Course to Steer with a Cross-Current
You are navigating a 20-nm leg to a waypoint that bears 090°T. Your vessel's speed through the water is 10.0 kts. A current is setting 180°T at 2.0 kts, and you expect it to remain steady for the leg. You have no GPS course-to-steer function available, so you will solve it manually. The chart shows no other hazards. Fuel burn is based on speed through the water, not speed over ground. You must arrive at the waypoint, not merely steer 090°T and accept a track error.
1. Compute the course to steer and the speed made good.
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Let H be the course steered. The current sets south at 2.0 kts, so the vessel's north component through the water must cancel it:
10 cos H = 2.0 → cos H = 0.20 → H = 78.5°T.
East component = 10 sin 78.5° = 9.80 kts. Course to steer is about 078.5°T; speed made good is about 9.8 kts. Steering 090°T would not work because the current would set the vessel south of track.
2. Compute the ETA and the track error if you steered 090°T.
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At 9.8 kts over ground, time = 20 nm ÷ 9.8 kts = 2.04 hours, or about 2 h 02 m.
If you steered 090°T, the current would set you south at 2.0 kts for 2.04 hours: 2.0 × 2.04 = 4.1 nm south of the waypoint. That is a large miss, so the course must be offset upstream.
3. If the current increases to 3.0 kts, how does the solution change?
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Now 10 cos H = 3.0 → cos H = 0.30 → H = 72.5°T. Speed made good = 10 sin 72.5° = 9.54 kts. ETA = 20 ÷ 9.54 = 2.10 hours, or about 2 h 06 m.
A stronger cross-current requires a larger upstream offset and reduces SMG because more of the vessel's speed through the water is used to fight the set.