A signal tower chain fails far more often from hills, sleeping watchers and false alarms than from the curve of the Earth: at 15 km spacing two stations need eye heights of only 4.41 m each to see over the curve, while 9 relay stations that each work with probability 0.95 deliver an alarm only about 63% of the time. Adding a second sight line that skips one station lifts that to over 98%, at the cost of requiring stations to see twice as far.
Part of Signal Systems and Communication
A signal tower chain is a line of stations on high ground, each in sight of the next, that relays one alarm along its length; this article works out the spacing, the delay and the reliability from first principles.
Sources And Assumptions
The geometry uses the Earth as a sphere of radius R = 6,371 km (the standard mean radius, recalled) and the horizon formula d = √(2Rh), which is derivable from the right triangle between the observer’s eye, the Earth’s centre and the tangent point. Light is taken to travel at 3 x 10^8 m/s (recalled). Every other figure here, the 3-minute reaction time, the per-station reliability and the false-alarm rate, is an assumed example input chosen to show the method; nothing in this corpus measures them. Replace them with your own test results.
Geometry: What The Curve Costs
With h in metres and d in kilometres, d = √(2 x 6,371,000 x h) / 1,000 = 3.57 x √h. For two points the ranges add: d = 3.57 x (√h1 + √h2). For two stations of equal eye height h a distance D apart the sight line just clears the curve when each horizon is D / 2, so h = (D / (2 x 3.57))^2, which equals the mid-span bulge D^2 / (8R).
| Spacing D | Equal eye height needed | Mid-span bulge D^2/(8R) |
|---|---|---|
| 5 km | 0.49 m | 0.49 m |
| 10 km | 1.96 m | 1.96 m |
| 15 km | 4.41 m | 4.41 m |
| 20 km | 7.85 m | 7.85 m |
| 30 km | 17.7 m | 17.7 m |
| 50 km | 49 m | 49 m |
The two columns agree, as they must. At the parent’s 5 to 15 km spacing the Earth costs less than 4.5 m of height. A station on a hill is needed to clear terrain and trees between stations, not the curvature. The geometric limit is also not the practical one: a signal must be visible and readable at that range, and a campfire is put at 5 to 10 km on a dark night (Light Discipline), so a spacing near 10 km is limited by the signal, and a mirror or large bonfire is needed for more. Standard refraction extends geometry by roughly 7% (Observation Posts).
Delay: Almost All Of It Is People
Light crosses 50 km in 50,000 / (3 x 10^8) = 167 microseconds. The delay is the station’s reaction and signalling time, t. For n hops the delay is n x t. At 3 minutes per hop and 10 km spacing, 100 km is 10 hops: 30 minutes. The parent article cites the British Admiralty shutter-telegraph chain from London to Portsmouth, about 100 km and 15 minutes. That implies 1.5 minutes per hop if the line had 10 hops; I recall about 15 stations, which would be 1 minute each, but I have not been able to check either figure. Because a station should read the whole pattern before sending it on, a four-pulse code with pulses of 2 seconds each takes about 8 seconds to receive and another 8 to send, 16 seconds per hop on top of the reaction time.
Reliability: A Chain Is Only As Strong As Its Product
Let p be the probability that a given station is awake, in sight and acts on a given alarm. For m interior stations in a plain chain the alarm gets through with probability p^m.
| p | m = 5 (60 km at 10 km spacing) | m = 9 (100 km) |
|---|---|---|
| 0.90 | 0.59 | 0.39 |
| 0.95 | 0.77 | 0.63 |
| 0.99 | 0.95 | 0.91 |
Even stations that work 95% of the time deliver an alarm only about 63% of the time over 100 km. Reliability falls with length, so a long chain needs redundancy.
Skip links. If each station can also see the station two ahead, so that a dead station is bypassed, the chain fails only when two adjacent interior stations both fail. Counting the arrangements in which no two consecutive stations fail, the result for p = 0.95 is 0.990 for m = 5 and 0.981 (over 98%) for m = 9, against 0.77 and 0.63 for the plain chain. For p = 0.90 it is 0.963 and 0.928 against 0.59 and 0.39. The skip needs a sight line of twice the spacing: for 10 km spacing that is 20 km, which with equal heights needs 7.85 m (table above). Everything is achievable on modest hills.
False Alarms Travel The Whole Way
If each station has probability q of a false trigger in a season, a chain of n stations produces a chain-wide false alarm with probability 1 - (1 - q)^n. For q = 0.01, n = 6: 1 - 0.99^6 = 0.059; for n = 10: 0.096. A longer chain is more likely to cry wolf. Two defences: require the alarm to be authenticated (Authentication and Security) and allow any station to send a “cancel” that propagates the same way (Priority and Urgency Codes).
Method: Designing A Chain
Step 1: Fix the end points and the straight line, then walk it looking for hills and gaps.
Step 2: Choose sites on ground that clears terrain, and verify each pair by day and by night with the actual signal, not by map.
Step 3: Compute the geometry from the table and check every site against it.
Step 4: Choose p honestly from drills, then compute p^m and the skip-link figure. If the answer is below what the settlement needs, add skip links or shorten the chain.
Step 5: Test daily with a fixed-hour test message (Relay Networks).
Diagnosing Failures
- Alarm stops at one station. That station is asleep, blocked or misread the signal; the daily test finds the position.
- Alarm arrives garbled. One station repeated a pulse wrongly; use even-parity patterns (Code Systems).
- Works by day, not by night. The night signal is too faint at that spacing; shorten the span or enlarge the fire.
Common Mistakes
- Reading spacing from the horizon table. The geometric figure is a ceiling.
- Assuming p is close to 1. Test it.
- No skip links. One sleeper breaks the line.
Safety
Fire, height and storms at hill stations
A hilltop station needs a cleared ring for its fire (Fire and Smoke Signals), a railing on any tower, and a way for the watcher to leave the exposed site in a storm.
What’s Next
- Learn next: Telegraph Network — the wire-based successor to a staffed chain
- Combine with: Runner Systems — the slower channel for detail