Where the window comes from
The arch-duration formula follows directly from the geometry:
arch seconds = 2 · depth · tan(cone/2) / (speed × 1.467 ft/s per mph)
The numerator is the full width of the beam footprint at the fish's depth — the distance the boat travels while the fish stays in the beam. Divide by speed and you get duration in seconds. Multiply by 15 pings per second and you get the number of columns on screen.
Two limits define the window:
- Too slow (arch > 6 s). The arch spreads so wide that the rise from tail to peak is only a fraction of a display column. It reads as a flat line at depth.
- Too fast (arch < 0.5 s). The fish is in the beam for fewer than 8 pings. That is not enough columns to draw a visible curve — the mark is a short dash or absent.
Ideal speed window by depth and cone
The range of speeds that produce an arch between 0.5 s and 6 s, for a fish directly under the track. A fish off to the side has a narrower beam chord and needs the slow end of this range.
| Depth | 12° cone | 20° cone | 30° cone | 40° cone | 60° cone |
|---|---|---|---|---|---|
| 8 ft | 0.19–2.3 mph | 0.32–3.8 mph | 0.49–5.8 mph | 0.66–7.9 mph | 1.0–12.6 mph |
| 12 ft | 0.29–3.4 mph | 0.48–5.8 mph | 0.73–8.8 mph | 0.99–11.9 mph | 1.6–18.9 mph |
| 20 ft | 0.48–5.7 mph | 0.80–9.6 mph | 1.2–14.6 mph | 1.7–19.9 mph | 2.6–31.5 mph |
| 30 ft | 0.72–8.6 mph | 1.2–14.4 mph | 1.8–21.9 mph | 2.5–29.8 mph | 3.9–47.2 mph |
| 50 ft | 1.2–14.3 mph | 2.0–24.0 mph | 3.0–36.5 mph | 4.1–49.6 mph | 6.6–78.7 mph |
Arch duration by depth and speed (20° cone)
Seconds in the beam for each combination. Under 0.5 s is a dash. Over 6 s flattens into a line. Bold entries are in the ideal window.
| Depth | 0.5 mph | 1 mph | 1.5 mph | 2 mph | 3 mph | 5 mph |
|---|---|---|---|---|---|---|
| 8 ft | 3.8 s | 1.9 s | 1.3 s | 1.0 s | 0.6 s | 0.4 s |
| 12 ft | 5.8 s | 2.9 s | 1.9 s | 1.4 s | 1.0 s | 0.6 s |
| 20 ft | 9.6 s | 4.8 s | 3.2 s | 2.4 s | 1.6 s | 1.0 s |
| 30 ft | 14.4 s | 7.2 s | 4.8 s | 3.6 s | 2.4 s | 1.4 s |
| 50 ft | >20 s | 12.0 s | 8.0 s | 6.0 s | 4.0 s | 2.4 s |
Bold = ideal arch range (0.5–6 s). Blanks outside the window show as lines or dashes.
Deeper fish need slower speeds
Depth and cone both appear in the numerator of the formula. Doubling the depth doubles the beam footprint, so you need to go half as fast to keep the same arch duration. A fish at 40 ft needs half the speed of a fish at 20 ft to get the same mark.
Wider cones also increase the footprint. A 40° cone at 20 ft has a footprint of about 15 ft. A 20° cone at 20 ft has about 7 ft. Switching from a 7-ft to a15-ft footprint nearly doubles the arch duration at the same speed.
What if the fish is moving?
The formula uses relative speed — boat speed minus fish swim speed, both in the same direction. A fish swimming away from you at 1 mph while you travel at 2 mph gives a relative speed of 1 mph: the arch lasts twice as long as if the fish were stationary. A fish swimming toward you at 1 mph gives 3 mph relative: the arch is two-thirds as long.
The arch simulator has a fish-swim-speed slider so you can see this directly.
What this leaves out
- Fish directly under the track only. An off-track fish has a shorter chord through the beam, so its arch is shorter and needs a lower speed. Thearch simulator handles arbitrary offsets.
- Sensitivity and fish size. A small fish at low sensitivity paints a narrower footprint than the nominal cone, so the arch is also shorter. Thecoverage calculator shows how footprint changes with fish size and sensitivity.
- Scroll speed. Some units compress or expand the horizontal time axis. The arch duration in seconds is real; how many pixels wide it appears depends on the unit's chart speed setting.
- Ping rate is fixed at 15/s here. Units vary; high-frequency and CHIRP units often ping faster, which expands the number of columns for the same duration.
More from the sonar model
- Why fish show as lines, not archesThe interactive simulator: drag a boat over a fish and see the mark form or flatten.
- Why sonar shows fish deeper than they areSlant-range geometry: how much deeper an off-track fish reads, by cone and depth.
- Cone coverage calculatorHow wide your beam is at depth, and how much of it a given fish actually paints.
- Can you tell fish size from an arch?Why a bigger fish off to the side draws the same mark as a small one underneath.