Imagine you have been asked to put a light on a headland. Nobody has told you how tall to make the tower, how bright the lamp must be, or what rhythm it should flash, and each of those answers comes from a different part of How Lighthouses Work 1 to 8: the horizon and the inverse square law, the lamps and lenses, the flash characters, the rock and the weather. In this chapter we turn those pieces into one small Python range calculator, use it to meet a written brief, choose a characteristic that cannot be confused with the lights next door, read a chart label the way a mariner does, and finish by putting the three Stevenson rock lights side by side.

The Brief

A planner starts with a statement of what the light is for. Mine is invented: a headland with a cliff top 20 m above the sea, a shipping lane offshore, and ships whose bridges are 8 m above the water. The light must be seen from 15 nautical miles in the standard weather of How Lighthouses Work 1 (meteorological visibility 10 nautical miles), and still from 6 nautical miles when the visibility falls to 2 nautical miles. Both figures are assumptions chosen to make the example work; a real brief would come from the authority and from the traffic.

Every decision in the brief answers to one of the earlier chapters, and the order matters, because the limits are independent: the horizon sets one, the brightness another, and what a sailor sees is the smaller of the two.

A Range Calculator

The calculator is three functions. The first is the geographic range of How Lighthouses Work 1, with the light and the eye both contributing a horizon. The second finds the luminous range by bisection: it keeps halving an interval until the illuminance given by Allard's law falls to 0.2 microlux, and it takes the visibility as the distance at which 5% of the light survives, as that chapter did. The third returns the smaller of the two. The table it prints crosses three heights with two lamps, 100,000 cd and 1,000,000 cd (both assumed), for an 8 m eye.

Three things stand out. At 20 m a lamp ten times brighter changes nothing in clear weather: both lamps are cut off by the horizon at 15.2 nm. At 60 m the dimmer lamp is the limit (19.8 nm against a horizon of 22.0 nm), so there more intensity helps. And in poor visibility the height makes no difference at all: the 2 nm column reads 5.6 nm for the 100,000 cd lamp and 6.9 nm for the 1,000,000 cd lamp at every height, because the air, not the horizon, is the limit. Ten times the intensity gave only about 1.2 times the range (6.9 divided by 5.6, from the rounded figures), which is the inverse square law at work again.

Meeting the Brief

Now run the calculator backwards. Given a required range, the focal height that just reaches it follows from solving the geographic formula for H, and the intensity that just reaches it can be found by searching on the luminous-range function itself; the search below halves the interval on a logarithmic scale, because the answer could be anywhere from a candle to millions of candela. The code does the clear-weather case and the poor-weather case separately, keeps the larger intensity, and then tests a tower of my choosing, 12 m on the 20 m cliff, against the horizon.

The horizon needs a focal height of only 19.1 m for 15 nm, and the 15 nm brightness needs only 13,805 cd. The poor-weather requirement is the hard one: 6 nm in 2 nm visibility needs 197,562 cd, about fourteen times as much (by my arithmetic from the printed figures). The line "the clear-weather case decides: False" says exactly that, and it is the lesson of the whole capstone: a lamp is sized for the bad night, a tower is sized for the good one. The 32 m focal height (20 m of cliff plus 12 m of tower) gives 17.7 nm, a margin of 2.7 nm. Doubling the minimum intensity to 395,125 cd lifts the nominal range to 23.3 nm but the poor-weather range only from 6.0 to 6.4 nm, and five times the minimum reaches 6.9 nm, so the extra lamp buys very little where it is needed. I take a round 400,000 cd, about twice the minimum and roughly a fifth of the Bell Rock's 1,900,000 cd (my arithmetic). A lens of the kind in How Lighthouses Work 4 would be what turns a lamp into that candlepower; the plan does not choose an order, and the sizes of lens in that chapter's model were illustrations, not measurements.

Choosing a Characteristic

A sailor identifies a light by its rhythm, so a new light must not look like its neighbours. For the example I place the headland between three lights with the characteristics of the Bell Rock, Skerryvore and Muckle Flugga (Fl W 5s, Fl W 10s and Fl(2) W 20s). This is an invented neighbourhood: the real lights are nowhere near each other. The code tests six candidate characteristics against two things: the rules for flashing lights quoted in How Lighthouses Work 5 (a single flash has a period of 2 to 15 s with darkness at least three times the flash; a group of two has a period of at most 20 s and a group of three or more at most 30 s), and a confusion test that is entirely my own rule: two lights are confusable if they have the same class and the same number of flashes and their periods are within 3 s of each other. The source gives no such number; it is a simple way to make the idea concrete.

Fl W 5s is the Bell Rock's own character and is rejected; Fl W 12s is within 3 s of Fl W 10s; Fl(2) W 20s is identical to Muckle Flugga's. That leaves Fl(2) W 10s, Fl(3) W 15s and Iso W 4s. All three would do; I choose Fl(3) W 15s, three flashes in 15 s, because a group of three is easy for a tired watchkeeper to count and no neighbour shows one. It is lit 20% of the time (three 1 s flashes in 15 s), the same share as the Fl(2) W 10s option, and its group of three needs a period of at most 30 s, which it meets easily.

Reading a Chart

The finished design appears on a chart as one line of text. The page on light characteristics gives the convention: the class and group, the colour, the period, the height of the light above the chart datum (usually based on high water), and a range in nautical miles written with an M. Its worked example, "Gp Oc(3) W 10s 15m 10M", is described as a group occulting white light with a period of 10 s, 15 m high, whose nominal range is 10 nautical miles. The page is not consistent about the range: in the list it calls the range "the range in which the light is visible", and in the example it calls it the nominal range. As How Lighthouses Work 1 explained, the nominal range ignores the horizon, so I follow the worked example. Our design therefore reads "Fl(3) W 15s 32m 23M": 32 m is the focal height, and 23 is the 23.3 nm of the 395,125 cd run, rounded.

The code below parses a label with a regular expression, works out how far a 5 m eye can see the light, and says whether the horizon or the brightness is the limit. It then does the reverse: given a list of the times at which flashes began, it finds the long gap in the pattern and counts the flashes between two long gaps.

For a 5 m eye the first three lights are limited by the horizon, at 15.7, 18.8 and 21.6 nm, against printed ranges of 18, 23 and 22 nm, which is the finding of How Lighthouses Work 6 again. Our own design is also limited by the horizon (16.4 nm for a 5 m eye, against 23 nm printed), which is a reminder that a bigger number on the chart is not a longer view. The stopwatch readings contain gaps of 2 s, 2 s and 11 s repeated, so the pattern is three flashes with a period of 2 + 2 + 11 = 15 s, which is the character we chose. This simple version looks only at when flashes begin, so it cannot tell an isophase light from an occulting one, or a short flash from a long one; a mariner would also note how long each flash lasts.

Three Scottish Lights Compared

Finally, put the planning tool to work on real lights. The data are the focal heights and listed ranges read from the pages, the characteristics from How Lighthouses Work 6, and the years from How Lighthouses Work 6 and 8. For each light the code finds the horizon for a 5 m eye, compares it with the listed range, works out how high an eye would have to be for the listed range to be the horizon, and counts the flashes an hour.

Muckle Flugga is the only one whose listed range matches the geometry (a ratio of 1.02, an eye of 5.9 m), which suggests that its range is a horizon figure for a low boat, while the Bell Rock and Skerryvore ranges are larger than any such horizon and would have to be nominal ranges or assume a high eye. I cannot tell which from my sources, as before. The tallest light, Muckle Flugga, is 2.36 times the height of the Bell Rock but sees only 1.38 times as far for a 5 m eye, the square root law at work. The Bell Rock's own infobox gives a flash of 0.1 s in a 5 s period, so the light is on for 2.0% of the time. If the lens turns once in 5 s, which I am assuming, that flash matches a beam 7.2 degrees wide, whereas the model hyper-radial beam of How Lighthouses Work 4 (0.86 degrees) would last only 0.012 s. The page does not give the beam width or the rotation period, so the tension noted in How Lighthouses Work 5 between narrow beams and second-long flashes remains unresolved. Most of the other numbers in this chapter are tidier than that one.