M005 · Moving water

Riverboat Navigation

Does the quickest crossing land straight opposite? The quickest crossing points the boat straight across the water, leaving the full water-relative speed available for crossing. The current carries it downstream. Landing opposite needs an upstream heading and takes longer, when a constant-heading solution exists.

Steer a boat through uniform, sheared or faster central currents. Compare the quickest crossing with a constant heading that lands opposite, and watch water-relative and ground velocities add.

Across-river progress $0\,\mathrm{m}$

Downstream drift $0\,\mathrm{m}$

Ground speed $3\,\mathrm{m\,s^{-1}}$

Predicted crossing time $4\,\mathrm{s}$

Heading to land opposite $0\,{}^{\circ}$

Shortest crossing time $4\,\mathrm{s}$

Time at opposite-bank heading $4\,\mathrm{s}$

Model time $0\,\mathrm{s}$

Predict a course, then watch it unfold

Fastest, or straight opposite?

Speed through water

v_b

3\,\mathrm{m\,s^{-1}}

Mean current speed

\bar u

1.2\,\mathrm{m\,s^{-1}}

River width

W

12\,\mathrm{m}

Heading upstream from straight across

\theta

10\,\mathrm{{}^{\circ}}

Current profile

j

2

Replay progress

\eta

0\,\mathrm{\%}

Heading is measured upstream from the bank normal. Current is downstream and depends only on the across-river position; the three profiles have the same mean. The boat is a point advected by the water plus its own prescribed velocity. Opposite-bank feasibility below refers only to a constant heading, not every possible route in a variable current.

Model assumptions

The boat keeps a prescribed speed and heading relative to a steady downstream current. Three profiles share the same channel-average speed. The model has no acceleration, drag, waves or hull dynamics. The opposite-bank solution is restricted to constant headings.

Start with the essentials

Focus question: What changes when the destination is the opposite point rather than anywhere on the far bank?
One-sentence intuition: The boat crosses fastest when all its water-relative speed points across. A constant upstream heading can cancel the total downstream drift when the mean current is small enough.

Core mathematical model

Water-relative and ground velocity

\begin{aligned}\mathbf v_g&=\mathbf v_b+\mathbf u\\v_{g,x}&=u(y)-v_b\sin\theta\\v_{g,y}&=v_b\cos\theta\end{aligned}

Downstream is positive horizontal distance. The heading is measured upstream from the across-river direction.

Crossing time and landing drift

\begin{aligned}T&=\frac{W}{v_b\cos\theta}\\X&=\frac{W(\bar u-v_b\sin\theta)}{v_b\cos\theta}\\\bar u&=\frac1W\int_0^W u(y)\,\mathrm dy\end{aligned}

The heading stays fixed during a trial. The across-river speed is constant; downstream drift integrates the flow through the full channel width.

The quickest arrival anywhere on the far bank

T_{\min}=\frac{W}{v_b},\qquad\theta=0

The water has no across-river component. No heading can provide more across-river speed than the boat full water-relative speed.

A constant heading that lands opposite

\begin{aligned}\sin\theta_{\mathrm{opp}}&=\frac{\bar u}{v_b}\\T_{\mathrm{opp}}&=\frac{W}{\sqrt{v_b^2-\bar u^2}}\end{aligned}

This finite-time solution requires the mean current to be smaller than the boat speed. In a variable current, other time-varying routes lie outside this constant-heading test.

Common difficulties

Aimed across does not mean moving straight across

Typical misconceptionThe boat follows the direction its bow points.

Better mental modelThe bow sets its velocity through water. Water motion adds another velocity relative to the bank.

Zero landing drift can have a curved path

Typical misconceptionLanding opposite means the boat stayed on the opposite-bank line throughout.

Better mental modelIn a shaped current, upstream and downstream drift can cancel over the whole trip. The path can bend and still end opposite.

One heading is one constraint

Typical misconceptionIf the constant-heading formula fails, every possible navigation plan fails.

Better mental modelThe test fixes one heading. A route that varies its heading or waits in a slower part of the stream is a different control problem.

Run the experiment

01
Watch the speed change across the stream
Choose Curved crossing and watch the gold water markers.
What to observe: The center flows faster than the edges. The blue boat keeps one water-relative heading while its ground path bends.
02
Cross as quickly as possible
Choose Quickest crossing.
What to observe: The full boat speed carries it across. It lands downstream, with a shorter crossing time.
03
Land opposite
Choose Land opposite or press Set opposite-bank course.
What to observe: The upstream heading cancels total drift. The crossing takes longer.
04
Keep the average, change the profile
Compare uniform, shear and central-jet profiles at the same mean current and heading.
What to observe: The final drift and crossing time agree, although the shapes of the paths differ.
05
Make the current too strong for one heading
Choose Current outruns boat.
What to observe: No finite constant-heading solution lands opposite in this uniform current. The simulator still shows a valid crossing downstream.
06
Read the velocity triangle
Pause partway across and inspect the vector diagram below the controls.
What to observe: The blue water-relative vector followed by the gold current vector ends at the teal ground-velocity tip.

Core mathematical model

Water-relative and ground velocity

Crossing time and landing drift

The quickest arrival anywhere on the far bank

A constant heading that lands opposite

Common difficulties

Aimed across does not mean moving straight across

Zero landing drift can have a curved path

One heading is one constraint

Run the experiment

Watch the speed change across the stream

Cross as quickly as possible

Land opposite

Keep the average, change the profile

Make the current too strong for one heading

Read the velocity triangle